A LOW MACH NUMBER LIMIT OF A DISPERSIVE
NAVIER-STOKES SYSTEM
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA∗
Abstract. We establish a low Mach number limit for classical solutions over the whole space
of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes
equations. The limiting system is a ghost effect system [26]. Our analysis builds upon the framework
developed by Métivier and Schochet [20] and Alazard [2] for non-dispersive systems. The strategy
involves establishing a priori estimates for the slow motion as well as a priori estimates for the fast
motion. The desired convergence is obtained by establishing the local decay of the energy of the fast
motion.
Key words. Low Mach number, compressible and viscous fluid, dispersive Navier-Stokes equations, kinetic theory, ghost effect systems.
AMS subject classifications. 35L65, 76N10, 35B45
1. Introduction. We establish a low Mach number limit for classical solutions
over the whole space of a compressible fluid dynamic system that includes dispersive
corrections to the Navier-Stokes equations. The limiting system is a so-called ghost
effect system [26] which is not derivable from the Navier-Stokes system of gas dynamics
but is derivable from kinetic equations. This work is part of a program [16, 17] that
aims to identify fluid dynamic regimes and to construct a unified model that captures
them all. Such a model can also be useful in transition regimes when classical fluid
equations are inadequate to describe the dynamics of fluids while computations using
kinetic models are expensive.
The governing equations for the dispersive system (DNS) under consideration are
as follows
∂t ρ + ∇x · (ρU ) = 0 ,
(1.1)
∂t (ρU ) + ∇x · (ρU ⊗ U ) + ∇x p(ρ, θ) = ∇x · Σ + ∇x · Σ̃ ,
∂t (ρe) + ∇x · (ρeU + ρΘU ) = ∇x · (ΣU − q) + ∇x · (Σ̃U + q̃) ,
(ρ, U, Θ)(x, 0) = (ρin , U in , Θin )(x) ,
where (ρ(x, t), U (x, t), Θ(x, t)) denote the density, velocity, and temperature of the
fluid at time t ∈ R+ and position x ∈ R3 respectively, whereas p = p(ρ, Θ), q = q(Θ),
Σ = Σ(U, Θ) denote the pressure, heat flux, and viscous stress tensor respectively
with
q(Θ) = −κ̂(Θ)∇x Θ ,
p(ρ, Θ) = ρΘ,
and Σ is the stress for a Newtonian fluid
2
Σ = µ̂(Θ)(∇x U + (∇x U )T − (∇x · U )I).
3
The total energy density is given by
ρe =
1
3
ρ|U |2 + ρΘ.
2
2
∗ K.T. was supported in part by the National Science Foundation under the grants DMS-1109397,
DMS-0807815 and PECASE DMS-0239063.
1
2
A Low Mach Number Limit of a Dispersive Navier-Stokes System
The quantities Σ̃ and q̃ denote dispersive corrections to the stress tensor and heat
flux respectively and are given by
Σ̃ = τ̂1 (ρ, Θ)(∇x2 Θ − 13 (∆x Θ)I) + τ̂2 (ρ, Θ)(∇x Θ ⊗ ∇x Θ − 31 |∇x Θ|2 I)
+τ̂3 (ρ, Θ)(∇x U (∇x U )T − (∇x U )T ∇x U ) ,
q̃ = τ̂4 (ρ, Θ)(∆x U + 13 ∇x ∇x · U ) + τ̂5 (ρ, Θ)∇x Θ · (∇x U + (∇x U )T − 23 (∇x · U )I)
(
)
+τ̂6 (ρ, Θ) ∇x U − (∇x U )T · ∇x Θ .
Here we assume that the transport coefficients τ̂1 , · · · , τ̂6 are C ∞ functions of
their variables and τ̂1 , τ̂4 > 0. By linearizing the DNS system (1.1) around constant
states one can see that the coupling of ∇x · Σ̃ and ∇x · q̃ forms a dispersive relation.
Compared with the original system derived from kinetic equations in [16], system
(1.1) is simplified in that terms involving second order derivatives of ρ are not included
within ∇x · Σ̃ and ∇x · q̃. In fact, unlike second order terms in Θ and U within these
dispersive terms, which can be controlled by the diffusive regularization, the state of
the art at the moment does not allow control of the second order terms in ρ. This is
due to the fact that ρ satisfies a purely hyperbolic equation.
One feature of the DNS system is that it possesses an entropy structure provided
the transport coefficients in Σ̃ and q̃ satisfy
( )
Θ
τ̂2
2τ̂5
τ̂4
(1.2)
τ̂4 = τ̂1 ,
+ 2 = ∂Θ
,
2
Θ
Θ
Θ2
such that
∇x U
∇x Θ
Σ̃ :
+ q̃ · 2 = ∇x ·
Θ
Θ
(
(
))
τ̂1
2
T
∇x Θ · ∇x U + (∇x U ) − (∇x · U )I
.
2Θ
3
Under assumption (1.2) one can show that the DNS system dissipates the Euler
entropy in the same way as the Navier-Stokes system. However we do not need
assumption (1.2) in this paper because we only consider local-in-time behavior of
the system. Indeed the low Mach number limit is established only by assuming that
τ̂1 , · · · , τ̂6 are C ∞ functions of their variables and τ̂1 , τ̂4 > 0.
The ghost effect system can be formally derived from kinetic equations using a
Hilbert expansion method (cf. Sone [26]). This is a system beyond classical fluid
equations that describes the phenomenon in which the temperature field of the fluid
has finite variations, and the flow is driven by the gradient of the temperature field
[26]. Let (ϱ, ϑ, P ∗ ) be the density, temperature, and pressure fields of the fluid. Then
the ghost effect system in its general form is given by
∇x (ϱϑ) = 0,
(1.3)
∂t ϱ + ∇x · (ϱu) = 0,
∂t (ϱu) + ∇x · (ϱu ⊗ u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃,
3
2 ∂t (ρϑ)
+ 52 ρϑ∇x · u = −∇x · q,
where P ∗ is the pressure, which is an independent variable for the ghost effect system.
P ∗ can be viewed as a Lagrangian multiplier as in the case of incompressible dynamics.
The viscous and heat conducting terms are written as
)
(
2
T
q = −κ̄(ϑ)∇x ϑ,
Σ = µ̄(ϑ) ∇x u + (∇x u) − (∇x · u)I ,
3
A Low Mach Number Limit of a Dispersive Navier-Stokes System
3
where µ̄(ϑ), κ̄(ϑ) > 0 are the viscosity and heat conductivity respectively. The thermal
stress tensor Σ̃ has the form
(
)
(
)
1
Σ̃ = τ̄1 (ϑ) ∇x2 ϑ − ∆x ϑ I + τ̄2 (ϑ) ∇x ϑ ⊗ ∇x ϑ − 13 |∇x ϑ|2 I ,
3
where τ̄1 (·), τ̄2 (·) ∈ C ∞ (R) are transport coefficients with τ̄1 > 0. The term ∇x · Σ̃
contains third-order derivatives and is not derivable from the Navier-Stokes system.
In order to derive (1.3) from (1.1) one introduces a small parameter ϵ to (1.1). In
kinetic theory, ϵ is the Knudsen number which measures the relative size of the mean
free path of gas molecules with respect to a typical macroscopic length. It satisfies
the von Kármen relation,
ϵ∝
Ma
.
Re
Here M a denotes the Mach number which is typically used to compare a typical flow
velocity Uref with a characteristic speed of sound c and is given by
(1.4)
M a :=
Uref
,
c
and Re is the Reynolds number. If one considers a fluid regime where the viscous and
inertial effects are comparable to each other, then the Reynolds number is of order
one. In such a regime, ϵ and M a are of the same scale. We thereby use ϵ to denote
the Mach number in this paper. In accordance with the kinetic theory of fluids
µ̂ = ϵµ̃,
τ̂i = ϵ2 τ̃i , i = 1, · · · , 6 .
κ̂ = ϵκ̃,
For the derivation of the ghost effect system we need to focus on the physical regime
in which
p ∼ p0 + O(ϵ) ,
U ∼ O(ϵ) ,
∇x Θ ∼ O(1) ,
p = ρΘ ,
where p0 is a constant. The existence of solutions to dispersive system (DNS) is
established for the case where ρ and θ are close to constants at infinity. Therefore
in our case the limiting pressure ϱϑ will be a constant that is independent of time.
Without loss of generality, we fix this constant p0 as 1. Thus, we consider the case
when p is around 1 while Θ has a finite variation. To emphasize the dependence of
(ρ, U, Θ) on ϵ, we denote the solution as (ρϵ , Uϵ , Θϵ ) and the pressure as pϵ .
In order to ensure the positivity of both pϵ and Θϵ , as usual we write (p, Uϵ , Θϵ )
as
(1.5)
ϵ
pϵ = eϵp ,
Uϵ = ϵuϵ ,
ϵ
Θϵ = eθ ,
ϵ
ρϵ = eρ ,
where the superscripts denote the fluctuation functions. In this regime, a longer time
scale needs to be considered in order to capture the evolution of the fluctuations,
namely
(1.6)
t=
1
t̂ .
ϵ
For simplicity we subsequently drop the hat from t̂ and use t in the system for the
fluctuations. The dispersive Navier-Stokes system (1.1) thereby transforms into its
scaled analogue
4
A Low Mach Number Limit of a Dispersive Navier-Stokes System
3
1
ϵ
ϵ
5 (∂t +u ·∇x )p + ϵ ∇x ·
((
)
)
( ϵ 2 −ϵpϵ ϵ
)
ϵ
u − 5e
κ(θ )∇x θϵ = 25 ϵe−ϵp Σ+ Σ̃ : ∇x uϵ +∇x · q̃
+ 25 e−ϵp κ(θϵ )∇x pϵ · ∇x θϵ ,
(
)
ϵ
ϵ
e−θ (∂t + uϵ · ∇x )uϵ + 1ϵ ∇x pϵ = e−ϵp ∇x · Σ + ∇x · Σ̃ ,
((
)
)
3
ϵ
ϵ
ϵ
ϵ
2 −ϵpϵ
Σ+
Σ̃
:
∇
u
+∇
·
q̃
(∂
+u
·∇
)θ
+∇
·u
=
ϵ
e
x
x
x
x
2 t
ϵ
(1.7)
−e−ϵp ∇x · q ,
in in
(pϵ , uϵ , θϵ )(x, 0) = (pin
ϵ , uϵ , θϵ )(x) ,
ϵ
where the constitutive relations inherited from those for (1.1) are
(
)
ϵ
Θϵ = eθ , q = −κ(θϵ )∇x θϵ , Σ = µ(θϵ ) ∇x uϵ + (∇x uϵ )T − 23 (∇x · uϵ )I ,
(
)
(
)
Σ̃ = τ1 (ϵpϵ , θϵ ) ∇x2 θϵ − 13 (∆x θϵ )I + τ2 (ϵpϵ , θϵ ) ∇x θϵ ⊗ ∇x θϵ − 13 |∇x θϵ |2 I
(
)
(1.8)
+ϵ2 τ3 (ϵpϵ , θϵ ) ∇x uϵ (∇x uϵ )T − (∇x uϵ )T ∇x uϵ ,
(
)
(
)
ϵ T
q̃ = τ4 (ϵpϵ , θϵ ) ∆x uϵ + 13 ∇x ∇x(· uϵ + τ6 (ϵpϵ , θϵ ) ∇x uϵ − (∇x u
) · ∇x θϵ
)
+τ5 (ϵpϵ , θϵ )∇x θϵ · ∇x uϵ + (∇x uϵ )T − 23 (∇x · uϵ )I ,
while
κ(θϵ ) = κ̃(Θϵ ) Θϵ ,
ϵ
(1.9)
µ(θϵ ) = µ̃(Θϵ ) ,
ϵ
ρϵ = Pϵ /Θϵ = eϵp
ϵ
ϵ
−θ ϵ
,
ϵ
τ1 (ϵp , θ ) = τ̃1 (ρϵ , Θϵ ) Θϵ ,
τ4 (ϵp , θ ) = τ̃4 (ρϵ , Θϵ ) ,
ϵ
τ2 (ϵp , θ ) = τ̃1 (ρϵ , Θϵ ) Θϵ + τ̃2 (ρϵ , Θϵ ) Θ2ϵ , τ3 (ϵpϵ , θϵ ) = τ̃3 (ρϵ , Θϵ ) ,
ϵ
τ5 (ϵpϵ , θϵ ) = τ̃5 (ρϵ , Θϵ )Θϵ ,
τ6 (ϵpϵ , θϵ ) = τ̃6 (ρϵ , Θϵ )Θϵ .
If we assume that for all indices α = (α1 , α2 , α3 ) such that |α| ≤ 3,
(1.10)
(∇x pϵ , ∇xα uϵ , ∇xα θϵ )(t, x) → (0, ∇xα u, ∇xα θ)(t, x)
as ϵ → 0,
then (1.7) converges formally to the system
(
)
∇x · 52 u − κ(θ)∇x θ = 0 ,
e−θ (∂t u + u · ∇x u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃ ,
(1.11)
3
2 (∂t
+ u · ∇x )θ + ∇x · u = −∇x · q ,
where P ∗ is the pressure which is an independent variable. By (1.10) we have
(1.12)
)
(
Σ = µ(θ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
q = −κ(θ)∇x θ ,
( 2
)
(
)
1
Σ̃ = τ1 (ϱ, θ) ∇x θ − 3 ∆x ΘI + τ2 (ϱ, θ) ∇x θ ⊗ ∇x θ − 31 |∇x θ|2 I .
Here µ(θ) is the viscosity, κ(θ) is the heat conductivity, and τ1 (ϱ, θ), τ2 (ϱ, θ) ∈ C ∞ (R×
R) are transport coefficients with τ1 > 0 as defined in (1.9).
We can write the limiting system (1.11) in the form of a ghost system as (1.3) by
taking (ϱ, u, ϑ) = (ϱ, u, eθ ) as the variables. System (1.11) then has the form
(1.13)
ϱϑ = 1 ,
∂t ϱ + ∇x · (ϱu) = 0 ,
∂t (ϱu) + ∇x · (ϱu ⊗ u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃ ,
(
)
∇x · 52 u − κ(θ)∇x ϑ = 0 ,
5
A Low Mach Number Limit of a Dispersive Navier-Stokes System
where Σ and Σ̃ are defined in (1.12).
Our goal in this paper is to rigorously justify the above formal calculation of the
low Mach number limit from the dispersive system (1.7) to the ghost effect system
(1.11) in the whole space.
We remark that the investigation of singular limits of compressible flows over
bounded domains is physically relevant and presents new challenges in the analysis.
Unlike the cases involving the whole domain or exterior domains where acoustic waves
are damped locally due to dispersive effects of the wave equation [1, 2, 6, 20, 29], the
main obstacle in the treatment of bounded domains is the persistency of the fast waves
over these domains. Therefore in general one can only expect weak convergence of the
solutions. It is worth noting [7] that there are situations where strong convergence
can be achieved due to the interaction of acoustic waves with the boundary of the
domain, where a thin boundary layer is created to damp the energy carried by these
fast oscillations. This phenomenon has been observed for both asymptotics of fluid
equations and hydrodynamic limits of kinetic equations [7, 21]. It is therefore natural
to ask the question whether similar phenomenon happens for the DNS system. Before
trying to answer this question, one needs to know what are the physical boundary
conditions that should be imposed on the DNS system. These boundary conditions are
typically derived from the underlying kinetic equations so that they are compatible
with the given boundary conditions for the kinetic equations. Deriving admissible
boundary conditions for the DNS system, establishing the well-posedness theory of
this system and investigating its asymptotics over bounded domains are the long terms
goals of this program.
The outline of this paper is as follows. In Section 2 we introduce the space
setting and present main result of our article. In Section 3 we state the local wellposedness theorems with brief explanations of their proofs for the dispersive NavierStokes system (1.1) (or equivalently system (1.7)) and the ghost effect system (1.3).
In Section 4 we prove a priori estimates for the slow motion ψ ϵ = (ϵpϵ , ϵuϵ , θϵ ). In
Section 5 we prove a priori estimates for the fast motion (pϵ , uϵ ) based on the bounds
derived in Section 3. In Section 6 we show the local decay of the energy of the fast
motion, and in Section 7 we show the convergence of (1.7) to (1.11).
2. Main result. Before stating the main theorem, we introduce the following
norms that appear naturally during the analysis.
Definition 2.1. Let α0 , θ be two constants and α0 > 0. For each ϵ, t, s > 0,
define the norms
(
)
|||(p, u, θ−θ)|||ϵ,s,t := sup ∥(p, u)(t)∥H s + ∥Λ2s+1
(ϵp, ϵu, θ−θ)(t)∥H s+1
ϵ
[0,t]
(∫
(2.1)
t(
∥∇x (u, p)∥2H s
+ α0
+
)
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1
ϵ
) 21
(τ )dτ ,
0
and
(2.2)
|||(p, u, θ−θ)|||ϵ,s,0 := ∥(pin , uin )∥H s + ∥Λ2s+1
(ϵpin , ϵuin , θin −θ)∥H s+1 ,
ϵ
where H s = H s (R3 ) is the usual Sobolev norm such that derivatives up to order s of
a function are square-integrable. The invertible operator Λϵ is defined by
(2.3)
for any ϵ > 0.
Λϵ := (I − ϵ2 ∆x )1/2 .
6
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Definition 2.2. Let (pϵ , uϵ , θϵ ) be the solution to (1.7). We call ψ = (ϵpϵ , ϵuϵ , θϵ )
as the slow motion because it varies on the time scale of order one. We call (pϵ , uϵ )
as the fast motion because it varies on the time scale of order ϵ.
The main theorem states
Theorem 2.3. Let s ≥ 6. Suppose that there exist positive constants M0 , θ, c0 , σ
in in
such that the initial data of the fluctuations (pin
ϵ , uϵ , θϵ ) satisfy
in
in
|||(pin
ϵ , uϵ , θϵ −θ)|||ϵ,s,0 ≤ M0 ,
(2.4)
in
in
s
3
(θϵin −θ, Π(e−θϵ uin
ϵ )) → (θ −θ, u ) in H (R ) ,
in
−1−σ
in
−2−σ
|θϵ − θ| ≤ c0 |x|
,
|∇x θϵ | ≤ c0 |x|
,
in
where Π is the projection onto the divergence free part of e−θϵ uin
ϵ . Then there exists
T > 0 such that for any ϵ ∈ (0, 1], the Cauchy problem for system (1.7) has a unique
solution
in
(pϵ , uϵ , θϵ −θ) ∈ C([0, T ]; H s1 (R3 )) ∩ L∞ (0, T ; H 3s+2 (R3 )) ∩ L2 (0, T ; H 3s+3 (R3 )) ,
for all s1 < 3s + 1 and there exists a positive constant M depending only on T, M0
such that
(2.5)
|||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ≤ M .
Furthermore, the sequence of solutions (pϵ , uϵ , θϵ ) to system (1.7) converges weakly in
s′
L∞ (0, T ; H s (R3 )) and strongly in L2 (0, T ; Hloc
(R3 )) for all s′ < s to the limit (0, u, θ),
where (u, θ) satisfies the ghost effect system (1.11).
Our analysis builds on the framework of Métivier and Schochet [20] and Alazard
[2]. In [20] Métivier and Schochet proved the incompressible limit for the nonisentropic Euler equations for classical solutions with general initial data. In [2]
Alazard proved the low Mach number limit for the compressible Navier-Stokes for
classical solutions with general initial data and in [3] he proved the low Mach number
limit for the compressible Navier-Stokes with general equations of state and a large
source term in the temperature equation. Due to the presence of the high-order dispersive terms, our result does not follow directly from [2, 3]. We need to take into account
of the anti-symmetric structure of those terms. Specifically, let U ϵ = (pϵ , uϵ , θϵ ) be
the solution to system (1.1) and ψ ϵ = (ϵpϵ , ϵuϵ , θϵ ), then (1.1) can be reformulated as
1
A1 (ψ ϵ )(∂t + u · ∇x )U ϵ + A2 (ψ ϵ )U ϵ = A3 (ψ ϵ )U ϵ + R ,
ϵ
where A1 (ψ ϵ ) is a diagonal matrix, A2 (ψ ϵ )U ϵ are formed by certain combinations of
the singular terms with the leading orders from the dispersive terms, A3 (ψ ϵ )U ϵ are
the dissipative terms, and R includes the rest of the dispersive terms. In the case
of compressible Navier-Stokes A2 (ψ ϵ ) is anti-symmetric. Here it is not readily antisymmetric but it is anti-symmetrizable by a certain symmetrizer matrix composed
of symmetric positive operators. Similar as in [2] we establish uniform bounds for
the solution by considering the interactions of its fast and slow motion parts. The
theorem for the uniform bounds states
Theorem 2.4. For each fixed ϵ > 0, let (pϵ , uϵ , θϵ ) ∈ C([0, T ]; H s (R3 )) be the
solution to the scaled DNS system (1.7). Let
Ω = |||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ,
in in
Ω0 = |||(pin
ϵ , uϵ , θϵ −θ)|||ϵ,s,0 .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
7
Then there exists an increasing function C(·) such that
√
T +ϵ)C(Ω)
|||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ≤ C(Ω0 )e(
,
which further indicates [20] that there exists T0 > 0 independent of ϵ such that
∥|(pϵ , uϵ , θϵ−θ)∥|ϵ,s,T are uniformly bounded in ϵ over [0, T0 ]. The proof for Theorem
2.4 in this paper is presented slightly differently from [2]. We directly derive bounds
for (pϵ , uϵ ) without separating the frequency space into high and low frequencies.
Once the uniform bounds are established, the convergence of the solutions is shown
by directly applying the local energy decay method for fast waves in the whole space
[20].
3. Local Well-Posedness. In this section we state the local well-posedness
theorems with brief explanation of their proofs for the dispersive Navier-Stokes system
(1.1) and its limiting system (1.11).
The proof of the local well-posedness of (1.1) follows directly by classical energy
method for hyperbolic-parabolic systems. More specifically, although we have thirdorder dispersive terms, the leading orders of these terms form an anti-symmetric
structure. Therefore they do not hamper the usual L2 -H s estimates. The rest of
the dispersive terms are of orders up to two. Although this is the same order as
the dissipation, they do not introduce extra difficulties because they are of order
O(ϵ2 ) while the dissipative terms are of order O(ϵ). When ϵ is small we have enough
control for these terms using dissipative regularization. Here we do need the viscosity
coefficient µ̂(Θ) and κ̂(Θ) to be bounded away from zero when Θ is bounded from
below. This assumption is consistent with the forms of µ̂ and κ̂ as derived from kinetic
equations. For most collision kernels, µ̂ and κ̂ are in the form of Θγ where γ > 0
depends on the specific type of the collision kernel. The local well-posedness theorem
states
Theorem 3.1. Let s ≥ 6. Suppose there exist two constants ρ0 , Θ0 > 0 such that
in
in
the initial data (ρin
ϵ , Uϵ , Θϵ ) satisfies that
in
in
∥(ρin
ϵ − ρ0 , Uϵ , Θϵ − Θ0 )∥H s (R3 ) ≤ C0 .
Then there exists ϵ0 > 0 such that for any fixed 0 < ϵ < ϵ0 , there exists α0 > 0
depending on ρ0 , Θ0 and Tϵ > 0 depending on ϵ and C0 such that (1.1) has a unique
solution which satisfies
ρϵ ∈ C([0, Tϵ ]; H s (R3 )) , (Uϵ , Θϵ −Θ0 ) ∈ C([0, Tϵ ]; H s (R3 )) ∩ L2 ([0, Tϵ ]; H s+1 (R3 )) ,
and
∫
sup ∥(ρϵ , Uϵ , Θϵ − Θ0 )∥
Hs
[0,Tϵ ]
+ α0
Tϵ
∥∇x (Uϵ , Θϵ )∥2H s+1 (τ ) dτ ≤ 2C0 .
0
We also need the well-posedness of the ghost effect system (1.11). Again local
well-posedness of classical solutions of this system can be established with the aid
of classical energy estimates for hyperbolic-parabolic equations. Note that although
there is a third-order term in θ in (1.11) and there is no anti-symmetric structure to
balance this term, it still does not give rise to any major difficulty because the leading
order of this term is in the form of a gradient, which can be incorporated into the
pressure term. By doing so the rest of the terms can be treated as perturbations.
8
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Similar proofs can be found for combustion models and Kazhikhov-Smagulov type
models (see [10, 27] for example). The theorem states
Theorem 3.2. Let s ≥ 5 and θ̄ be two constants. Let the initial data (θin , uin )
satisfy that
(3.1)
θin − θ̄ ∈ H s+1 (R3 ) ,
uin ∈ H s (R3 ) .
Then there exists T > 0 such that the ghost effect system (1.11) has a unique solution
(P, θ, u) with
θ − θ̄ ∈ C([0, T ]; H s+1 (R3 )) ∩ L2 ([0, T ]; H s+2 (R3 )) ∩ C ∞ ((0, T ) × R3 ) ,
(3.2)
u ∈ C([0, T ]; H s (R3 )) ∩ L2 ([0, T ]; H s+1 (R3 )) ∩ C ∞ ((0, T ) × R3 ) ,
∇x P ∈ C([0, T ]; H s−2 (R3 )) ∩ C ∞ ((0, T ) × R3 ) .
4. A Priori Estimates – Slow Motion. In this section we derive a priori
estimates for slow motions which varies on time scale of order one. Estimates in the
next section about fast motions depend heavily on those for slow motions.
4.1. Linearization. From now on, we drop the ϵ-index for the variables. Consider the following linearized system of (1.7), where we add forcing terms to each equation and linearize the system by replacing certain terms by a given state (p0 , u0 , θ0 ):
(4.1)
((
)
)
(
) 2
1
3
2
(∂
+
u
·∇
)p
+
∇
·
u
−
h(ϵp
)κ(θ
)
∇
θ
=
ϵh(ϵp
)
Σ
+
Σ̃
:
∇
u
+
∇
·
q̃
t
0
x
x
0
0
x
0
x
0
x
5
5
5
ϵ
− 25 h(ϵp0 )κ(θ0 )∇x p0 · ∇x θ + f1 ,
(
)
1
g(θ0 )(∂t + u0 ·∇x ) u + ∇x p = h(ϵp0 ) ∇x · Σ + ∇x · Σ̃ + f2 ,
ϵ
((
)
)
3
2 (∂t
+ u0 ·∇x ) θ + ∇x · u = ϵ2 h(ϵp0 )
Σ + Σ̃ : ∇x u0 +∇x · q̃
− h(ϵp0 )∇x · q + f3 ,
in
in
in
(p, u, θ)(x, 0) = (p , u , θ )(x) ,
where
q = −κ(θ0 )∇x θ , h(ϵp0 ) = e−ϵp0 , g(θ0 ) = e−θ0 ,
)
(
Σ = µ(θ0 ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
(
)
(
)
Σ̃ = τ1 (ϵp0 , θ0 ) ∇x2 θ − 13 (∆x θ)I + τ2 (ϵp0 , θ0 ) ∇x θ0 ⊗ ∇x θ − 31 (∇x θ0 · ∇x θ)I
(
)
+ ϵ2 τ3 (ϵp0 , θ0 ) (∇x u0 )(∇x u)T − (∇x u0 )T ∇x u ,
(
)
(
)
q̃ = τ4 (ϵp0 , θ0 ) ∆x u + 31 ∇x ∇x · u + τ5 (ϵp0 , θ0 )∇x θ0 · ∇x u + (∇x u)T − 23 (∇x · u)I
(
)
+ τ6 (ϵp0 , θ0 ) ∇x u − (∇x u)T · ∇x θ0 .
Assume there exist two constants α1 , α2 > 0 such that
(4.2)
0 < α1 < h(ϵp0 ), g(θ0 ), µ(θ0 ), κ(θ0 ), τ1 (p0 , θ0 ), τ4 (p0 , θ0 ) < α2 < ∞ .
We call (p, u) the fast motion and (ϵp, ϵu, θ) the slow motion, because (ϵp, ϵu, θ) varies
on the time scale of order O(1), while (p, u) on the time scale of order O(ϵ).
Definition 4.1. For any s ∈ R1 , define the weighted Sobolev norm ∥ · ∥Hϵs+1 as
∥w∥Hϵs+1 = ϵ∥∇x w∥H s + ∥w∥H s ,
A Low Mach Number Limit of a Dispersive Navier-Stokes System
9
for any w ∈ H s+1 (R3 ). By the definition of Λϵ in (2.3), it is clear that
∥w∥Hϵ1 ∼ ∥Λϵ w∥L2 ,
for any ϵ > 0 and w ∈ H s+1 .
First we prove a linear estimate for (ϵp, ϵu, θ) where (p, u, θ) is the solution of
(4.1). To this end, we work out the system for
(p̂, û, θ̂) = (ϵp − θ, ϵu, θ) .
(4.3)
Algebraic calculation using (4.1) shows that the system for (p̂, û, θ̂) has the form
(4.4)
1
(∂t + u0 · ∇x )p̂ + ∇x · û = 35 ϵf1 − 23 f3 ,
ϵ
1
1
g(θ0 )(∂t + u0 · ∇x ) û + ∇x p̂ + ∇x θ̂ = h(ϵp0 )∇x · Σ + ϵh(ϵp0 )∇x · Σ̃ + ϵf2 ,
ϵ
ϵ
1
3
2 (∂t + u0 · ∇x ) θ̂ + ϵ ∇x · û = ϵh(ϵp0 )∇x · q̃ + ϵh(ϵp0 ) Σ̃ : ∇x (ϵu0 )
+ h(ϵp0 )Σ : ∇x (ϵu0 ) − h(ϵp0 )∇x · q
+ f3 ,
(p, u, θ)(x, 0) = (pin , uin , θin )(x) .
where
(
)
q = −κ(θ0 )∇x θ̂ ,
Σ = µ(θ0 ) ∇x û + (∇x û)T − 23 (∇x · û)I ,
(
)
(
)
Σ̃ = τ1 (ϵp0 , θ0 ) ∇x2 θ̂ − 13 (∆x θ̂)I + τ2 (ϵp0 , θ0 ) ∇x θ0 ⊗ ∇x θ̂ − 13 (∇x θ0 · ∇x θ̂)I
(
)
+ τ3 (ϵp0 , θ0 ) ∇x (ϵu0 )(∇x û)T − (∇x (ϵu0 ))T ∇x û ,
(
)
(
)
q̃ = τ4 (ϵp0 , θ0 ) ∆x û + 13 ∇x ∇x · û + τ6 (ϵp0 , θ0 ) ∇x û − (∇x û)T · ∇x θ0 .
(
)
+ τ5 (ϵp0 , θ0 )∇x θ0 · ∇x û + (∇x û)T − 23 (∇x · û)I
Note that the equation for p̂ = ϵp − θ is essentially the linearized density equation.
4.2. Linear Estimate for the Slow Motion. The L2 -estimate for system (4.4)
states
Theorem 4.2. Let (p̂, û, θ̂) be a solution to the linear system (4.4). Then there
exist an increasing function C(·) and a constant α0 > 0 such that
∫ T
2
sup ∥(p̂, û, θ̂)∥Hϵ1 + α0
∥∇x (û, θ̂)∥2Hϵ1 (τ ) dτ
[0,T ]
(4.5)
0
∫
≤ C(r0 )eT C(r) ∥(ϵpin , ϵuin , θin )∥2Hϵ1 + C(r)
∫
∫
T
0
∫ T⟨
⟩
3
+ C(r)
ϵ f2 , Ψ1 û + Ψ0 û dτ ,
⟩
p̂, T2 ( 3 ϵf1 − 2 f3 ) (τ ) dτ
5
3
T
|⟨θ̂, T1 f3 ⟩|(τ ) dτ
|⟨û, T2 (ϵf2 )⟩| dτ + C(r)
+ C(r)
0
T⟨
0
0
where the operators T1 , T2 are defined in (4.9) and the constants r, r0 are given by
ϕ = (ϵp0 , θ0 ) ,
(4.6)
r0 = ∥ϕ(0)∥L∞ ,
r = sup (∥(ϕ, ∂t ϕ)∥W 2,∞ , ∥(u0 , p0 )∥W 1,∞ ) .
[0,T ]
10
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Proof. The main structure of system (4.4) becomes more transparent by rewriting
it in the following abstract way. We use Ψi to denote ith -order homogeneous differential operator with their coefficients depending on (ϵp0 , ϵu0 , θ0 ) and their derivatives
up to the second order to obtain
1
(∂t + u0 · ∇x )p̂ + ∇x · û = 35 ϵf1 − 23 f3 ,
ϵ
(
)
1
1
g(θ0 )(∂t +u0 ·∇x ) û + ∇x p̂ + ∇x θ̂ = h(ϵp0 )µ(θ0 ) ∆x + 31 ∇x ∇x · û
ϵ
ϵ
+ 23 ϵ∇x ∇x ·(h(ϵp0 )τ1 (ϵp0 , θ0 )∇x θ̂)
+ ϵΨu2 (û, θ̂) + Ψu1 (û, θ̂) + ϵf2 ,
(4.7)
1
3
2 (∂t +u0 ·∇x ) θ̂+ ϵ ∇x · û
= h(ϵp0 )κ(θ0 )∆x θ̂+ϵΨθ2 (û, θ̂)+Ψθ1 (û, θ̂)
+ 43 ϵ ∇x ·(∇x ·(h(ϵp0 )τ4 (ϵp0 , θ0 )∇x û))
+ f3 ,
(p̂, û, θ̂)(x, 0) = (p̂in , ûin , θ̂in )(x),
where
Ψu2 (û, θ̂) = h(ϵp0 ) ∇x τ1 (ϵp0 , θ0 ) · (∇x2 θ̂ − 13 (∆x θ̂)I)
− 23 ∇x (h(ϵp0 ) τ1 (ϵp0 , θ0 )) · (∇x2 θ̂ + (∆x θ̂)I)
+ 32 h(ϵp0 ) τ2 (ϵp0 , θ0 )∇x θ0 · ∇x2 θ̂
(
)
+ h(ϵp0 ) τ3 (ϵp0 , θ0 ) ∇x (ϵu0 ) : ∇x2 û − ∇x (ϵu0 ) : ∇x (∇x û)T ,
Ψu1 (û, θ̂) = − 23 ϵ∇x2 (h(ϵp0 ) τ1 (ϵp0 , θ0 )) · ∇x θ̂
(
)
+ ϵh(ϵp0 ) ∇x τ2 (ϵp0 , θ0 ) · ∇x θ0 ⊗ ∇x θ̂ − 31 (∇x θ0 · ∇x θ̂)I
)
(
+ ϵh(ϵp0 ) τ2 (ϵp0 , θ0 ) · (∆x θ0 )∇x θ̂ − 13 ∇x2 θ0 · ∇x θ̂
(
)
+ ϵh(ϵp0 ) ∇x τ3 (ϵp0 , θ0 ) · (∇x (ϵu0 )) (∇x û)T − (∇x (ϵu0 ))T (∇x û)
(
)
+ τ3 (ϵp0 , θ0 ) ∆x (ϵu0 )(∇x û)T − (∇x ∇x · (ϵu0 ))T ∇x û ,
(
)
+ h(ϵp0 )∇x µ(θ0 ) · ∇x û + (∇x û)T − 32 (∇x · û)I ,
Ψθ2 (û, θ̂) = h(ϵp0 ) ∇x τ4 (ϵp0 , θ0 ) · (∆x û + 13 ∇x ∇x · û)
− 43 ∇x (h(ϵp0 ) τ4 (ϵp0 , θ0 )) · (∇x ∇x · û + ∆x û)
(
)
+ h(ϵp0 )τ5 (ϵp0 , θ0 )∇x θ0 · ∆x û + 13 ∇x ∇x · û
Ψθ1 (û, θ̂)
+ h(ϵp0 )τ6 (ϵp0 , θ0 )∇x θ0 · (∆û − ∇x ∇x · û) ,
(
)
= ϵh(ϵp0 )(∇x τ5 (ϵp0 , θ0 ) ⊗ ∇x θ0 ) : ∇x û + (∇x û)T − 23 (∇x · û)I
(
)
+ h(ϵp0 )τ5 (ϵp0 , θ0 )∇x2 θ0 : ∇x (û) + (∇x (û))T − 23 (∇x · (û))I
+ ϵh(ϵp0 )(∇x τ6 (ϵp0 , θ0 ) ⊗ ∇x θ0 ) : (∇x û − (∇x û)T )
− 34 ϵ∇x2 (h(ϵp0 ) τ4 (ϵp0 , θ0 )) : ∇x û + ∇x κ(θ0 ) · ∇x θ .
11
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Combining the leading order dispersive terms with the singular terms, we have
(4.8)
1
(∂t + u0 · ∇x )p̂ = − ∇x · û + 35 ϵf1 − 23 f3 ,
ϵ
) )
1
1 ((
g(θ0 )(∂t + u0 · ∇x ) û + ∇x p̂ = − ∇x I − 23 ϵ2 ∇x · (h(ϵp0 )τ1 (ϵp0 , θ0 )∇x ) θ̂
ϵ
ϵ
(
)
+ h(ϵp0 )µ(θ0 ) ∆x + 31 ∇x ∇x · û + ϵΨu2 (û, θ̂)
+ Ψu1 (û, θ̂) + ϵf2 ,
((
) )
1
3
I − 43 ϵ2 ∇x · (h(ϵp0 )τ4 (ϵp0 , θ0 )∇x ) û
2 (∂t + u0 · ∇x ) θ̂ = − ϵ ∇x ·
+ h(ϵp0 )κ(θ0 )∆x θ̂ + ϵΨθ2 (û, θ̂) + Ψθ1 (û, θ̂) + f3 ,
(p̂, û, θ̂)(x, 0) = (p̂in , ûin , θ̂in )(x) .
Defining the operators T1 , T2 as
(4.9)
T1 = I − 23 ϵ2 ∇x · (h(ϵp0 )τ1 (θ0 )∇x ) ,
T2 = I − 43 ϵ2 ∇x · (h(ϵp0 )τ4 (θ0 )∇x )
it is apparent that T1 , T2 are symmetric positive operators. Adopting this notation,
system (4.8) now has the form
(4.10)
1
(∂t + u0 · ∇x )p̂ + ∇x · û = 35 ϵf1 − 23 f3 ,
ϵ
(
)
1
1
g(θ0 )(∂t + u0 · ∇x ) û + ∇x p̂ + ∇x (T1 θ̂) = h(ϵp0 )µ(θ0 ) ∆x + 31 ∇x ∇x · û
ϵ
ϵ
+ ϵΨu2 (û, θ̂) + Ψu1 (û, θ̂) + ϵf2 ,
1
θ
3
2 (∂t + u0 · ∇x ) θ̂ + ϵ ∇x · (T2 û) = h(ϵp0 )κ(θ0 )∆x θ̂ + ϵΨ2 (û, θ̂)
+ Ψθ1 (û, θ̂) + f3 ,
(p̂, û, θ̂)(x, 0) = (p̂in , ûin , θ̂in )(x) .
In order to perform the L2 energy estimate, we multiply T2 p̂, T2 û, T1 θ̂ to the equations
for p̂, û, θ̂ respectively, integrate over R3 , and sum up the three integrated equations.
Note that although T1 , T2 are positive symmetric operators, they have variable coefficients. Thus they do not commute either with time derivatives or spatial derivatives.
To make notations shorter, let
∫
2
4
g1 (ϕ0 ) = 3 h(ϵp0 )τ1 (ϵp0 , θ0 ) , g2 (ϕ0 ) = 3 h(ϵp0 )τ4 (ϵp0 , θ0 ) , ⟨·⟩ =
· dx .
R3
First, the time derivative term in the integrated p̂-equation is
(4.11)
d
⟨p̂, T2 p̂⟩ = ⟨∂t p̂, T2 p̂⟩ + ⟨p̂, ∂t (T2 p̂)⟩ = 2⟨∂t p̂, T2 p̂⟩ + ⟨p̂, [∂t , T2 ]p̂⟩ ,
dt
where
[∂t , T2 ]p̂ = −ϵ2 ∇x · ((∂t g2 (ϕ0 ))∇x p̂) .
Therefore we have
⟨
⟩
p̂, [∂t , T2 ]p̂ ≤ C(r)∥ϵ∇x p̂∥2L2 ≤ C(r)∥p̂∥2Hϵ1 ,
12
A Low Mach Number Limit of a Dispersive Navier-Stokes System
where C(r) = C1 (∥∂t g2 (ϕ0 )∥L∞ ) is a positive increasing function in its variable. The
definition of r is given by (4.6). Thus, we have
⟨∂t p̂, T2 p̂⟩ =
(4.12)
1
2
d
⟨p̂, T2 p̂⟩ − 12 ⟨p̂, [∂t , T2 ]p̂⟩ ,
dt
with
|⟨p̂, [∂t , T2 ]p̂⟩| ≤ C(r)∥p̂∥2Hϵ1 .
⟨
⟩
The second term in the estimate for p̂ is p̂, T2 (u0 · ∇x p̂) , which has the bound
⟨
⟩
⟩ ⟨
⟩ ⟨
p̂, T2 (u0 · ∇x p̂) ≤ p̂, (u0 · ∇x p̂) + p̂, ϵ2 ∇x · (g2 (ϕ0 )∇x (u0 · ∇x p̂)) (4.13)
where
⟨
⟩
p̂, (u0 · ∇x p̂) =
and
1
2
∫
2
≤ C(r) ∥p̂∥2 2 ,
(∇
·
u
)
|p̂|
dx
L
3 x 0
R
⟨
⟨
⟩
⟩
p̂, ϵ2 ∇x · (g2 (ϕ0 )∇x (u0 · ∇x p̂)) = ϵ2 ∇x p̂, g2 (ϕ0 )∇x (u0 · ∇x p̂) ∫
∫
≤ ϵ2
|g2 (ϕ0 )∇x u0 | |∇x p̂|2 dx + 21 ϵ2
|∇x · (g2 (ϕ0 ) u0 )||∇x p̂|2 dx .
R3
Therefore,
R3
⟨
⟩
(
)
p̂, T2 (u0 · ∇x p̂) ≤ C(r) ∥p̂∥2 2 + ∥ϵ∇x p̂∥2 2 ≤ C(r)∥p̂∥2 1 .
L
L
Hϵ
(4.14)
Here C(r) = C2 (∥u0 ∥W 1,∞ , ∥g2 ∥W 1,∞ ).
The third term in the estimate for the p̂-equation is
⟨
⟩
1
1
(4.15)
p̂, T2 ∇x · û = ⟨T2 p̂, ∇x · û⟩ .
ϵ
ϵ
Adding (4.12), (4.14), and (4.15) together, we obtain
(4.16)
1
2
⟨
⟩
d
1
⟨p̂, T2 p̂⟩ + ⟨T2 p̂, ∇x · û⟩ ≤ C(r) ∥p̂∥2Hϵ1 + p̂, T2 ( 35 ϵf1 − 32 f3 ) ,
dt
ϵ
where C(r) = C3 (∥∂t g2 ∥L∞ , ∥(u0 , g2 )∥W 1,∞ ).
Next, we estimate each term in the integrated û-equation. First, the time derivative term is treated in the similar way as in (4.11). In order to make the operator
symmetric, we consider
⟩
√
d⟨ √
û, g(θ0 )T2 ( g(θ0 )û)
dt
(4.17)
⟩
⟩
√
d⟨
d⟨ √
=
û, T2 (g(θ0 )û) +
û, [ g(θ0 ), T2 ]( g(θ0 )û) ,
dt
dt
The first term on the right-hand side of (4.17) is
⟩ ⟨
⟩ ⟨
⟩
d⟨
û, T2 (g(θ0 )û) = ∂t û, T2 (g(θ0 )û) + û, ∂t (T2 (g(θ0 )û))
dt
⟨
⟩
(
)⟩ ⟨
(4.18)
= 2 û, T2 g(θ0 )∂t û − ∂t û, [g(θ0 ), T2 ]û
⟨
⟩
+ û, [∂t , T2 (g(θ0 ) ·)]û ,
A Low Mach Number Limit of a Dispersive Navier-Stokes System
13
where
[∂t , T2 (g(θ0 ) ·)] û = ∂t (g(θ0 )) û − ϵ2 ∇x · (∂t g2 (ϕ0 ) ∇x (g(θ0 ) û))
− ϵ2 ∇x · (g2 (ϕ0 ) ∇x ((∂t g(θ0 )) û)) .
⟨
⟩
Hence by integration by parts, the bound for û, [∂t , T2 (g(θ0 ) ·)]û is
(4.19)
⟨
⟩
)
(
û, [∂t , T2 (g(θ0 ) ·)]û ≤ C(r) ∥û∥2L2 + ∥ϵ∇x û∥2L2 ≤ C(r) ∥û∥2Hϵ1 ,
with C(r) = C4 (∥∂t g∥W 1,∞
⟨ , ∥g∥W 1,∞ , ∥∂t g2⟩∥L∞ ).
In order to estimate ∂t û, [g(θ0 ), T2 ]û , first we write
[g(θ0 ), T2 ]û = ϵ2 ∇x · (g2 (ϕ0 ) ∇x g(θ0 ) ⊗ û) + ϵ2 g2 (ϕ0 )∇x g(θ0 ) · ∇x û
, ϵ2 (Ψ1 û + Ψ0 û) .
Here again Ψ1 and Ψ0 are homogeneous operators of order 1 and 0 respectively. By
the û-equation,
ϵ2 ∂t û = −ϵ2 u0 · ∇x û − ϵg(θ0 )−1 ∇x p̂ − ϵg(θ0 )−1 ∇x (T1 θ̂)
)
(
+ ϵ2 g(θ0 )−1 h(ϵp0 )µ(θ0 ) ∆x + 13 ∇x ∇x · û + ϵ3 Ψu2 (û, θ̂) + ϵ2 Ψu1 (û, θ̂) + ϵ3 f2
= −ϵ2 u0 · ∇x û − ϵg(θ0 )−1 ∇x p̂ − ϵg(θ0 )−1 ∇x (T1 θ̂) + ϵ2 Ψu2 û + ϵ3 Ψu2 θ̂
+ ϵ2 Ψu1 (û, θ̂) + ϵ3 f2 .
⟨
⟩
The bounds for each term in ϵ2 ∂t û, Ψ1 û are
⟨
⟩
2
ϵ u0 · ∇x û, Ψ1 û + Ψ0 û ≤ C(r) ∥ϵû∥2L2 + C(r) ∥ϵ∇x û∥2L2 ,
⟨
⟩
ϵg(θ0 )−1 ∇x p̂, Ψ1 û + Ψ0 û ≤ C(r) ∥ϵ∇x p̂∥2L2 + C(r) ∥û∥2L2 + 81 α0 ∥∇x û∥2L2 ,
⟨
⟩
ϵg(θ0 )−1 ∇x (T1 θ̂), Ψ1 û + Ψ0 û ⟨
⟩ ⟨
⟩
≤ ϵg(θ0 )−1 ∇x θ̂, Ψ1 û + Ψ0 û + ϵ3 g(θ0 )−1 ∇x ∇x · (g1 (ϕ0 )∇x θ̂), Ψ1 û + Ψ0 û (
)
(
)
≤ C(r) ∥ϵ∇x θ̂∥2L2 + ∥û∥2L2 + 81 α0 ∥∇x û∥2L2 + ϵ C(r) ∥ϵ∆x û∥2L2 + ∥ϵ∆x θ̂∥2L2 ,
⟨
⟩
2
ϵ Ψ2 û, Ψ1 û + Ψ0 û ≤ C(r)
⟨
⟩
3
ϵ Ψ2 θ̂, Ψ1 û + Ψ0 û ≤ C(r)
⟨
⟩
2
ϵ Ψ1 (û, θ̂), Ψ1 û + Ψ0 û ≤ C(r)
(
)
∥ϵ∇x û∥2L2 + ∥ϵû∥2L2 + 81 α0 ∥ϵ∆x û∥2L2 ,
)
(
∥ϵ∇x û∥2L2 + ∥ϵû∥2L2 + ϵ ∥ϵ∆x θ̂∥2L2 ,
)
(
∥ϵ∇x û∥2L2 + ∥ϵû∥2L2 + ∥ϵ∇x θ̂∥2L2 ,
Combining these bounds we obtain
⟨
)
⟩
(
∂t û, [g(θ0 ), T2 ]û ≤ C(r) ∥ϵ∇x p̂∥2L2 + ∥û∥2Hϵ1 + ∥ϵ∇x θ̂∥2L2
(4.20)
+ 14 α0 ∥∇x û∥2L2 + 14 α0 ∥ϵ∆x û∥2L2 + 41 α0 ∥ϵ∆x θ̂∥2L2 ,
⟨
⟩
+ ϵ3 f2 , Ψ1 û + Ψ0 û 14
A Low Mach Number Limit of a Dispersive Navier-Stokes System
where α0 is less than the lower bound of the viscosity coefficients such that the terms
associated with α0 can be absorbed into the dissipation terms and ϵ is chosen to be
small enough such that ϵC(r) ≤ 18 α0 . Here
C(r) = C5 (∥u0 ∥L∞ , ∥(g, g1 , g2 )∥W 2,∞ , ∥g −1 ∥L∞ , ∥(τ1 , h)∥W 1,∞ ) .
Combining (4.19) and (4.20), we have
⟩
⟨
⟨
⟩
(
)⟩
d⟨
û, T2 (g(θ0 )û) + 12 û, [g(θ0 ), T2 ]∂t û
û, T2 g(θ0 )∂t û = 21
dt⟨
(4.21)
⟩
− 21 û, [∂t , T2 (g(θ0 ) ·)]û ,
with
⟨
⟩ ⟨
⟩
1
2 û, [g(θ0 ), T2 ]∂t û + 21 û, [∂t , T2 (g(θ0 ) ·)]û (
)
≤ C(r) ∥û∥2Hϵ1 + ∥ϵ∇x p̂∥2L2 + ∥ϵ∇x θ̂∥2L2 + 14 α0 ∥∇x û∥2L2
⟨
⟩
+ 41 α0 ∥ϵ∆x û∥2L2 + 14 α0 ∥ϵ∆x θ̂∥2L2 + ϵ3 f2 , Ψ1 û + Ψ0 û ,
where
C(r)
= C6 (∥∂t g∥W 1,∞ , ∥∂t g2 ∥L∞ , ∥u0 ∥L∞ , ∥(g, g1 , g2 )∥W 2,∞ , ∥g −1 ∥L∞ , ∥(τ1 , h)∥W 1,∞ ) .
The second term on the right-hand side of (4.17) is bounded in a similar way. By
integration by parts,
⟨
⟩
√
√
û, [ g(θ0 ), T2 ]( g(θ0 )û) = ϵ2 ⟨û, Ψ1 û⟩ + ϵ2 ⟨û, Ψ0 û⟩ = ϵ2 ⟨û, Ψ0 û⟩ ,
where Ψ1 , Ψ0 are homogeneous operators of order 1 and 0 respectively. Their coefficients depend on ∇xα (g(θ0 ), g2 (ϵp0 , θ0 )) with |α| ≤ 2. Thus,
⟩ ⟨
√
⟩
d⟨ √
(4.22)
û, [ g(θ0 ), T2 ]( g(θ0 )û) = ϵ2 ∂t û, Ψ0 û + ϵ2 ⟨û, Ψ0 û⟩ .
dt
The equation for û yields
ϵ2 ∂t û = ϵΨ1 p̂ + ϵ2 Ψ2 û + ϵ2 Ψ1 û + ϵ3 Ψ2 θ̂ + ϵ2 Ψ2 θ̂ + ϵ2 Ψ1 θ̂ + ϵ3 g −1 (θ0 )f2 .
Hence the bound for the first term on the right-hand side of (4.22) is
⟨ 2
⟩
ϵ ∂t û, Ψ0 û ≤ C(r) ∥(p̂, û, θ̂)∥2 1 + 1 α0 ∥∆x (ϵû)∥2 2 + ϵ3 |⟨û, g −1 (θ0 )f2 ⟩| .
L
Hϵ
16
Thus we have the bound for the second term of the right-hand side of (4.17) as
⟨
⟩
√
√
d
dt û, [ g(θ0 ), T2 ]( g(θ0 )û) (4.23)
≤ C(r) ∥(p̂, û, θ̂)∥2Hϵ1 +
Combining (4.21) and (4.23), we have
⟨
(
)⟩
(4.24)
û, T2 g(θ0 )∂t û =
2
1
16 α0 ∥∆x (ϵû)∥L2
1
2
+ ϵ3 |⟨û, g −1 (θ0 )f2 ⟩| .
⟩
d⟨
û, T2 (g(θ0 )û) + R1 ,
dt
A Low Mach Number Limit of a Dispersive Navier-Stokes System
15
with
⟨
⟩
R1 ≤ C(r) ∥(p̂, û, θ̂)∥2Hϵ1 + 14 α0 ∥∇x û∥2Hϵ1 + 14 α0 ∥ϵ∆x θ̂∥2L2 + ϵ3 f2 , Ψ1 û + Ψ0 û .
The convection term is bounded as
|⟨û, T2 (g(θ0 )u0 · ∇x û)⟩|
(4.25)
≤ |⟨û, g(θ0 )u0 · ∇x û⟩| + ϵ2 |⟨∇x · (g2 (ϕ0 )∇x û), g(θ0 )u0 · ∇x û⟩|
≤ C(r)∥û∥2Hϵ1 ,
by integration by parts. Here C(r) = C8 (∥(g, g2 , u0 )∥W 1,∞ ).
The next two terms in the integrated û-equation are
⟨
⟩ ⟨
⟩
⟨
⟩ ⟨
⟩
1
1
1
1
(4.26) û, ∇x (T2 p̂) + û, T2 ∇x (T1 θ̂) = − ∇x · û, (T2 p̂) + T2 û, ∇x (T1 θ̂) .
ϵ
ϵ
ϵ
ϵ
Next, the dissipative term satisfies
⟨
(
(
) )⟩
− û, T2 h(ϵp0 )µ(θ0 ) ∆x + 13 ∇x ∇x · û
(
)
(4.27)
≥ 2α0 ∥∇x û∥2L2 + ∥ϵ∆x û∥2L2 − C(r) ∥û∥2L2 .
The bounds for the remaining terms in the û-equation are
⟩
⟨
⟩ ⟨ 1
û, ϵT2 Ψ2 (û, θ̂) + T2 Ψ1 (û, θ̂) + ϵ T2 f2 + û, [T2 , ∇x ]p̂ ϵ
⟨
⟨
⟩
⟩
3
≤ û, ϵΨ2 (û, θ̂) + Ψ1 (û, θ̂ + ϵ ∇x · (g2 (ϕ0 )∇x û), Ψ2 (û, θ̂) ⟨
⟩
+ ϵ3 ∇x · (g2 (ϕ0 )∇x û), Ψ1 (û, θ̂) + ϵ |⟨û, [∇x · (g2 (ϕ0 )∇x ), ∇x ]p̂⟩|
(4.28)
+ |⟨û, ϵT2 f2 ⟩|
≤ C(r)∥û∥L2 ∥∆x (ϵû, ϵθ̂)∥L2 + ϵC(r)∥∇x (ϵû)∥L2 ∥∆x (ϵû, ϵθ̂)∥L2
+ ϵC(r)∥∆x (ϵû, ϵθ̂)∥2L2 + ϵ |⟨û, ∇x · (∇x p̂ ⊗ ∇x g2 )⟩| + |⟨û, T2 f2 ⟩|
(
)
≤ C(r) ∥(p̂, û, θ̂)∥2Hϵ1 + 18 α0 ∥∇x û∥2Hϵ1 + ∥∆x (ϵθ̂)∥2L2 + |⟨û, ϵ T2 f2 ⟩| ,
1
α0 . Here C(r) = C9 (∥(h, τ1 , τ2 , τ3 , g2 )∥W 1,∞ ).
by choosing ϵ small such that ϵC(r) ≤ 16
Combining (4.24), (4.25), (4.26), (4.27), and (4.28), we have
(4.29)
⟩ ⟨
⟩
⟨
⟩ ⟨
1
1
1 d
û,
T
(g(θ
)û)
−
∇
·
û,
(T
p̂)
+
T
û,
∇
(T
θ̂)
+ 32 α0 ∥∇x û∥2Hϵ1
2
0
x
2
2
x
1
2 dt
ϵ
ϵ
⟨
⟩
1
≤ C(r)∥(p̂, û, θ̂)∥2Hϵ1 + α0 ∥ϵ∆x θ̂∥2L2 + |⟨û, ϵ T2 f2 ⟩| + ϵ3 f2 , Ψ1 û + Ψ0 û .
2
Similarly, the θ̂-equation yields
⟩
⟩ ⟨
1
d ⟨3
θ̂, T1 θ̂ − T2 û, ∇x (T1 θ̂) + 32 α0 ∥∇x θ̂∥2Hϵ1
dt 2
ϵ
⟨
⟩
1
≤ C(r)∥(p̂, û, θ̂)∥2Hϵ1 + α0 ∥ϵ∆x û∥2L2 + θ̂, T1 f3 .
2
1
2
(4.30)
16
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Upon adding up (4.16), (4.29), and (4.30), the singular terms are canceled out,
and the energy inequality is
(4.31)
(⟨
⟩ ⟨ √
⟩ ⟨
⟩)
√
1 d
p̂, T2 p̂ + û, g(θ0 )T2 ( g(θ0 )û) + 32 θ̂, T1 θ̂ + α0 ∥∇x (û, θ̂)∥2Hϵ1
2 dt
⟨
⟩
⟨
⟩
≤ C(r) ∥(p̂, û, θ̂)∥2Hϵ1 + p̂, T2 ( 35 ϵf1 − 23 f3 ) + |⟨û, T2 (ϵf2 )⟩| + θ̂, T1 f3 ⟨
⟩
+ ϵ3 f2 , Ψ1 û + Ψ0 û .
By Gronwall’s inequality and the equivalence of norms
⟨
⟩ ⟨ √
⟩ ⟨
⟩
√
p̂, T2 p̂ + û, g(θ0 )T2 ( g(θ0 )û) + 23 θ̂, T1 θ̂ ∼ ∥(p̂, û, θ̂)∥2Hϵ1 ∼ ∥(ϵp, ϵu, θ)∥2Hϵ1 ,
we have
(∫
sup ∥(p̂, û, θ̂)∥Hϵ1 + α0
[0,t]
≤ C(r0 )e
T C(r)
(4.32)
∫
)1/2
t
0
∥∇x (û, θ̂)∥2Hϵ1 (τ ) dτ
∫
∥(ϵp , ϵu , θ )∥
in
in
in
Hϵ1
∫
T
0
T
|⟨û, T2 (ϵf2 )⟩|(τ ) dτ + C(r)
+ C(r)
0
+ C(r)
T
+ C(r)
∫ T⟨
⟩
3
ϵ f2 , Ψ1 û + Ψ0 û dτ ,
⟨
⟩
p̂, T2 ( 3 ϵf1 − 2 f3 ) (τ ) dτ
5
3
|⟨θ̂, T1 f3 ⟩|(τ ) dτ
0
0
which completes the proof.
4.3. A Priori Estimate for Slow Motions. In this part we establish the a
priori estimate for the nonlinear system (1.7). First, for t > 0 and s > 5, recall the
norm ||| · |||ϵ,s,t defined as
(4.33)
(
)
|||(p, u, θ)|||ϵ,s,t := sup ∥(p, u)∥H s + ∥Λϵ2s+1 (ϵp, ϵu, θ)∥H s+1
[0,t]
(∫
+ α0
t
(
∥∇x (u, p)∥2H s
+
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1
ϵ
)
)1/2
(τ ) dτ
,
0
where α0 > 0 is given by Theorem 4.2. We also define
(4.34)
(ϵp, ϵu, θ)∥H s+1 ,
R = ∥(p, u)∥H s + ∥Λ2s+1
ϵ
R′ = ∥∇x u∥H s + ∥∇x Λ2s+1
(ϵu, θ)∥H s+1 .
ϵ
The a priori estimate for the slow motion (ϵp, ϵu, θ) is stated in the following theorem.
Theorem 4.3. Let (p, u, θ) be a solution to the system (1.7). For t > 0, and
s > 5, define the norm ||| · |||♭ϵ,s,t as
|||(p, u, θ)|||♭ϵ,s,t : = sup ∥Λ2s+1
(ϵp, ϵu, θ)∥H s+1
ϵ
[0,t]
(∫
(4.35)
)1/2
t
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1 (τ ) dτ
ϵ
+ α0
0
,
A Low Mach Number Limit of a Dispersive Navier-Stokes System
17
where α0 > 0 is given by Theorem 4.2. Then for all T ∈ [0, 1], we have
(4.36)
|||(p, u, θ)|||♭ϵ,s,T ≤ C(Ω0 ) e(
√
T +ϵ)C(Ω)
,
where
(4.37)
Ω = |||(p, u, θ)|||ϵ,s,T ,
Ω0 = |||(p, u, θ)|||ϵ,s,0 .
The proof combines the linear estimate (4.5) with the estimates for commutator
terms, which are based on the following lemmas:
Lemma 4.4 (Calculus Inequality-Commutator). Let ψm ∈ OP S m be a Fourier
multiplier with m > 0. Then for all σ ≥ 0 and f, g ∈ H m+σ there exists a generic
constant c0 such that
(4.38)
∥[f, ψm ] g∥H σ (R3 ) ≤ c0 (∥f ∥W 1,∞ ∥g∥H σ+m−1 + ∥f ∥H σ+m ∥g∥L∞ ) .
As a consequence, for any 0 ≤ ϵ ≤ 1, α ≥ 0, and f, g ∈ H σ+m+|α|+1 we have
α
α
(4.39) ∥Λα
ϵ [f, ψm ] g∥H σ (R3 ) ≤ c0 (∥f ∥W 1,∞ ∥Λϵ g∥H σ+m−1 + ∥Λϵ f ∥H σ+m ∥g∥L∞ ) ,
where the operator Λϵ is defined by (2.3).
Lemma 4.5 (Calculus Inequalities-Composition). Let G ∈ C ∞ such that G(0) =
0. Let v ∈ H s (R3 ) with s > 5. Then G(v) ∈ H s (R3 ) and there exists a constant c0
which depends on s such that
s−1
s
∥∇xk G(v)∥L2 ≤ c0 ∥∇v G∥C s−1 ∥v∥L
∞ ∥v∥H s ≤ c0 ∥∇v G∥C s−1 ∥v∥H s ,
for all 1 ≤ k ≤ s.
We will frequently estimate terms of the form |⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩| in the following calculations where Tϵ = I − ϵ2 ∇x · (a(x)∇x ). Here a(x) ∈ W 1,∞ , the index m is
positive, and ν = (ν1 , ν2 , ν3 ) is a multi-index such that |ν| = k. We collect its bounds
in the following lemma.
Lemma 4.6. Let m > 0 and ν = (ν1 , ν2 , ν3 ) be a multi-index such that |ν| = k.
Then for all σ ≥ 0 and f, g ∈ H m+k+σ , there exists a generic constant c0 such that
(4.40)
m
∥[f, ∂xν Λm
ϵ ] g∥H σ (R3 ) ≤ c0 ϵ (∥f ∥W 1,∞ ∥g∥H σ+k+m−1 + ∥f ∥H σ+k+m ∥g∥L∞ )
+ c0 (∥f ∥W 1,∞ ∥g∥H σ+k−1 + ∥f ∥H σ+k ∥g∥L∞ ) ,
and for all ϕ1 , ϕ2 ∈ H k+m+2 ,
(4.41)
|⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
(
)
≤ c0 ϵm ∥ϕ1 ∥Hϵ1 ∥a∥L∞ ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk+m + ∥f ∥Hϵk+m+1 ∥ϕ2 ∥L∞
(
)
+ c0 ∥ϕ1 ∥Hϵ1 ∥a∥L∞ ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk + ∥f ∥Hϵk+1 ∥ϕ2 ∥L∞ ,
and
|⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
(4.42)
(
)
≤ c0 ϵm ∥a∥W 1,∞ ∥∇x ϕ1 ∥Hϵ1 ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk+m−1 + ∥f ∥Hϵk+m ∥ϕ2 ∥L∞
(
)
+ c0 ∥a∥W 1,∞ ∥∇x ϕ1 ∥Hϵ1 ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk−1 + ∥f ∥Hϵk ∥ϕ2 ∥L∞ .
18
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Proof. The commutator term on the left-hand side of (4.40) can be rewritten as
ν
m
m/2
[f, ∂xν Λm
)(1 + ϵm (−∆x )m/2 )−1 Λm
ϵ ] g = [f, ∂x (1 + ϵ (−∆x )
ϵ ]g.
Define the Fourier multiplier ψϵ as
ψϵ = (1 + ϵm (−∆x )m/2 )−1 Λm
ϵ .
Then ψϵ ∈ S 0 and its seminorms are uniformly bounded in ϵ. Therefore by (4.38)
ν
m
ν
m/2
∥[f, ∂xν Λm
ψϵ ] g∥H σ
ϵ ] g∥H σ ≤ ∥[f, ∂x ψϵ ] g∥H σ + ϵ ∥[f, ∂x (−∆x )
m
≤ c0 ϵ (∥f ∥W 1,∞ ∥g∥H σ+k+m−1 + ∥f ∥H σ+k+m ∥g∥L∞ )
+ c0 (∥f ∥W 1,∞ ∥g∥H σ+k−1 + ∥f ∥H σ+k ∥g∥L∞ ) .
Estimate (4.41) follows directly from (4.40) by Hölder inequality. Indeed,
|⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
2
ν m
≤ |⟨ϕ1 , [f, ∂xν Λm
ϵ ] ϕ2 ⟩| + ϵ |⟨a∇x ϕ1 , ∇x [f, ∂x Λϵ ] ϕ2 ⟩|
)
(
≤ c0 ϵm ∥ϕ1 ∥Hϵ1 ∥a∥L∞ ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk+m + ∥f ∥Hϵk+m+1 ∥ϕ2 ∥L∞
)
(
+ c0 ∥ϕ1 ∥Hϵ1 ∥a∥L∞ ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk + ∥f ∥Hϵk+1 ∥ϕ2 ∥L∞ .
Estimate (4.42) follows by integration by part to transfer one derivative to ϕ1 .
(4.43)
|⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
2
ν m
≤ |⟨ϕ1 , [f, ∂xν Λm
ϵ ] ϕ2 ⟩| + ϵ |⟨a∇x ϕ1 , ∇x [f, ∂x Λϵ ] ϕ2 ⟩| .
We have
ν2 m
ν1
ν2 m
|⟨ϕ1 , [f, ∂xν Λm
ϵ ] ϕ2 ⟩| ≤ |⟨∂ν1 ϕ1 , [f, ∂x Λϵ ] ϕ2 ⟩| + |⟨ϕ1 , (∂x f ) ∂x Λϵ ϕ2 ⟩|
ν1
ν4 m
≤ |⟨∂ν1 ϕ1 , [f, ∂xν2 Λm
ϵ ] ϕ2 ⟩| + |⟨∂ν3 ϕ1 , [∂x f, ∂x Λϵ ] ϕ2 ⟩|
ν5
ν6 m
+ |⟨ϕ1 , (∂x f ) ∂x Λϵ ϕ2 ⟩| ,
where |ν1 | = |ν3 | = 1, |ν5 | = 2, |ν4 | = |ν6 | = k − 2, and |ν2 | = k − 1. Therefore,
|⟨ϕ1 , [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
(4.44)
≤ c0 ϵm ∥∇x ϕ1 ∥L2 (∥f ∥W 2,∞ ∥ϕ2 ∥H k+m−2 + ∥f ∥H k+m−1 ∥ϕ2 ∥L∞ )
+ c0 ∥∇x ϕ1 ∥L2 (∥f ∥W 1,∞ ∥ϕ2 ∥H k−2 + ∥f ∥H k−1 ∥ϕ2 ∥L∞ )
+ c0 ∥ϕ1 ∥L2 ∥f ∥W 2,∞ ∥Λm
ϵ ϕ2 ∥H k−2 .
The last term in (4.43) is bounded as
ϵ2 |⟨a∇x ϕ1 , ∇x [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
(4.45)
2
ν m
≤ ϵ2 |⟨a∆x ϕ1 , [f, ∂xν Λm
ϵ ] ϕ2 ⟩| + ϵ |⟨∇x a · ∇x ϕ1 , [f, ∂x Λϵ ] ϕ2 ⟩|
≤ c0 ϵm+1 ∥a∥W 1,∞ ∥ϵ∇x ϕ1 ∥H 1 (∥f ∥W 1,∞ ∥ϕ2 ∥H k+m−1 + ∥f ∥H k+m ∥ϕ2 ∥L∞ )
+ c0 ϵ∥a∥W 1,∞ ∥ϵ∇x ϕ1 ∥H 1 (∥f ∥W 1,∞ ∥ϕ2 ∥H k−1 + ∥f ∥H k ∥ϕ2 ∥L∞ ) .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
19
Combining (4.44) and (4.45) we have
|⟨ϕ1 , Tϵ [f, ∂xν Λm
ϵ ] ϕ2 ⟩|
(
)
≤ c0 ϵm ∥a∥W 1,∞ ∥∇x ϕ1 ∥Hϵ1 ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk+m−1 + ∥f ∥Hϵk+m ∥ϕ2 ∥L∞
(
)
+ c0 ∥a∥W 1,∞ ∥∇x ϕ1 ∥Hϵ1 ∥f ∥W 1,∞ ∥ϕ2 ∥Hϵk−1 + ∥f ∥Hϵk ∥ϕ2 ∥L∞ .
The rest of this section is devoted to the proof of Theorem 4.3.
Proof. [Proof of Theorem 4.3] Let β = (β1 , β2 , β3 ) be a multi-index such that
|β| = s + 1. Let
(
)
β 2s
β 2s
(pβ , uβ , θβ ) , ∂xβ Λ2s
ϵ (ϵp − θ), ∂x Λϵ (ϵu), ∂x Λϵ θ .
Then (pβ , uβ , θβ ) satisfy the nonlinear system
(4.46)
1
(∂t + u · ∇x )pβ + ∇x · uβ = h1 ,
ϵ
1
1
g(θ)(∂t + u · ∇x ) uβ + ∇x pβ + ∇x θβ = h(ϵp)∇x · Σβ + ϵh(ϵp)∇x · Σ̃β + h2 ,
ϵ
ϵ
1
3
2 (∂t + u · ∇x ) θβ + ϵ ∇x · uβ = ϵh(ϵp)∇x · q̃β + h(ϵp)∇x · qβ + h3
+ ϵh(ϵp)Σβ : ∇x u + ϵ2 h(ϵp) Σ̃β : ∇x u,
where
(
)
qβ = κ(θ)∇x θβ ,
Σβ = µ(θ) ∇x uβ + (∇x uβ )T − 23 (∇x · uβ )I ,
(
)
(
)
Σ̃β = τ1 (ϵp, θ) ∇x2 θβ − 31 (∆x θβ )I + τ2 (ϵp, θ) ∇x θ ⊗ ∇x θβ − 13 (∇x θ · ∇x θβ )I
(
)
+ τ3 (ϵp, θ) ∇x (ϵu)(∇x uβ )T − (∇x ϵu)T ∇x uβ ,
(
)
(
)
q̃β = τ4 (ϵp, θ) ∆x uβ + 31 ∇x ∇x · uβ + τ6 (ϵp, θ) ∇x (ϵu) − (∇x (ϵu))T · ∇x θβ ,
(
)
+ τ5 (ϵp, θ)∇x θ · ∇x uβ + (∇x uβ )T − 23 (∇x · uβ )I
and (h1 , h2 , h3 ) are the commutator terms given by
h1 = −[∂xβ Λ2s
ϵ , u] · ∇x (ϵp − θ) ,
[ β 2s
]
(
)
h2 = − ∂x Λϵ , g(θ) ∂t (ϵu) + ∂xβ Λ2s
ϵ (h(ϵp)∇x ·Σ) − h(ϵp)∇x · Σβ
(
(
)
)
[
]
β 2s
− ∂xβ Λ2s
ϵh(ϵp)∇x · Σ̃ − ϵh(ϵp)∇x · Σ̃β ,
ϵ , g(θ)u · ∇x (ϵu) + ∂x Λϵ
[
]
( β 2s
)
h3 = − ∂xβ Λ2s
ϵ , u · ∇x θ + ∂x Λϵ (ϵh(ϵp)∇x · q̃) − ϵh(ϵp)∇x · q̃β
(
)
+ ∂xβ Λ2s
ϵ (h(ϵp)∇x · q) − h(ϵp)∇x · qβ
)
(
+ ∂xβ Λ2s
ϵ (ϵh(ϵp)Σ : ∇x u) − ϵh(ϵp)Σβ : ∇x u
(
)
)
(
2
2
+ ∂xβ Λ2s
ϵ
h(ϵp)
Σ̃
:
∇
u
−
ϵ
h(ϵp)
Σ̃
:
∇
u
,
x
β
x
ϵ
where
(
)
q = κ(θ)∇x θ , Σ = µ(θ) ∇x (ϵu) + (∇x ϵu)T − 23 (∇x · (ϵu))I ,
(
)
(
)
Σ̃ = τ1 (θ) ∇x2 θ − 31 (∆x θ)I + τ2 (θ) ∇x θ ⊗ ∇x θ − 13 |∇x θ|2 I
(
)
+ τ3 (ϵp, θ) ∇x (ϵu)(∇x (ϵu))T − (∇x (ϵu))T ∇x (ϵu) ,
(
)
(
)
q̃ = τ4 (θ) ∆x (ϵu) + 13 ∇x ∇x · (ϵu) + τ6 (ϵp, θ) ∇x (ϵu) − (∇x (ϵu))T · ∇x θ
(
)
+ τ5 (θ)∇x θ · ∇x (ϵu) + (∇x (ϵu))T − 23 (∇x · (ϵu))I .
20
A Low Mach Number Limit of a Dispersive Navier-Stokes System
By the linear estimate (4.5), we
⟨need to estimate the
⟩ following terms: |⟨pβ , T2 h1 ⟩|,
2
|⟨uβ , T2 h2 ⟩|, |⟨θβ , T1 h3 ⟩|, and ϵ h2 , Ψ1 uβ + Ψ0 uβ . First,
|⟨pβ , T2 h1 ⟩| ≤ C(R)∥pβ ∥Hϵ1 ∥h1 ∥Hϵ1 ,
and by (4.40),
∥h1 ∥Hϵ1 = ∥[u, ∂xβ Λ2s
ϵ ] · ∇x (ϵp − θ)∥Hϵ1
(
)
≤ c0 ϵ2s ∥u∥W 1,∞ ∥∇x (ϵp − θ)∥Hϵ3s+1 + ∥u∥Hϵ3s+2 ∥∇x (ϵp − θ)∥L∞
(
)
+ c0 ∥u∥W 1,∞ ∥∇x (ϵp − θ)∥Hϵs+1 + ∥u∥Hϵs+2 ∥∇x (ϵp − θ)∥L∞ .
Since
ϵ2s ∥∇x (ϵp, θ)∥Hϵ3s+1 ≤ ∥Λ2s+1
∇x (ϵp, θ)∥H s ≤ ∥Λ2s+1
(ϵp, θ)∥H s+1 ≤ R ,
ϵ
ϵ
∥∇x (ϵp, θ)∥Hϵs+1 ≤ ∥Λϵ ∇x (ϵp, θ)∥H s ≤ ∥Λϵ (ϵp, θ)∥H s+1 ≤ R ,
′
ϵ2s ∥u∥Hϵ3s+2 ≤ ∥Λ2s+1
u∥H s+1 ≤ ∥u∥H s+1 + ϵ∥Λ2s
ϵ
ϵ ∇x u∥H s+1 ≤ R + R ,
∥u∥Hϵs+2 ≤ ∥u∥H s+1 + ∥∇x (ϵu)∥H s+1 ≤ R + R′ ,
We have
∥h1 ∥Hϵ1 ≤ C(R)(1 + R′ ) ,
and
|⟨pβ , T2 h1 ⟩| ≤ C(R)(1 + R′ ) .
For the first term of h2 , we have
(4.47)[
]
[ β 2s
]
[ β 2s
] ( −1
)
− ∂xβ Λ2s
(θ)∇x p
ϵ , g(θ) ∂t (ϵu) = ∂x Λϵ , g(θ) (u · ∇x (ϵu)) + ∂x Λϵ , g(θ) g
] ( −1
)
[
(θ)h(ϵp)∇x · Σ
− ∂xβ Λ2s
ϵ , g(θ) g
)
] ( −1
[
ϵg (θ)h(ϵp)∇x · Σ̃ .
− ∂xβ Λ2s
ϵ , g(θ)
We have the following bounds related to these terms in (4.47). First by (4.42)
⟨
[
]
⟩
uβ , T2 ∂xβ Λ2s
ϵ , g(θ) (u · ∇x (ϵu))
(
)
≤ C(R) ϵ2s ∥∇x uβ ∥Hϵ1 ∥u · ∇x (ϵu)∥Hϵ3s + ∥g(θ) − g(0)∥Hϵ3s+1
(
)
+ C(R) ∥∇x uβ ∥Hϵ1 ∥u · ∇x (ϵu)∥Hϵs + ∥g(θ) − g(0)∥Hϵs+1
)
(
≤ C(R) ∥∇x uβ ∥Hϵ1 Λ2s+1
(ϵu)H s + Λ2s+1
θH s
ϵ
ϵ
≤ C(R)R′ .
The estimate related to the second term on the right-hand side of (4.47) is
⟨
[
] ( −1
)⟩
uβ , T2 ∂xβ Λ2s
(θ)∇x p ϵ , g(θ) g
(
)
≤ C(R) ϵ2s ∥∇x uβ ∥Hϵ1 ∥g −1 (θ)∇x p∥Hϵ3s + ∥g(θ) − g(0)∥Hϵ3s+1
(
)
+ C(R) ∥∇x uβ ∥Hϵ1 ∥g −1 (θ)∇x p∥Hϵs + ∥g(θ) − g(0)∥Hϵs+1
(
)
2s+1
≤ C(R) ∥∇x uβ ∥Hϵ1 Λ2s
θ∥H s + ∥p∥H s
ϵ (ϵp) H s+1 + ∥Λϵ
≤ C(R)R′ .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
21
We show the estimates related to the leading order terms in the rest of the terms in
(4.47). The lower order terms are bounded similarly. The leading order term for the
third term on the right-hand side of (4.47) satisfies
(4.48) ⟨
[
] −1
⟩
uβ , T2 ∂xβ Λ2s
(θ)h(ϵp)µ(θ) ∆x (ϵu) ϵ , g(θ) g
(
)
≤ C(R) ϵ2s ∥∇x uβ ∥Hϵ1 ∥g −1 (θ)h(ϵp)µ(θ) ∆x (ϵu)∥Hϵ3s + ∥g(θ) − g(0)∥Hϵ3s+1
(
)
+ C(R) ∥∇x uβ ∥Hϵ1 ∥g −1 (θ)h(ϵp)µ(θ) ∆x (ϵu)∥Hϵs + ∥g(θ) − g(0)∥Hϵs+1
)
(
θ∥H s
(ϵu)H s+1 + ∥Λ2s+1
≤ C(R) Λ2s+1
ϵ
ϵ
≤ C(R)R′ .
The estimate for the leading order term in the last term of (4.47) is
(4.49) ⟨
[
] −1
⟩
uβ , T2 ∂xβ Λ2s
(θ)h(ϵp)τ1 (θ) ∆x ∇x θ ϵ , g(θ) ϵg
(
)
≤ C(R) ϵ2s ∥∇x uβ ∥Hϵ1 ∥ϵg −1 (θ)h(ϵp)τ1 (θ) ∆x ∇x θ∥Hϵ3s + ϵ∥g(θ) − g(0)∥Hϵ3s+1
(
)
+ C(R) ∥∇x uβ ∥Hϵ1 ∥ϵg −1 (θ)h(ϵp)τ1 (θ) ∆x ∇x θ∥Hϵs + ϵ∥g(θ) − g(0)∥Hϵs+1
(
)
≤ C(R)R′ ϵ∥∇x Λ2s+1
θ∥H s+1 + ϵ∥Λ2s+1
θ∥H s
ϵ
ϵ
≤ C(R)R′ (1 + ϵ R′ ) .
Thus, the leading order terms are all bounded by C(R)R′ (1 + ϵR′ ). The lower order
terms are estimated in the same way. Therefore the first term in h2 is bounded by
C(R)R′ (1 + ϵR′ )
⟨
[
]
⟩
uβ , T2 ∂xβ Λ2s
≤ C(R)R′ (1 + ϵR′ ) .
(4.50)
ϵ , g(θ) ∂t (ϵu)
The third term of h2 has a similar structure as h1 . Therefore its estimate is similar
to that for |⟨pβ , T2 h1 ⟩|, which gives
⟨
]
⟩
[
≤ C(R)(1 + R′ ) .
uβ , T2 ∂xβ Λ2s
(4.51)
ϵ , g(θ)u · ∇x (ϵu)
We are left with the estimates for the second and fourth terms on h2 . Again we show
only
bounds of their leading orders.
The leading order term of the second term
( β the
)
∂x Λ2s
(h(ϵp)∇
·Σ)
−
h(ϵp)∇
·
Σ
is
x
x
β
ϵ
(
)
β 2s
1
[h(ϵp), ∂xβ Λ2s
ϵ ]µ(θ) + h(ϵp) [µ(θ), ∂x Λϵ ] (∆x (ϵu) + 3 ∇x ∇x · (ϵu)) .
Its estimate is similar to (4.48). Therefore,
⟨
(
)⟩
uβ , T2 ∂xβ Λ2s
≤ C(R)R′ .
(4.52)
ϵ (h(ϵp)∇x ·Σ) − h(ϵp)∇x · Σβ
(
(
)
)
The leading order term in ∂xβ Λ2s
ϵh(ϵp)∇x · Σ̃ − ϵh(ϵp)∇x · Σ̃β is
ϵ
(
) 2
β 2s
[ϵh(ϵp), ∂xβ Λ2s
ϵ ]τ1 (ϵp, θ) + ϵh(ϵp) [τ1 (ϵp, θ), ∂x Λϵ ] ( 3 ∇x ∆x θ) .
Its estimate is similar to (4.49). Therefore,
⟨
(
(
)
)⟩
(4.53)
ϵh(ϵp)∇x · Σ̃ − ϵh(ϵp)∇x · Σ̃β ≤ C(R)R′ (1 + ϵR′ ) .
uβ , T2 ∂xβ Λ2s
ϵ
22
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Combining (4.50), (4.51), (4.52), and (4.53) we have
|⟨uβ , T2 h2 ⟩| ≤ C(R) R′ (1 + ϵR′ ) .
(4.54)
The θ-equation has similar structures to the ϵu-equation. Hence the estimate for h3
is similar to the estimates for h2
|⟨θβ , T1 h3 ⟩| ≤ C(R) R′ (1 + ϵR′ ) .
(4.55)
By the linear estimate for the slow motion (4.31), we have
⟩)
⟩ ⟨
⟩ ⟨
√
√
d (⟨
pβ , T2 pβ + uβ , g(θ)T2 ( g(θ)uβ ) + 32 θβ , T1 θβ
dt
+ α0 ∥∇x (uβ , θβ )∥2Hϵ1
1
2
(4.56)
≤ C(R) + C(R)R′ + ϵC(R)(R′ )2 .
By the equivalence of the norms
⟨
⟩ ⟨
⟩ ⟨
⟩
√
√
∥(pβ , uβ , θβ )∥Hϵ1 ∼ pβ , T2 pβ + uβ , g(θ)T2 ( g(θ)uβ ) + 32 θβ , T1 θβ ,
we have
∫
|||(p, u, θ)|||♭ϵ,s,T ≤ C(Ω0 ) + C(Ω)T + C(Ω)
T
R′ (s) ds + ϵC(Ω)
√0
≤ C(Ω0 ) + C(Ω)T + C(Ω) T + ϵC(Ω)
√
≤ C(Ω0 ) + ( T + ϵ)C(Ω)
≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
∫
T
(R′ )2 (s)ds
0
,
for T ∈ (0, 1].
Remark: From the proof of Theorem 4.3 it is clear that for any m ∈ N, a similar
estimate as in (4.36) holds for ∥Λm+1
(ϵp, ϵu, θ)∥H s+1 . In particular, the estimate for
ϵ
s+1
∥Λm+1
(ϵp,
ϵu,
θ)∥
is
closed
in
terms
of ∥(u, p)∥H s and ∥(ϵp, ϵu, θ)∥H s+1 for all
H
ϵ
m ∈ N.
4.4. A Priori Estimate of curl(e−θ u). In addition to (ϵp, ϵu, θ), we have one
part of u which varies on the time scale of order one as in [2]. By taking curl on both
sides of the u-equation, we have the equation for curl(e−θ u) as
(
(
))
(
(
))
(∂t + u · ∇x ) curl e−θ u = curl e−ϵp ∇x · Σ + ∇x · Σ̃ + curl(e−θ u ∂t θ)
(4.57)
+ [curl, u] · ∇x (e−θ u) + curl(e−θ u(u · ∇x θ)) .
Because there is no singular terms in the above equation, the Sobolev norms of
curl(e−θ u) is done by directly differentiating the equations.
Lemma 4.7. Let s > 5. Let (p, u, θ) be a solution to the fluctuation equations
(1.7). Then there exists an increasing function such that
−θ
(4.58) sup ∥curl(e
[0,T ]
∫
u))∥2H s−1 +α0
0
T
√
∥∇x curl(e−θ u)∥2H s−1 (τ ) dτ ≤ C(Ω0 )e
T C(Ω)
.
23
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Proof. We rewrite (4.57) in terms of curl(e−θ u):
(
(
))
(4.59)
(∂t + u · ∇x ) curl e−θ u = ∇x · (e−ϵp µ(θ)∇x (curl(e−θ u)) + R1 ,
where
R1 = ∇x (e−ϵp µ(θ)) · ∇x (curl(e−θ u)) + ∇x (e−ϵp µ(θ)) × (∆x u + 13 ∇x (∇x · u))
(4.60)
+ curl(e−ϵp µ′ (θ)∇x θ · (∇x u + (∇x u)T − 23 (∇x · u)I))
( (
))
+ ∇x (e−ϵp τ1 )×( 32 ∇x ∆x θ) + curl∇x · τ2 ∇x θ ⊗ ∇x θ − 13 (∇x θ·∇x θ)I
( (
))
+ curl∇x · τ3 ∇x (ϵu)(∇x u)T − (∇x (ϵu))T ∇x u )
+ curl(e−θ u ∂t θ) + [curl, u] · ∇x (e−θ u) + curl(e−θ u(u · ∇x θ)) .
Notice that we do not have a fourth-order term from ∇x · Σ̃ because the leading order
term of curl(∇x · Σ̃) vanishes. We do have a fourth-order term from curl(e−θ u ∂t θ)
but it is pre-multiplied by ϵ. Thus the leading order term of curl(e−θ u ∂t θ) becomes
third-order of Λϵ (ϵu). Now apply ∂xσ on both sides with 1 ≤ |σ| ≤ s − 1:
(
(
))
(∂t + u · ∇x )∂xσ curl e−θ u
(4.61)
= ∇x · (e−ϵp µ(θ)∇x ∂xα (curl(e−θ u)) + ∂xσ R1 + R2 ,
where
(4.62)
R2 = [u, ∂xσ ] · ∇x (curl(e−θ u)) + [∇x ·(e−ϵp µ(θ)), ∂xσ ] ∇x (curl(e−θ u)) .
By standard energy estimates, multiplying both sides by ∂xσ (curl(e−θ u)) and integrate
over R3 , we have
−θ
sup ∥curl(e
[0,T ]
(4.63)
∫
u))∥2H s−1
T
+ α0
∥∇x curl(e−θ u)∥2H s−1 (τ ) dτ
0
−θ
≤ c0 ∥curl(e
(∫
u)(0, ·)∥2H s−1
)2
T
(∥∂xσ R1 ∥L2
+ c0
+ ∥R2 ∥L2 )(τ ) dτ
0
√ ∫
≤ c0 ∥curl(e−θ u)(0, ·)∥2H s−1 + c0 T
T
(∥∂xσ R1 ∥2L2 + ∥R2 ∥2L2 )(τ ) dτ .
0
Notice that the leading order terms in R1 are third-order terms in (Λϵ (ϵu), θ) and
second-order terms in u. The leading order terms in R2 are terms of order s + 1 in u
and of order s + 1 in (ϵp, θ). Therefore,
∫
T
(∥∂xσ R1 ∥2L2 + ∥R2 ∥2L2 )(τ ) dτ ≤ C(R) ,
0
which combined with (4.63) gives (4.58).
4.5. Some Additional Bounds. In this subsection we show some bounds of
2(s−k)
terms ∥Λϵ
(ϵ∂t )k (ϵp, ϵu, θ)∥H s+1−k and ∥(ϵ∂t )k (u, p)∥L2 for 1 ≤ k ≤ s. These
bounds will be used in the next section. Recall the following basic calculus inequality:
for |k1 | + |k2 | + · + |kn | = s with s > d + 2 = 5 ,
(4.64)
∥∇xk1 u1 ∇xk2 u2 · · · ∇xkn un ∥L2 ≤ ∥u1 ∥H s ∥u2 ∥H s · · · ∥un ∥H s ,
∀n ∈ N .
24
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Recall the definitions (4.34) of R, R′ :
R : = ∥(p, u)∥H s + ∥Λ2s+1
(ϵp, ϵu, θ)∥H s+1 ,
ϵ
R′ : = ∥∇x (u, θ)∥H s + ∥∇x Λ2s+1
(ϵu, θ)∥H s+1 .
ϵ
2(s−k)
First we show the bounds for ∥Λϵ
(ϵ∂t )k (ϵp, ϵu, θ)∥H s+1−k . These are immediate consequences of the bound for the slow motion in Theorem 4.3.
Lemma 4.8. Let (p, u, θ) be the solution to (1.7) and ψ = (ϵp, ϵu, θ). Then for
all 1 ≤ k ≤ s,
∥Λ2(s−k)
(ϵ∂t )k ψ∥H s+1−k ≤ C(R) ,
ϵ
∥∇x Λ2(s−k)
(ϵ∂t )k ψ∥H s+1−k ≤ C(R)(1 + R′ ) ,
ϵ
sup ∥Λ2(s−k)
(ϵ∂t )k ψ∥H s+1−k ≤ C(Ω0 )e(
ϵ
(4.65)
∫
0
√
T +ϵ)C(Ω)
,
[0,T ]
T
∥∇x Λ2(s−k)
(ϵ∂t )k ψ∥2H s+1−k (τ ) dτ ≤ C(Ω0 )e(
ϵ
√
T +ϵ)C(Ω)
,
where Ω0 , Ω are defined in (4.37).
Proof. We simply use equation (1.7) to write (ϵ∂t )ψ in terms of combinations of
the spatial derivatives of ψ and apply Theorem 4.3. Roughly speaking, each ϵ∂t
corresponds to the spatial derivatives which are at most of order (ϵ∇x )α ∇x with
|α| = 2. More specifically, we claim that (ϵ∂t )k ψ is a summation of products of
k
(j)
(j)
functions in the form of fj (ψ) or ∇x i (ϵ∇x )mi ψ, that is,
(
)
∑
∏ k(j)
(j)
(ϵ∂t )k ψ =
fj (ψ)
∇x i (ϵ∇x )mi ψ ,
(4.66)
j
i
where fj is a smooth function in ψ for all j and each summand satisfies that
∑ (j)
∑ (j)
|k
|
≤
k
,
|mi | ≤ 2k .
i
(4.67)
i
i
This claim can be proved by induction. First, (4.66) and (4.67) hold for k = 1 by
(1.7). Suppose (4.66) holds for k. Then for 1 ≤ k + 1 ≤ s, we have
(
)
∏ k(j)
(j)
mi
i
(ϵ∂t ) fj (ψ)
∇x (ϵ∇x )
ψ
i
= ∇ψ fj · (ϵ∂t )ψ + fj (ψ)
∑


) ∏ (j)
( (j)
(j)
(j)
k
 ∇xki (ϵ∇x )mi (ϵ∂t )ψ ·
∇x l (ϵ∇x )ml ψ  .
i
l̸=i
Using (1.7) to replace (ϵ∂t )ψ in the above equation shows that (4.66) and (4.67) hold
for k + 1. Thus (4.66) and (4.67) hold for all 1 ≤ k ≤ s. This further implies that
inequalities in (4.65) hold by Theorem 4.3.
Lemma 4.9. Let (p, u, θ) be the solution to (1.7). Let 1 ≤ k ≤ s. Then
(4.68)
∥(ϵ∂t )k (p, u)∥H s−k ≤ C(R) ,
∥(ϵ∂t )k (p, u)∥H s+1−k + ∥(ϵ∂t )k−1 ∂t θ∥H s+1−k ≤ C(R)(1 + R′ ) .
25
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Proof. The idea of the proof of (4.68) is similar as that of (4.65): we use equation
(1.7) to write (ϵ∂t )(p, u) in terms of combinations of the spatial derivatives of (p, u)
and ψ and use norms of (p, u) and ψ given by R in (4.34). By (4.64), we only need
to check the leading order terms in (ϵ∂t )k (u, p). We claim that these terms are of the
following forms:
(4.69)
f (ψ)∇xα u ,
f (ψ)∇xα p ,
f (ψ)∇xβ ψ ,
f (ψ)(ϵ∇x )γ ∇xβ ψ ,
where f is a smooth function in ψ. The multi-indices α, β, γ satisfy |α| = k, |β| = k+1,
and |γ| = 2k − 1. Again we prove this claim by induction. First, (4.69) holds for k = 1
by (1.7). Now suppose (4.69) holds for k. Then the leading-orders in (ϵ∂t )k (u, p) are
the leading-orders in the following terms:
f (ψ)∇xα (ϵ∂t )u ,
f (ψ)∇xα (ϵ∂t )p ,
f (ψ)∇xβ (ϵ∂t )ψ ,
f (ψ)(ϵ∇x )γ ∇xβ (ϵ∂t )ψ .
Using (1.7) to replace (ϵ∂t )(u, p, ψ) gives that (4.69) holds for k + 1. Thus (4.69) holds
for all 1 ≤ k ≤ s. Therefore,
∥(ϵ∂t )k (p, u)∥H s−k ≤ C(∥(p, u)∥H s , ∥Λ2s−1
ψ∥H s+1 ) ≤ C(R) ,
ϵ
and
∥(ϵ∂t )k (p, u)∥H s+1−k + ∥(ϵ∂t )k−1 ∂t θ∥H s+1−k
(4.70)
≤ C(∥(p, u)∥H s , ∥Λ2s−1
ψ∥H s+1 ) · (∥∇x (p, u)∥H s + ∥Λ2s−1
∇x ψ∥H s+1 )
ϵ
ϵ
≤ C(R)(1 + R′ ) ,
by the definitions of R, R′ in (4.34).
Given Lemma 4.8 and 4.9, we recall the following bound in [2] (Remark 6.21):
Lemma 4.10 ([2]). Let F ∈ C ∞ (R1 × R3 × R1 ) such that F (0) = 0. Let ψ =
(ϵp, ϵu, θ) be the solution to (1.7). Then
(4.71)
∥F (ψ)∥L∞ (0,T ;H s ) ≤ C(Ω0 ) +
√
√
T C(Ω) ≤ C(Ω0 )e(
T +ϵ)C(Ω)
.
We will use this bound in next section for estimates of the fast motion.
5. A priori estimate – fast motion. In this section we establish estimates
for p and the acoustic part of u. The proof is presented slightly differently from [2]
where the frequency space is divided into high and low regimes and estimates are
obtained for these regimes respectively. Here using the various bounds obtained for
the slow motion in Section 4, we can show Sobolev bounds of (∇x·u, ∇x p) by studying
estimates for (ϵ∂t )k (p, u) with 1 ≤ k ≤ s.
26
A Low Mach Number Limit of a Dispersive Navier-Stokes System
5.1. Linear Estimate for the Fast Motion. First we prove a linear estimate
for the fast motion. Recall system (4.1) for (p, u, θ)
(5.1)
((
)
)
(
) 2
1
2
3
(∂
+
u
·∇
)p+
∇
·
u−
h(ϵp
)κ(θ
)∇
θ
=
ϵh(ϵp
)
Σ
+
Σ̃
:
∇
u
+
∇
·
q̃
0
x
x
0
0
x
0
x 0
x
5 t
5
5
ϵ
2
− 5 h(ϵp0 )κ(θ0 )∇x p0 · ∇x θ + f1 ,
(
)
1
g(θ0 )(∂t + u0 ·∇x )u + ∇x p = h(ϵp0 ) ∇x · Σ + ∇x · Σ̃ + f2 ,
ϵ
((
)
)
3
2 (∂t
+ u0 ·∇x )θ + ∇x · u = ϵ2 h(ϵp0 ) Σ + Σ̃ : ∇x u0 + ∇x · q̃
+ h(ϵp0 )∇x · (κ(θ0 )∇x θ) + f3 ,
(p, u, θ)(x, 0) = (pin , uin , θin )(x) ,
where
(5.2)
q = κ(θ0 )∇x θ , h(ϵp0 ) = e−ϵp0 , g(θ0 ) = e−θ0 ,
)
(
Σ = µ(θ0 ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
(
)
(
)
Σ̃ = τ1 (ϵp0 , θ0 ) ∇x2 θ − 13 (∆x θ)I + τ2 (ϵp0 , θ0 ) ∇x θ0 ⊗ ∇x θ̂ − 13 (∇x θ0 · ∇x θ)I
(
)
+ ϵ2 τ3 (ϵp0 , θ0 ) (∇x u0 )(∇x u)T − (∇x u0 )T ∇x u ,
(
)
(
)
q̃ = τ4 (ϵp0 , θ0 ) ∆x u + 13 ∇x ∇x · u + τ6 (ϵp0 , θ0 )∇x θ0 · ∇x u − (∇x u)T ,
(
)
+ τ5 (ϵp0 , θ0 )∇x θ0 · ∇x u + (∇x u)T − 32 (∇x · u)I
The estimate for (p, u) states
Theorem 5.1. Let (p, u, θ) be a solution to the linear system (5.1). Then there
exists an increasing function C(·) such that
∫ T
(
)
sup ∥p̃∥2L2 + ∥ũ∥2Hϵ1 + α0
∥∇x ũ∥2Hϵ1 (τ ) dτ
[0,T ]
0
∫
)
≤ C(r0 )eT C(r) ∥p̃in ∥2L2 + ∥ũin ∥2Hϵ1 + C(r)
(
(∫
(5.3)
+
0
)2
T
∥(f1 (τ ), f2 (τ )∥Hϵ1 dτ
T
0
∥(ϵf2 (τ ), f3 (τ )∥2Hϵ1 dτ
+ T C(r) sup ∥∇x (ϵu, θ̃)∥2Hϵ1
∫
+ (∥F0 (ψ0 )∥L∞ + ∥G0 (ψ0 )∥L∞ )
0
[0,T ]
T
∥∆x θ̃∥2Hϵ1 (τ )dτ ,
where (p̃, ũ, θ̃) is defined in (5.5) and F0 (·), G0 (·) are defined in (5.32) and (5.34)
respectively.
Proof. Similarly as in the slow motion estimate, we combine leading order term
in the dispersive term ∇x · q̃ in the p-equation with the singular term. In that way,
the p-equation is transformed into
)
(
1
3
2
5 (∂t + u0 · ∇x )p + ϵ ∇x · T3 u − 5 h(ϵp0 )κ(θ0 )∇x θ
= ϵΨ2 u + ϵΨ1 u + ϵΨ2 θ + Ψ1 θ + f1 ,
where the symmetric positive operator T3 is
(8
)
(5.4)
T3 = I − ϵ2 ∇x · 15
h(ϵp0 )τ4 (ϵp0 , θ0 )∇x , I − ϵ2 ∇x · (g3 (ϵp0 , θ0 )∇x ) .
27
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Again we use Ψi to denote a homogeneous ith -order differential operator. Here
the coefficients of Ψ1 and Ψ2 depends on ∇xα1 (h, τ1 , τ2 , τ3 , τ4 , θ0 ) with |α1 | ≤ 2 and
∇xα2 (u0 , p0 ) with |α2 | ≤ 1.
As suggested by the singular term in the above p-equation, we consider the system
for
(
(
) )
(5.5)
(p̃, ũ, θ̃) = p, u − T3−1 52 h(ϵp0 )κ(θ0 )∇x θ , θ .
The equation for p̃ is
(5.6)
1
+ u0 · ∇x )p̃ + ∇x · (T3 ũ) = ϵΨ2 u + ϵΨ1 u + ϵΨ2 θ̃ + Ψ1 θ̃ + f1 ,
ϵ
3
5 (∂t
Next, we compute the equation for ũ. By (5.1) the equation for ∇x θ̃ is
(5.7)
=
3
2 (∂t + u0 · ∇x )∇x θ̃ + ∇x ∇x · u
ϵ2 Ψ3 u + ϵ2 Ψ2 u + ϵ2 Ψ1 u + 43 ϵ2 h(ϵp0 )τ4 (θ0 )∇x ∆x ∇x
·u
+ h(ϵp0 )κ(θ0 )∇x ∆x θ̃ + ϵ Ψ3 θ̃ + Ψ2 θ̃ + ∇x f3 .
2
Therefore, we have
(5.8)
3
2 (∂t
+ u0 · ∇x )∇x θ̃ + ∇x ∇x · (T2 u) = Ψ3 θ̃ + Ψ2 θ̃ + Ψ1 θ̃ + ϵ2 Ψ3 u
+ ϵ2 Ψ2 u + ϵ2 Ψ1 u + ∇x f3 .
Thus,
(5.9)
(∂t + u0 · ∇x )
(
2
5 h(ϵp0 )κ(θ0 )∇x θ̃
)
+
4
15 h(ϵp0 )κ(θ0 )∇x ∇x
= Ψ3 θ̃ + Ψ2 θ̃ + Ψ1 θ̃ + ϵ2 Ψ3 u + ϵ2 Ψ2 u + ϵ2 Ψ1 u +
· T2 ũ
4
15 h(ϵp0 )κ(θ0 )∇x f3
.
Now apply T3−1 to (5.9) to obtain the equation for T3−1 ( 25 h(ϵp0 )κ(θ0 )∇x θ̃).
)
(
(4
)
h(ϵp0 )κ(θ0 )∇x ∇x · T2 ũ
(∂t + u0 · ∇x )T3−1 52 h(ϵp0 )κ(θ0 )∇x θ̃ + T3−1 15
(5.10)
= T3−1 Ψ3 θ̃ + T3−1 Ψ2 θ̃ + T3−1 Ψ1 θ̃ + ϵ2 T3−1 Ψ3 u + ϵ2 T3−1 Ψ2 u
)
(
)
(4
h(ϵp0 )κ(θ0 )∇x f3 + [∂t , T3−1 ] 25 h(ϵp0 )κ(θ0 )∇x θ̃
+ T3−1 15
(
)
+ ϵ2 T3−1 Ψ1 u + [u0 · ∇x , T3−1 ] 25 h(ϵp0 )κ(θ0 )∇x θ̃ .
The last two commutator terms satisfy that
(
)
(
)
[∂t , T3−1 ] 25 h(ϵp0 )κ(θ0 )∇x θ̃ + [u0 · ∇x , T3−1 ] 25 h(ϵp0 )κ(θ0 )∇x θ̃
(
)
= T3−1 Ψ̃0 52 h(ϵp0 )κ(θ0 )∇x θ̃ ,
where Ψ̃0 is a zeroth order pseudo-differential operator which maps H s to H s for
all s ∈ R. The L2 -norm of this operator depends on ∥∂t (h, τ4 )∥W 1,∞ and ∥u0 ∥W 2,∞ .
Recall that we have the u-equation as
(
)
1
(5.11)
g(θ0 )(∂t + u0 · ∇x )u + ∇x p̃ = h(ϵp0 ) ∇x · Σ + ∇x · Σ̃ + f2 .
ϵ
28
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Subtracting g(θ0 )× (5.10) from the u-equation, we obtain the equation for ũ
(5.12)
(4
)
1
g(θ0 )(∂t + u0 · ∇x )ũ + ∇x p̃ − g(θ0 )T3−1 15
h(ϵp0 )κ(θ0 )∇x ∇x · T2 ũ
ϵ
= h(ϵp0 )µ(θ0 )(∆x + ∇x ∇x ·)ũ + ϵ2 Ψ2 u + Ψ1 u + ϵ2 g(θ0 )T3−1 (Ψ3 u + Ψ2 u + Ψ1 u)
+ g(θ0 )T3−1 (Ψ3 θ̃ + Ψ2 θ̃ + Ψ1 θ̃) + Ψ3 θ̃ + Ψ2 (T3−1 Ψ1 θ̃) + Ψ2 θ̃ + Ψ1 θ̃
(4
)
+ g(θ0 )T3−1 Ψ̃0 ( 52 h(ϵp0 )κ(θ0 )∇x θ̃) − g(θ0 )T3−1 15
h(ϵp0 )κ(θ0 )∇x f3 + f2 ,
The norms of Ψ1 , Ψ2 , Ψ3 and Ψ̃0 depend on ∥(h, τ4 , τ5 , τ6 )∥W 1,∞ , ∥∂t (h, τ4 )∥W 1,∞ , and
∥u0 ∥W 2,∞ . We summarize the system for (p̃, ũ) given by (5.6) and (5.12):
(5.13)
1
3
5 (∂t + u0 · ∇x )p̃ + ϵ ∇x · (T3 ũ) = ϵΨ2 u + ϵΨ1 u + ϵΨ2 θ̃ + ϵΨ1 θ̃ + f1 ,
(4
)
1
g(θ0 )(∂t + u0 · ∇x )ũ + ∇x p̃ − g(θ0 )T3−1 15
h(ϵp0 )κ(θ0 )∇x ∇x · T2 ũ
ϵ
= h(ϵp0 )µ(θ0 )(∆x + ∇x ∇x ·)ũ + ϵ2 Ψ2 u + Ψ1 u + ϵ2 g(θ0 )T3−1 (Ψ3 u + Ψ2 u + Ψ1 u)
+ g(θ0 )T3−1 (Ψ3 θ̃ + Ψ2 θ̃ + Ψ1 θ̃) + Ψ3 θ̃ + Ψ2 (T3−1 Ψ1 θ̃) + Ψ2 θ̃ + Ψ1 θ̃
(4
)
h(ϵp0 )κ(θ0 )∇x f3 + f2 .
+ g(θ0 )T3−1 Ψ̃0 ( 52 h(ϵp0 )κ(θ0 )∇x θ̃) − g(θ0 )T3−1 15
We still keep the notation u in some terms on the right-hand side and recall that ũ
and u are related by (5.5).
Multiplying the p̃-equation in (5.13) by p̃ and integrating the resulting equation
over R3 we have
⟨
⟩
1
2
3 d
10 dt ∥p̃∥L2 + p̃, ϵ ∇x · (T3 ũ) = − ⟨p̃, u0 · ∇x p̃⟩ + ⟨p̃, ϵΨ2 ũ⟩ + ⟨p̃, ϵΨ1 ũ⟩
(5.14)
⟨
⟩ ⟨
⟩
+ p̃, ϵΨ2 θ̃ + p̃, Ψ1 θ̃ + ⟨p̃, f1 ⟩ .
The estimates for the right-hand side of (5.14) are
|⟨p̃, u0 · ∇x p̃⟩| ≤ C(r)∥p̃∥2L2 ,
⟨
⟩
p̃, ϵ(Ψ2 + Ψ1 )(ũ, θ̃) ≤ C(r)(∥p̃∥2L2 + ∥(ũ, θ̃)∥2Hϵ1 ) + ∥∆x (ϵθ̃)∥2L2 +
⟨
⟩
(
)
p̃, Ψ1 θ̃ ≤ C(r) ∥p̃∥2L2 + ∥∇x θ̃∥2L2 .
Therefore we have for the p̃-equation
⟨
⟩
1
2
3 d
∥p̃∥
+
p̃,
∇
·(T
ũ)
x
3
L2
10 dt
ϵ
(5.15)
≤ C(r)(∥p̃∥2L2 + ∥(ũ, θ̃)∥2Hϵ1 ) + ∥∇x θ∥2Hϵ1 +
2
1
16 ∥∇x ũ∥Hϵ1
2
1
16 ∥∇x ũ∥Hϵ1
,
+ ∥p̃∥L2 ∥f1 ∥L2 .
To estimate the Hϵ1 -norm of ũ, multiply T3 ũ to the ũ-equation in (5.13) and integrate
over R3 . We bound each term similarly as in those estimates for the slow motion.
First,
(5.16)
d
√
√
√
√
⟨T3 ( g ũ), g ũ⟩ = 2⟨T3 ( g ũ), g ∂t ũ⟩ + R1
dt
√
√
= 2⟨T3 ũ, g ∂t ũ⟩ + 2⟨[T3 , g], g ∂t ũ⟩ + R1 ,
29
A Low Mach Number Limit of a Dispersive Navier-Stokes System
where
√
√
R1 = ⟨(I − ϵ2 ∇x · ((∂t g3 ) ∇x ))( g ũ), g ũ⟩
√
√
+ ⟨(I − ϵ2 ∇x · (g3 ∇x ))((∂t g)ũ), gũ⟩
√
√
+ ⟨(I − ϵ2 ∇x · (g3 ∇x ))( g ũ), (∂t g)ũ⟩ .
Therefore,
|R1 | ≤ C(r)∥ũ∥2Hϵ1 ,
(5.17)
with C(r) depends on ∥(∂t g3 , ∂t g)∥W 1,∞ . We estimate 2⟨[T3 ,
ũ-equation. First
√
[T3 , g]ũ = ϵ2 Ψ1 ũ + ϵ2 Ψ0 ũ ,
√
g],
√
g ∂t ũ⟩ using the
with the coefficients of Ψ1 , Ψ0 depending on ∥(g, g3 )∥W 2,∞ . We show here the esti√
√
mates for the leading order terms in 2⟨[T3 , g], g ∂t ũ⟩. The lower order terms obey
similar bounds. First we have
ϵ2 ⟨Ψ1 ũ,
1 −1
(θ0 )∇x p̃⟩
ϵg
= ϵ⟨Ψ1 ũ, ∇x p̃⟩ .
Hence,
(5.18)
2
ϵ ⟨Ψ1 ũ,
1
ϵ ∇x p̃⟩
≤ C(r)∥∇x (ϵp̃)∥2L2 +
2
1
16 α0 ∥∇x ũ∥L2
,
where C(r) depends on ∥(g, g3 , τ4 )∥W 2,∞ . Next,
2
(
)
ϵ ⟨Ψ1 ũ, g(θ0 )T −1 4 h(ϵp0 )κ(θ0 )∇x ∇x · T2 ũ ⟩
3
15
4
= ⟨ϵ2 ∇x ∇x · ( 15
h(ϵp0 )κ(θ0 )T3−1 Ψ1 ũ), T2 ũ⟩
(5.19)
≤ C(r) ∥Ψ1 ũ∥Hϵ1 ∥ũ∥Hϵ1 + ϵ C(r) ∥Ψ1 ũ∥Hϵ1 ∥∇x ũ∥Hϵ1
≤ C(r) ∥ũ∥2Hϵ1 +
2
1
16 α0 ∥∇x ũ∥Hϵ1
+ ϵ C(r)∥∇x ũ∥2Hϵ1 ,
and
(5.20)
2
ϵ ⟨Ψ1 ũ, Ψ2 ũ⟩ = ⟨Ψ1 (ϵũ), Ψ2 (ϵũ)⟩ ≤ C(r)∥ũ∥2 1 +
Hϵ
2
1
16 α0 ∥∇x ũ∥Hϵ1
.
Third,
2
ϵ ⟨Ψ1 ũ, Ψ3 θ̃⟩ = |ϵ⟨Ψ1 (ϵũ), Ψ3 θ̃⟩|
(5.21)
≤ C(r)∥(ϵũ, θ̃)∥2L2 + ϵC(r)∥∇x ũ∥2Hϵ1 + ϵC(r)∥∆x θ̃∥2L2
≤ C(r)∥(ϵũ, θ̃)∥2L2 +
2
1
16 α0 ∥∇x ũ∥Hϵ1
+ ϵC(r)∥∆x θ̃∥2L2 ,
by choosing ϵ small enough. Combining (5.18), (5.19), (5.20), and (5.21) we have
√
√
|⟨[T3 , g], g ∂t ũ]⟩| ≤ C(r)∥(p̃, ũ, θ̃)∥2Hϵ1 + 14 α0 ∥∇x ũ∥2Hϵ1 + ϵC(r)∥∆x θ̃∥2L2 ,
+ ⟨∇x · ( 25 h(ϵp0 )κ(θ0 )T3−1 (Ψ1 (ϵũ) + Ψ0 (ϵũ))), ϵf3 ⟩
+ ⟨Ψ1 (ϵũ) + Ψ0 (ϵũ), ϵf2 ⟩ .
(5.22)
By (5.16),
(5.23)
⟨T3 ũ, g ∂t ũ⟩ =
1
2
d
√
√
√
√
⟨T3 ( g ũ), g ũ⟩ − ⟨T3 ( gũ), g ∂t ũ⟩ − 12 R1 ,
dt
30
A Low Mach Number Limit of a Dispersive Navier-Stokes System
√
√
where ⟨T3 ( g ũ), g ∂t ũ⟩ satisfies (5.22) and R1 satisfies (5.17).
The rest of the terms are estimated similarly. First by integration by parts the
convection term has the bound
|⟨T3 ũ, g(θ0 )u0 · ∇x ũ⟩| ≤ C(r)∥ũ∥2Hϵ1 ,
(5.24)
where C(r) depends on ∥(g, g3 , u0 )∥W 1,∞ . The pressure term has the form
⟨T3 ũ,
(5.25)
1
1
∇x p̃⟩ = ⟨T3 ũ, ∇x p̃⟩ .
ϵ
ϵ
The third term on the left-hand side of the ũ-equation in (5.13) gives
(4
)
− ⟨T3 ũ, g(θ0 )T3−1 15
h(ϵp0 )κ(θ0 )∇x ∇x · T2 ũ ⟩
(5.26)
1
α0 ∥∇x ũ∥2Hϵ1 − C(r)∥ũ∥2L2 .
≥ 2α0 ∥∇x · ũ∥2Hϵ1 − 16
The dissipative term satisfies
−⟨T3 ũ, h(ϵp0 )µ(θ0 )(∆x + ∇x ∇x ·)ũ⟩ ≥ 2α0 ∥∇x ũ∥2Hϵ1 − C(r)∥ũ∥2L2 .
(5.27)
The rest of the terms related to u have the bounds
(5.28)
|⟨T3 ũ, ϵ2 Ψ2 u⟩| ≤ C(r)∥(ϵu, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu, θ̃)∥2Hϵ1 .
|⟨T3 ũ, Ψ1 u⟩| ≤ C(r) ∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x θ̃∥2Hϵ1 +
|⟨T3 ũ, ϵ
2
g(θ0 )T3−1 (Ψ3 u
+ Ψ2 u + Ψ1 u)⟩| ≤
C(r) ∥(ũ, θ̃)∥2Hϵ1
+
2
1
16 ∥∇x ũ∥Hϵ1
2
1
16 ∥∇x ũ∥Hϵ1
,
+ C(r) ∥∇x (ϵu, θ̃)∥2Hϵ1
.
Terms involving θ̃ are bounded as
|⟨T3 ũ, (Ψ2 θ̃ + Ψ1 θ̃)⟩| ≤ C(r)∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu, θ̃)∥2Hϵ1
2
1
16 α0 ∥∇x u∥Hϵ1 ,
C(r)∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu,
+
(5.29)
|⟨T3 ũ, g(θ0 )T3−1 (Ψ2 θ̃ + Ψ1 θ̃)⟩| ≤
θ̃)∥2Hϵ1
1
+ 16
α0 ∥∇x u∥2Hϵ1 .
We need to be more specific with the bounds for terms involving Ψ3 θ̃. The reason
will be clear when we do the estimates for high-order derivatives for (p, u). There are
two terms that have third-order derivatives in θ̃. The term g(θ0 )T3−1 Ψ3 θ̃ comes from
(5.10). It has the form
g(θ0 )T3−1 Ψ3 θ̃ = g(θ0 )T3−1 (h(ϵp0 )κ(θ0 )∆x ∇x θ̃) .
Therefore,
|⟨T3 ũ, g(θ0 )T3−1 Ψ3 θ̃⟩| = |⟨T3 ũ, g(θ0 )T3−1 (h(ϵp0 )κ(θ0 )∆x ∇x θ̃)⟩|
(5.30)
= |⟨T3−1 (g(θ0 )T3 ũ), h(ϵp0 )κ(θ0 )∆x ∇x θ̃⟩|
≤ |⟨g(θ0 )ũ, h(ϵp0 )κ(θ0 )∆x ∇x θ̃⟩|
+ |⟨T3−1 [g(θ0 ), T3 ]ũ), h(ϵp0 )κ(θ0 )∆x ∇x θ̃⟩| .
31
A Low Mach Number Limit of a Dispersive Navier-Stokes System
The first term on the right-hand side of (5.30) has the bound
|⟨g(θ0 )ũ, h(ϵp0 )κ(θ0 )∆x ∇x θ̃⟩|
≤ C(r)∥ũ∥L2 + C(r)∥∇x θ̃∥2L2 + ∥F0 (ψ0 )∥L∞ ∥∆x θ̃∥2L2 +
where F0 (ψ0 ) =
satisfies
8
α0 g(θ0 )h(ϵp0 )κ(θ0 ).
2
1
16 α0 ∥∇x ũ∥L2
,
The second term on the right-hand side of (5.30)
|⟨T3−1 [g(θ0 ), T3 ]ũ), h(ϵp0 )κ(θ0 )∆x ∇x θ̃⟩| ≤ C(r)∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu, θ̃)∥2Hϵ1
+
2
1
16 α∥∇x ũ∥Hϵ1
.
Therefore we have
|⟨T3 ũ, g(θ0 )T3−1 Ψ3 θ̃⟩|
≤ C(r)∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu, θ̃)∥2Hϵ1 + 18 α∥∇x ũ∥2Hϵ1
(5.31)
+ ∥F0 (ψ0 )∥L∞ ∥∆x θ̃∥2L2 ,
with
F0 (ψ0 ) =
(5.32)
8
α0 g(θ0 )h(ϵp0 )κ(θ0 )
The other term that has the third-order derivative of θ̃ is from the dispersive term in
(5.11). It has the form
Ψ3 θ̃ = h(ϵp0 )τ1 (ϵp0 , θ0 )∆∇x θ̃ .
Following similar estimates as we have done for (5.31), we have
(5.33)
|⟨T3 ũ, Ψ3 θ̃⟩| ≤ C(r)∥(ũ, θ̃)∥2Hϵ1 + C(r)∥∇x (ϵu, θ̃)∥2Hϵ1 + 81 α∥∇x ũ∥2Hϵ1
+ ∥G0 (ψ0 )∥L∞ ∥∆x θ̃∥2L2 ,
where
G0 (ψ0 ) =
(5.34)
4
α0 (1
+ g3 (ϵp0 , θ0 ))h(ϵp0 )τ1 (ϵp0 , θ0 ) ,
and g3 is defined in (5.4).
Finally the forcing terms satisfy
(
)
⟨T3 ũ, g(θ0 )T3−1 25 h(ϵp0 )κ(θ0 )∇x f3 ⟩ + ⟨T3 ũ, f2 ⟩
(5.35)
= −⟨ 52 h(ϵp0 )κ(θ0 )∇x · (T3−1 (g(θ0 )T3 ũ)), f3 ⟩ + ⟨T3 ũ, f2 ⟩ .
Combining (5.23) through (5.35), the energy estimate for ũ is
1
2
⟩
√
d⟨ √
1
ũ, g(θ0 ) T3 ( g(θ0 )ũ) + α0 ∥∇x ũ∥2Hϵ1 + ⟨T3 ũ, ∇x p̃⟩
dt (
ϵ
)
≤ C(r) ∥p̃∥2L2 + ∥(ũ, θ̃)∥2Hϵ1 + C(r) ∥∇x (ϵu, θ̃)∥2Hϵ1
(5.36)
+ (∥F0 (ψ0 )∥L∞ + ∥G0 (ψ0 )∥L∞ ) ∥∆x θ̃∥2Hϵ1 + ⟨Ψ1 (ϵũ) + Ψ0 (ϵũ), ϵf2 ⟩
+ ⟨T3 ũ, f2 ⟩ + ⟨ 52 h(ϵp0 )κ(θ0 )∇x · (T3−1 (g(θ0 )T3 ũ)), f3 ⟩
+ ⟨∇x · ( 52 h(ϵp0 )κ(θ0 )T3−1 (Ψ1 (ϵũ) + Ψ0 (ϵũ)), ϵf3 ⟩ .
32
A Low Mach Number Limit of a Dispersive Navier-Stokes System
By (5.15) and (5.36) the energy estimate for (p̃, ũ) is
⟨ √
⟩)
√
d (3
2
∥p̃∥
ũ,
g(θ
)
T
(
g(θ
)ũ)
+ 32 α0 ∥∇x ũ∥2Hϵ1
2 +
0
3
0
L
dt 5 (
)
≤ C(r) ∥p̃∥2L2 + ∥(ũ, θ̃)∥2Hϵ1 + (∥F0 (ψ0 )∥L∞ + ∥G0 (ψ0 )∥L∞ ) ∥∆x θ̃∥2Hϵ1
+ C(r) ∥∇x (ϵu, θ̃)∥2Hϵ1 + ⟨Ψ1 (ϵũ) + Ψ0 (ϵũ), ϵf2 ⟩ + ⟨T3 ũ, f2 ⟩ + ⟨p̃, f1 ⟩
(5.37)
+ ⟨∇x · ( 52 h(ϵp0 )κ(θ0 )T3−1 (Ψ1 (ϵũ) + Ψ0 (ϵũ)), ϵf3 ⟩
+ ⟨ 52 h(ϵp0 )κ(θ0 )∇x · (T3−1 (g(θ0 )T3 ũ)), f3 ⟩
(
)
≤ C(r) ∥p̃∥2L2 + ∥(ũ, θ̃)∥2Hϵ1 + C(r) ∥∇x (ϵu, θ)∥2Hϵ1
+ (∥F0 (ψ0 )∥L∞ + ∥G0 (ψ0 )∥L∞ ) ∥∆x θ̃∥2Hϵ1 + ∥ϵf2 ∥2L2
+ ∥ũ∥Hϵ1 ∥f2 ∥Hϵ1 + ∥p̃∥L2 ∥f1 ∥L2 + C(r)∥f3 ∥2Hϵ1 + 12 α0 ∥∇x ũ∥2Hϵ1 .
By Gronwall’s inequality and the equivalence of norms
⟨ √
⟩
√
ũ, g(θ0 )T3 ( g(θ0 )ũ) ∼ ∥ũ∥2Hϵ1 ,
we have
∫
(
)
sup ∥p̃∥2L2 + ∥ũ∥2Hϵ1 + α0
[0,T ]
T
0
∥∇x ũ∥2Hϵ1 (τ ) dτ
∫
(
)
≤ C(r0 )eT C(r) ∥p̃in ∥2L2 + ∥ũin ∥2Hϵ1 + C(r)
(∫
+
0
)2
T
∥(f1 (τ ), f2 (τ )∥Hϵ1 dτ
T
0
∥(ϵf2 (τ ), f3 (τ )∥2Hϵ1 dτ
+ T C(r) sup ∥∇x (ϵu, θ̃)∥2Hϵ1
∫
+ (∥F0 (ψ0 )∥L∞ + ∥G0 (ψ0 )∥L∞ )
0
[0,T ]
T
∥∆x θ̃∥2Hϵ1 (τ )dτ .
5.2. A Priori Estimate for the Fast Motion. In this subsection we show
a priori estimates for the fast motion (u, p). First we apply Theorem 5.1 to obtain
bounds for (ϵ∂t )γ (p, u), which then lead us to the bounds for the pressure and the
acoustic part of the velocity field. The estimates for (ϵ∂t )γ (p, u) with 1 ≤ γ ≤ s are
stated in the following theorem:
Theorem 5.2. Let (p, u, θ) be a solution to (5.1). For s > 5, define
(pγ , uγ , θγ ) := (ϵ∂t )γ (p, u, θ) ,
(5.38)
where 1 ≤ γ ≤ s . Then there exists an increasing function C(·) such that
∫ T
)
(
√
∥∇x (pγ , uγ )∥2L2 dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) .
(5.39) sup ∥pγ ∥2L2 + ∥uγ ∥2Hϵ1 + α0
[0,T ]
0
Before proving Theorem 5.2, we show how to use it to obtain the bounds for ∇x · u
and ∇x p. To this end, we directly solve from equation (5.1) to get
∇x · u = − 35 (ϵ∂t )p − 53 (ϵu) · ∇x p + 25 e−ϵp ∇x · (κ(θ)∇x θ) + 25 ϵ e−ϵp Σ : ∇x (ϵu)
(5.40)
+ 52 ϵ e−ϵp Σ̃ : ∇x (ϵu) + 25 ϵ2 e−ϵp ∇x · q̃ ,
(
)
∇x p = −e−θ (ϵ∂t )u − e−θ (ϵu) · ∇x u + ϵe−ϵp ∇x · Σ + ∇x · Σ̃ .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
33
First we use induction to show that for every 0 ≤ k ≤ s,
(
)
√
k α
2
k α′
2
sup
∥(ϵ∂
)
∇
p∥
+
∥(ϵ∂
)
∇
∇
·
u∥
≤ C(Ω0 )e( T +ϵ)C(Ω) ,
2
2
t
t
x
x
x
(5.41)
L
L
[0,T ]
∫
T
′
∥∇x (ϵ∂t )k ∇xα ∇x · u∥2L2 (τ ) dτ ≤ C(Ω0 )e(
(5.42)
√
T +ϵ)C(Ω)
.
0
∫
T
∥∇x (ϵ∂t )k ∇xα p∥2L2 (τ ) dτ ≤ C(Ω0 )e(
(5.43)
√
T +ϵ)C(Ω)
.
0
where |α| = s − k, and |α′ | = s − k − 1. Notice that Theorem 5.2 shows (5.41) holds
for k = s. Now suppose (5.41) holds for k + 1 and we show that it also holds for k.
Apply (ϵ∂t )k ∇xβ with |β| = s − k − 1 to (5.40). The resulting equations are
(5.44)
(ϵ∂t )k ∇xβ ∇x · u = − 53 (ϵ∂t )k+1 ∇xβ p − 35 (ϵ∂t )k ∇xβ ((ϵu) · ∇x p)
(
)
(
)
+ 52 (ϵ∂t )k ∇xβ e−ϵp ∇x · (κ(θ)∇x θ) + 25 (ϵ∂t )k ∇xβ ϵ e−ϵp Σ : ∇x (ϵu)
(
)
(
)
+ 25 (ϵ∂t )k ∇xβ ϵ e−ϵp Σ̃ : ∇x (ϵu) + 25 (ϵ∂t )k ∇xβ ϵ2 e−ϵp ∇x · q̃ ,
(5.45)
(
)
(
)
(ϵ∂t )k ∇xβ ∇x p = −(ϵ∂t )k ∇xβ e−θ (ϵ∂t )u − (ϵ∂t )k ∇xβ e−θ (ϵu) · ∇x u
(
(
))
+ (ϵ∂t )k ∇xβ ϵe−ϵp ∇x · Σ + ∇x · Σ̃
.
We estimate each term on the right-hand side (RHS) of (5.44). The estimates for p
is similar. First, by the induction assumption, the first term on the RHS of (5.44)
satisfies
sup ∥(ϵ∂t )k+1 ∇xβ ∇x p∥L2 ≤ C(Ω0 )e(
(5.46)
√
T +ϵ)C(Ω)
.
[0,T ]
By Lemma 4.9, the second term on the RHS of (5.44) satisfies
sup ∥(ϵ∂t )k ∇xβ ((ϵu) · ∇x p) ∥L2 ≤ ϵC(Ω) .
(5.47)
[0,T ]
The rest of the terms on the RHS of (5.44) are all in terms of ψ = (ϵp, ϵu, θ) and
their leading orders are at most (ϵ∂t )k (ϵ∇x )∇xβ ∇x2 with |β| = s −√k − 1. Therefore by
Lemma 4.8, the L2 -norm of these terms are bounded by C(Ω0 )e( T +ϵ)C(Ω) . Together
with (5.46) and (5.47), we have the desired bound for u in (5.41). The bounds (5.41)
for p and (5.42), (5.43) are all shown in a similar way.
Combining (5.41), (5.42), and (5.43) we have the following theorem for p and the
acoustic part of u (given Theorem 5.2):
Theorem 5.3. Let (p, u, θ) be a solution to the fluctuation equations (1.7). Then
sup
[0,T ]
(
∥p∥2H s
√
≤ C(Ω0 )e(
+ ∥∇x ·
u∥2H s−1
)
∫
T
(∥∇x ∇x · u∥2H s−1 + ∥∇x p∥2H s )(τ ) dτ
+
0
T +ϵ)C(Ω)
.
34
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Now we proceed to prove Theorem 5.2.
Proof. [Proof of Theorem 5.2] Apply the operator (ϵ∂t )γ to the system (5.1).
Recall the notation (pγ , uγ , θγ ) = ((ϵ∂t )γ p, (ϵ∂t )γ u, (ϵ∂t )γ θ). Then (pγ , uγ , θγ ) satisfies
(5.48)
((
)
)
(
) 2
3
1
2
(∂
+
u·∇
)p
+
∇
·
u
−
h(ϵp)κ(θ)
∇
θ
=
ϵh(ϵp)
Σ
+
Σ̃
:
∇
u
+
∇
·
q̃
t
x
γ
x
γ
x
γ
γ
γ
x
x
γ
5
ϵ
5
5
+ 25 h(ϵp)κ(θ)∇x p · ∇x θγ + g1 ,
(
)
g(θ)(∂t + u·∇x ) uγ + 1ϵ ∇x pγ = h(ϵp) ∇x · Σγ + ∇x · Σ̃γ + g2 ,
((
)
)
2
3
(∂
θ
+
u·∇
θ
)
+
∇
·
u
=
ϵ
h(ϵp)
Σ
+
Σ̃
:
∇
u+∇
·
q̃
t
γ
x
γ
x
γ
γ
γ
x
x
γ
2
+ h(ϵp)∇x · (κ(θ)∇x θγ ) + g3 ,
where
(
)
Σγ = µ(θ) ∇x uγ + (∇x uγ )T − 23 (∇x · uγ )I ,
(
)
(
)
Σ̃γ = τ1 (ϵp, θ) ∇x2 θγ − 13 (∆x θγ )I + τ2 (ϵp, θ) ∇x θ ⊗ ∇x θγ − 13 ∇x θ · ∇x θγ I ,
(
)
+ τ3 (ϵp, θ) (∇x (ϵu))(∇x (ϵuγ ))T − (∇x (ϵu))T ∇x (ϵuγ ) ,
and
(
)
(
)
q̃γ = τ4 (ϵp, θ) ∆x uγ + 13 ∇x ∇x · uγ + τ6 (ϵp, θ) ∇x uγ − (∇x uγ )T · ∇x θ
(
)
+ τ5 (ϵp, θ)∇x θ · ∇x uγ + (∇x uγ )T − 23 (∇x · uγ )I , .
The commutator terms are
[
]
[
]
1
g1 = 35 u, (ϵ∂t )γ · ∇x p − ∇x · 25 h(ϵp)κ(θ), (ϵ∂t )γ ∇x θ
ϵ
+ ϵ [Ψ2 + Ψ1 , (ϵ∂t )γ ] u + ϵ [Ψ2 + Ψ1 , (ϵ∂t )γ ] θ
]
[
− ϵ 52 h(ϵp)τ4 (θ), (ϵ∂t )γ ∆x ∇x · u ,
(5.49)
g2 = [g(θ), (ϵ∂t )γ ]∂t u + [g(θ)u, (ϵ∂t )γ ] · ∇x u + [Ψ2 + Ψ1 , (ϵ∂t )γ ]u
− [h(ϵp)τ1 (θ), (ϵ∂t )γ ]∆x ∇x θ + [Ψ2 + Ψ1 , (ϵ∂t )γ ]θ ,
g3 = [u, (ϵ∂t )γ ]∇x θ + [Ψ2 + Ψ1 , (ϵ∂t )γ ] θ
− ϵ2 [h(ϵp)τ4 (θ), (ϵ∂t )γ ] ∆x ∇x · u + ϵ2 [Ψ2 + Ψ1 , (ϵ∂t )γ ] u .
Here Ψ2 and Ψ1 are homogeneous differential operators of order 2 and 1 respectively.
Terms involving Ψ2 in g1 , g2 come from the dissipative and dispersive terms and their
structures are
(5.50)
[δ(ψ)∂xk ψ, (ϵ∂t )γ ] ∂xi xj ψ
[δ(ψ), (ϵ∂t )γ ] ∂xi xj u ,
or
where ψ = (ϵp, ϵu, θ) and δ is a smooth function in ψ. Typical terms in (5.50) are
(
)(
)(
)
Ck1 ,k2 ,k3 (ϵ∂t )k1 δ(ψ) (ϵ∂t )k2 ∂xk ψ (ϵ∂t )k3 ∂xi xj ψ ,
and
(
)(
)
Ck4 ,k5 (ϵ∂t )k4 δ(ψ) (ϵ∂t )k5 ∂xi xj u ,
where Ck1 ,k2 ,k3 and Ck4 ,k5 are generic constants with
k1 + k2 + k3 = |γ| = k ,
k4 + k5 = k ,
k1 + k2 ≥ 1 ,
k4 ≥ 1 .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
35
To estimate the Hϵ1 -bounds of terms in (5.50), we only need to consider the leadingorders. These terms are
(
)
δ(ψ) ((ϵ∂t )γ ∂xk ψ) ∂xi xj ψ , δ(ψ) ((ϵ∂t )∂xk ψ) (ϵ∂t )γ−1 ∂xi xj ψ ,
and
(
)
δ(ψ) (ϵ∂t )γ−1 ∂xi xj u .
By Lemma 4.8 and 4.9,
∥(ϵ∂t )γ ∂xk ψ∥Hϵ1 + ∥(ϵ∂t )γ−1 ∂xi xj ψ∥Hϵ1 ≤ C(R) ,
∥(ϵ∂t )γ ∂xi xj u∥Hϵ1 ≤ C(R)(1 + R′ ) .
Therefore,
[δ(ψ)∂x ψ, (ϵ∂t )γ ] ∂xi xj ψ 1 ≤ C(R)(1 + R′ ) .
k
H
(5.51)
ϵ
By the linear estimate (5.3), we need the bounds of ∥(g1 , g2 , g3 )∥Hϵ1 . Again by (4.64)
we only need to check the leading-orders in each term of g1 , g2 , g3 . The leading-order
in the first term of g1 has the form
(
)
γ
γ−1
3
3
∇x p .
5 ((ϵ∂t ) u) · ∇x p + 5 ((ϵ∂t )u) · (ϵ∂t )
By Lemma 4.8 and 4.9,
∥ 53 ((ϵ∂t )γ u) · ∇x p +
3
5
(
)
((ϵ∂t )u) · (ϵ∂t )γ−1 ∇x p ∥Hϵ1 ≤ C(R)(1 + R′ ) .
Therefore,
∥
(5.52)
[3
5 u,
]
(ϵ∂t )γ · ∇x p∥Hϵ1 ≤ C(R)(1 + R′ ) .
The leading-order in the second term of g1 is
− 25 (∂t (h(ϵp)κ(θ))) ((ϵ∂t )γ−1 ∆θ) −
2
5
(
)
(ϵ∂t )γ−1 ∂t ∇x (h(ϵp)κ(θ)) · ∇x θ .
By Lemma 4.8 and 4.9,
∥−
2
5
∥ − 2 (∂t (h(ϵp)κ(θ))) (ϵ∂t )γ−1 ∆θ∥Hϵ1 ≤ C(R) ,
( 5 γ−1
)
(ϵ∂t )
∂t ∇x (h(ϵp)κ(θ)) · ∇x θ∥Hϵ1 ≤ C(R)(1 + R′ ) .
Therefore,
(5.53)
∥ − 1ϵ ∇x ·
[2
5 h(ϵp)κ(θ),
]
(ϵ∂t )γ ∇x θ∥Hϵ1 ≤ C(R)(1 + R′ ) .
The terms involving Ψ2 are bounded as in (5.51). The leading-order in the last term
in g1 is
(
)
−ϵ 25 ((ϵ∂t )γ (h(ϵp)τ4 (θ))) (∆x ∇x · u) − ϵ 25 (∂t (h(ϵp)τ4 (θ))) (ϵ∂t )γ−1 (∆x ∇x · u) ,
where by Lemma 4.9,
∥ − ϵ 25 (ϵ∂t )γ (h(ϵp)τ4 (θ)) (∆x ∇x · u) ∥Hϵ1 ≤ C(R) ,
36
A Low Mach Number Limit of a Dispersive Navier-Stokes System
and
∥ − ϵ 25 (∂t (h(ϵp)τ4 (θ))) (ϵ∂t )γ−1 (∆x ∇x · u) ∥Hϵ1 ≤ C(R)(1 + R′ ) .
This gives
(5.54)
∥−ϵ
[2
5 h(ϵp)τ4 (θ),
]
(ϵ∂t )γ ∆x ∇x · u∥Hϵ1 ≤ C(R)(1 + R′ ) .
Terms involving Ψ1 in g1 are commutators from Σ, Σ̃, and q̃. They have the form
[δ(ψ)∂xk u, (ϵ∂t )γ ] ∂xi ψ ,
or
[δ(ψ)∂xk p, (ϵ∂t )γ ] ∂xi ψ ,
or
[
]
δ(ψ)∂xk ψ ∂xj ψ, (ϵ∂t )γ ∂xi ψ .
Thus the leading-orders in Ψ1 -terms are
(
)
δ(ψ)∂xk ψ∂xj ψ ((ϵ∂t )γ ∂xi ψ) and
(δ(ψ)∂xi ψ) ((ϵ∂t )γ ∂xk (u, p)) .
By Lemma 4.8 and 4.9,
(
)
∥ δ(ψ)∂xk ψ∂xj ψ ((ϵ∂t )γ ∂xi ψ) ∥Hϵ1 ≤ C(R) ,
∥ (δ(ψ)∂xi ψ) ((ϵ∂t )γ ∂xk (u, p)) ∥Hϵ1 ≤ C(R)(1 + R′ ) .
Therefore,
∥[Ψ1 , (ϵ∂t )γ ]ψ∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.55)
Combining (5.52), (5.53), (5.51), (5.55), and (5.54), we obtain that
∥g1 ∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.56)
The bound of ∥g3 ∥Hϵ1 can be obtained by considering the equation for ϵp − θ:
3
5 ϵg1
(5.57)
− 23 g3 = [u, (ϵ∂t )γ ] · ∇x (ϵp − θ) .
The leading-orders in this term are
((ϵ∂t )γ u) · ∇x (ϵp − θ)
and
(
)
((ϵ∂t )u) · (ϵ∂t )γ−1 ∇x (ϵp − θ) .
By Lemma 4.8 and 4.9,
(
)
∥ ((ϵ∂t )γ u) · ∇x (ϵp − θ)∥Hϵ1 + ∥((ϵ∂t )u) · (ϵ∂t )γ−1 ∇x (ϵp − θ) ∥Hϵ1 ≤ C(R) .
By (5.57) and (5.56),
(5.58)
∥g3 ∥Hϵ1 ≤ C(R)(1 + ϵR′ ) .
Now we check the leading order terms in g2 . The leading-orders in the first term of
g2 are
(
)
(ϵ∂t )γ−1 ∂t g(θ) (∂t (ϵu)) and (∂t g(θ)) (ϵ∂t )γ−1 ∂t (ϵu) .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
37
By Lemma 4.8 and 4.9,
(
)
∥ (ϵ∂t )γ−1 ∂t (g(θ)) (∂t (ϵu))∥Hϵ1 ≤ C(R) ,
∥(∂t g(θ)) ((ϵ∂t )γ u) ∥Hϵ1 ≤ C(R) .
Thus,
∥[g(θ), (ϵ∂t )γ ]∂t u∥Hϵ1 ≤ C(R) .
(5.59)
Next, the second term of g2 has a similar structure as the first term in g1 . Therefore,
∥[g(θ)u, (ϵ∂t )γ ] · ∇x u∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.60)
The Ψ1 terms in g2 are from Σ and Σ̃. They have similar structures as the Ψ1 terms
in g1 . Thus,
∥[Ψ1 , (ϵ∂t )γ ]θ∥Hϵ1 + ∥[Ψ1 , (ϵ∂t )γ ]u∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.61)
The Ψ2 terms in g2 are bounded by (5.51). The leading-orders in the second last term
in g2 are
− ((ϵ∂t )γ (h(ϵp)τ1 (θ))) (∆x ∇x θ) and
− ((ϵ∂t )(h(ϵp)τ1 (θ))) (ϵ∂t )γ−1 (∆x ∇x θ) .
By Lemma 4.8,
∥ − ((ϵ∂t )γ (h(ϵp)τ1 (θ))) (∆x ∇x θ)∥Hϵ1 ≤ C(R) ,
∥ − ((ϵ∂t )(h(ϵp)τ1 (θ))) (ϵ∂t )γ−1 (∆x ∇x θ)∥Hϵ1 ≤ C(R)(1 + R′ ) .
Thus,
∥ − [h(ϵp)τ1 (θ), (ϵ∂t )γ ]∆x ∇x θ∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.62)
Combining (5.59), (5.60), (5.61), (5.51), and (5.62), we have
∥g2 ∥Hϵ1 ≤ C(R)(1 + R′ ) .
(5.63)
By the linear estimate (5.3), we have
(
sup
[0,T ]
∥pγ ∥2L2
≤ C(r0 )e
T C(r)
(∫
+
(
∥ũγ ∥2Hϵ1
2
∥pin
γ ∥L2
∫
)
+
0
∥∇x ũγ ∥2Hϵ1 (τ ) dτ
0
2
∥ũin
γ ∥Hϵ1
∥(g1 (τ ), g2 (τ )∥Hϵ1 dτ
∫
)
T
+ C(r)
0
)2
T
+
T
+ α0
∥(ϵg2 (τ ), g3 (τ )∥2Hϵ1 dτ
+ T C(r) sup ∥∇x (ϵuγ , θγ )∥2Hϵ1
∫
+ (∥F0 (ψ)∥L∞ + ∥G0 (ψ)∥L∞ )
0
[0,T ]
T
∥∆x θγ ∥2Hϵ1 (τ )dτ ,
where
ψ = (ϵp, ϵu, θ) ,
ũγ = uγ − T3−1 ( 52 h(ϵp)κ(θ)∇x θγ ) .
38
A Low Mach Number Limit of a Dispersive Navier-Stokes System
By the estimates for (g1 , g2 , g3 ) and (F0 (ψ), G0 (ψ)), we then have
∫
(
)
sup ∥pγ ∥2L2 + ∥uγ ∥2Hϵ1 + α0
[0,T ]
≤ C(Ω0 )e
T C(r)
(
+ T C(r) sup
[0,T ]
2
∥pin
γ ∥L2
+
T
0
2
∥ũin
γ ∥Hϵ1
∥∇x (ϵuγ , θγ )∥2Hϵ1
∥∇x uγ ∥2Hϵ1 (τ ) dτ
)
+ (T + ϵ2 )C(Ω)
√
( T +ϵ)C(r)
∫
T
+ C(Ω0 )e
+ sup ∥T3−1 ( 52 h(ϵp)κ(θ)∇x θγ )∥2Hϵ1 + α0
0
∫
[0,T ]
∥∆x θγ ∥2Hϵ1 (τ )dτ
T
0
∥∇x T3−1 ( 25 h(ϵp)κ(θ)∇x θγ )∥2Hϵ1 (τ ) dτ .
Estimates of terms involving (ϵuγ , θγ ) in the last four terms in the above inequality
are as follows: by Lemma 4.8 we have
T C(r) sup ∥∇x (ϵuγ , θγ )∥2Hϵ1 ≤ T C(Ω) ,
[0,T ]
∫
T
0
√
∥∆x θγ ∥2Hϵ1 (τ ) dτ + sup ∥T3−1 ( 25 h(ϵp)κ(θ)∇x θγ )∥Hϵ1 ≤ C(Ω0 )e(
T +ϵ)C(Ω)
,
[0,T ]
and
∫
T
0
∥∇x T3−1 ( 25 h(ϵp)κ(θ)∇x θγ )∥2Hϵ1 (τ ) dτ
√
≤ C(Ω0 )e(
T +ϵ)C(Ω)
.
Therefore,
(5.64)
∫
(
)
sup ∥pγ ∥2L2 + ∥uγ ∥2Hϵ1 + α0
[0,T ]
T
∥∇x uγ ∥2L2 dτ ≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
.
0
∫T
To conclude, we need the bound for 0 ∥∇x pγ ∥2L2 dτ . To this end, we multiply the
uγ -equation in (5.48) by ϵ∇x pγ and integrate over R3 × [0, T ]. The resulting equation
is
∫ T
∫ T∫
2
∥∇x pγ ∥L2 (τ ) dτ = −
g(θ) ∂t ((ϵ∂t )γ (ϵu)) · ∇x pγ (x, τ ) dx dτ
0
0
R3
(5.65)
∫ T∫
+
0
where
(5.66)
R3
Γ · (∇x pγ )(x, τ ) dx dτ ,
(
)
Γ = −g(θ)(ϵu)·∇x uγ + ϵh(ϵp) ∇x · Σγ + ∇x · Σ̃γ + ϵg2 .
Note that by (5.64), the first term in Γ satisfies
√
∥ − g(θ)(ϵu)·∇x uγ ∥L2 (R3 ×[0,T ]) ≤ ϵC(Ω) · C(Ω0 )e(
T +ϵ)C(Ω)
√
≤ C(Ω0 )e(
By Lemma 4.8, the second term in Γ satisfies
(
)
√
∥ϵh(ϵp) ∇x · Σγ + ∇x · Σ̃γ ∥L2 (R3 ×[0,T ]) ≤ C(Ω0 )e( T +ϵ)C(Ω) .
T +ϵ)C(Ω)
.
A Low Mach Number Limit of a Dispersive Navier-Stokes System
39
By the estimate for g2 in (5.63), the last term in Γ satisfies
(∫
∥ϵg2 ∥L2 (R3 ×[0,T ]) ≤ ϵ
)1/2
T
′
≤ ϵC(Ω) ,
2
(C(R)(1 + R )) (τ ) dτ
0
by the definitions of R, R′ , Ω in (4.34) and (4.37). Combining these bounds we have
∫ ∫
T
Γ · (∇x pγ )(x, τ ) dx dτ ≤ ∥Γ∥L2 (R3 ×[0,T ]) ∥∇x pγ ∥L2 (R3 ×[0,T ])
3
0
R
(5.67)
√
T +ϵ)C(Ω)
≤ C(Ω0 )e(
+ 41 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) .
To estimate the first term on the right-hand side of (5.65), we integrate by parts in
both t and x. Then
(5.68)
∫ T∫
g(θ) ∂t ((ϵ∂t )γ (ϵu)) · ∇x pγ (x, τ ) dx dτ
0
R3
∫
∫
= (g(θ)uγ ) · ((ϵ∂t )γ (∇x (ϵp)))(x, T ) dx − (g(θ)uγ ) · ((ϵ∂t )γ (∇x (ϵp)))(x, 0) dx
R3
R3
∫ T∫
∫ T∫
+
0
R3
0
R3
(ϵ∂t g(θ))(∇x · uγ ) pγ dx dτ +
0
∫ T∫
+
∫
R3
T∫
(g(θ)(∇x · uγ )) (ϵ∂t pγ ) dx dτ +
((ϵ∂t ∇x g(θ)) · uγ ) pγ dx dτ
((∇x g(θ)) · uγ ) (ϵ∂t pγ ) dx dτ .
R3
0
Estimates of each term are as follows. First, for each t ∈ [0, T ], by Lemma 4.8, Lemma
4.10, and (5.64),
√
T +ϵ)C(Ω)
∥((ϵ∂t )γ (∇x (ϵp)))(·, t)∥L2 (R3 ) ≤ C(Ω0 )e(
√
( T +ϵ)C(Ω)
∥uγ (·, t)∥L2 (R3 ) ≤ C(Ω0 )e
,
,
√
( T +ϵ)C(Ω)
∥g(θ)(·, t)∥L∞ (R3 ) ≤ C(Ω0 )e
.
Therefore for t = 0 and t = T ,
∫
√
(g(θ)uγ ) · ((ϵ∂t )γ (∇x (ϵp)))(x, t) dx ≤ C(Ω0 )e( T +ϵ)C(Ω) .
(5.69)
R3
Similarly, by the θ-equation, Lemma 4.8 and (5.64),
∥pγ ∥L2 (R3 ×[0,T ]) ≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
∥∇x · uγ ∥L2 (R3 ×[0,T ]) ≤ C(Ω0 )e
(5.70)
,
√
( T +ϵ)C(Ω)
∥(ϵ∂t )∇x g(θ)∥L∞ (R3 ×[0,T ]) ≤ C(Ω0 )e
Therefore,
,
√
( T +ϵ)C(Ω)
.
∫ ∫
T
√
(ϵ∂t g(θ))(∇x · uγ ) pγ dx dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) ,
0 R3
∫ ∫
T
√
((ϵ∂t ∇x g(θ)) · uγ ) pγ dx dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) .
0 R3
40
A Low Mach Number Limit of a Dispersive Navier-Stokes System
To estimate the last two terms in (5.68) we need to use the pγ -equation in (5.48),
which is
(
)
ϵ∂t pγ = − 53 (ϵu)·∇x pγ − 35 ∇x · uγ − 25 h(ϵp)κ(θ) ∇x θγ + ϵg1
((
)
)
(5.71)
+ 23 ϵ2 h(ϵp) Σγ + Σ̃γ : ∇x u + ∇x · q̃γ + 23 h(ϵp)κ(θ)∇x (ϵp)·∇x θγ .
Using (5.71) we show bounds for the second last term in (5.68). First, by Lemma 4.8
and (5.64), the contribution from the first term in ϵ∂t pγ is
∫ ∫
T
5
(g(θ)(∇x · uγ )) ( 3 (ϵu)·∇x pγ ) dx dτ 0 R3
≤ 35 ∥g(θ)(ϵu)∥L∞ (R3 ×[0,T ]) ∥∇x · uγ ∥L2 (R3 ×[0,T ]) ∥∇x pγ ∥L2 (R3 ×[0,T ])
≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
∥∇x pγ ∥L2 (R3 ×[0,T ])
+ 14 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) .
The contribution from the term 53 ∇x · uγ in ϵ∂t pγ is
∫ ∫
T
5
(g(θ)(∇x · uγ )) ( 3 ∇x · uγ ) dx dτ ≤ 53 ∥g(θ)∥L∞ (R3 ×[0,T ]) ∥∇x · uγ ∥2L2 (R3 ×[0,T ])
0 R3
≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
.
By (5.56), the contribution from the term ϵg1 in ϵ∂t pγ is
∫ ∫
T
(g(θ)(∇x · uγ )) (ϵg1 ) dx dτ 0 R3
≤ ∥g(θ)∥L∞ (R3 ×[0,T ]) ∥∇x · uγ ∥L2 (R3 ×[0,T ]) ∥ϵg1 ∥L2 (R3 ×[0,T ])
√
≤ C(Ω0 )e(
T +ϵ)C(Ω)
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
(ϵC(Ω))
.
The rest of the terms in ϵ∂t pγ are all for the slow motion (ϵ∂t )γ ψ = (ϵp, ϵu, θ). The
2
. Therefore by Lemma 4.8, their L2 (R3 ×
highest order for these terms are (ϵ∇x )∇
√x
( T +ϵ)C(Ω)
[0, T ])-norms are bounded by C(Ω0 )e
. Hence,
their contribution to the
√
( T +ϵ)C(Ω)
second last term of (5.68) is also bounded by C(Ω0 )e
. Altogether we have
∫ ∫
T
√
(g(θ)(∇x · uγ )) (ϵ∂t pγ ) dx dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) + 14 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) .
0 R3
The last term in (5.68) is bounded in a similar way. Therefore, we have at the end
∫ T
√
∥∇x pγ ∥2L2 (τ ) dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) + 12 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) ,
0
which gives
∫
T
∥∇x pγ ∥2L2 (τ ) dτ ≤ C(Ω0 )e(
0
We thereby finish the proof of Theorem 5.2.
√
T +ϵ)C(Ω)
.
A Low Mach Number Limit of a Dispersive Navier-Stokes System
41
Conclusion. Combining Theorem 4.3, Lemma 4.7, and Theorem 5.3, we complete the proof of Theorem 2.4.
6. Decay of the Local Energy. Given the uniform bound for (p, u, θ), we show
in this section the local strong convergence of the fast wave (pϵ , uϵ ). The theorem
states as
Theorem 6.1. Let T > 0, and s > 6. Assume that the functions (pϵ , uϵ , θϵ )
satisfy the system (1.7), and
(6.1)
sup (∥(pϵ , uϵ )∥H s + ∥θϵ ∥H s+1 ) < ∞ .
[0,T ]
σ
Assume further that θϵ converges strongly in C 0 ([0, T ]; Hloc
(R3 ) for some σ > 5/2, to
3
a limit θ satisfying for all (t, x) ∈ [0, T ] × R
(6.2)
|θ(t, x) − θ| ≤ C0 |x|−1−γ ,
|∇x θ(t, x)| ≤ C0 |x|−2−γ ,
for some given positive constants C0 , γ, and θ. Then
(6.3)
pϵ → 0
(
)
ϵ
∇x · uϵ − 52 e−ϵp κ(θϵ )∇x θϵ → 0
′
in
s
L2 ([0, T ]; Hloc
(R3 )) ,
in
s −1
L2 ([0, T ]; Hloc
(R3 ))
′
for all 1 ≤ s′ < s.
The proof of Theorem 6.1 is based on the following theorem via dispersive estimates by Métivier and Schochet [20] which is reformulated in [2].
Theorem 6.2. Let T > 0, and let wϵ be a bounded sequence in C 0 ([0, T ]; H 2 (R3 ))
such that
ϵ2 ∂t (aϵ ∂t wϵ ) − ∇x · (bϵ ∇x wϵ ) = cϵ ,
where cϵ converges to 0 strongly in L2 (0, T ; L2 (R3 )). Assume further that, for some
σ > 5/2, the coefficients (aϵ , bϵ ) are uniformly bounded in C 0 ([0, T ]; H σ (R3 )) and
σ
converges in C 0 ([0, T ]; Hloc
(R3 )) to a limit (a, b) satisfying the decay estimate
|a(t, x) − a| ≤ c0 |x|−1−γ ,
|∇x a(t, x)| ≤ c0 |x|−2−γ ,
|b(t, x) − b| ≤ c0 |x|−1−γ ,
|∇x b(t, x)| ≤ c0 |x|−2−γ ,
for some given positive constants a, b, c0 and γ. Then the sequence wϵ converges to 0
in L2 (0, T ; L2loc (R3 )).
s′
Proof. [Proof of Theorem 6.1] First we show that pϵ → 0 in L2 ([0, T ]; Hloc
(R3 )).
By the boundedness assumption (6.1) and interpolation arguments, it suffices to verify
that pϵ → 0 in L2 ([0, T ]; L2loc (R3 )). The equation for pϵ is
(
)
1
ϵ
ϵ
ϵ
ϵ
3
2 −ϵpϵ
∇
·
u
−
(∂
+
u
·
∇
)p
+
e
κ(Θ
)∇
θ
x
t
x
ϵ
x
5
5
((
) ϵ
)
ϵ
2 −ϵpϵ
Σ + Σ̃ : ∇x u + ∇x · q̃ .
= 5 ϵe
Apply the operator ϵ2 ∂t to the pϵ -equation. We have
)
(
)
(
ϵ
ϵ2 ∂t 53 (∂t + uϵ · ∇x )pϵ + ϵ∂t ∇x · uϵ − 25 e−ϵp κ(Θϵ )∇x θϵ
((
)
))
(
(6.4)
ϵ
= 25 ϵ3 ∂t e−ϵp
Σ + Σ̃ : ∇x uϵ + ∇x · q̃
.
42
A Low Mach Number Limit of a Dispersive Navier-Stokes System
By the uϵ -equation and θϵ -equation in system (1.7), the above wave equation (6.4)
has the form
( ϵ
)
(
)
(6.5)
ϵ2 ∂t 35 ∂t pϵ + ∇x · eθ ∇x pϵ = ϵF (pϵ , uϵ , θϵ ) ,
where F is a smooth function in its variables with F (0) = 0. By the boundedness
assumption (6.1) on (pϵ , uϵ , θϵ ), and the convergence assumption for θϵ , we have
ϵF (pϵ , uϵ , θϵ ) → 0 strongly in L2 (0, T ; L2 (R3 )) ,
)
( ϵ
ϵ
and the coefficient eθ in ∇x · eθ ∇x pϵ satisfies the requirement of bϵ in Theorem 6.2.
Therefore, by Theorem 6.2, we have
pϵ → 0 strongly in L2 ([0, T ]; L2loc (R3 )) ,
which also implies
′
s
pϵ → 0 strongly in L2 ([0, T ]; Hloc
(R3 )) ,
for all 1 ≤ s′ < s by interpolation.
(
)
ϵ
The convergence of ∇x · uϵ − 52 e−ϵp κ(θϵ )∇x θϵ is shown in the same way. Let
(
)
ϵ
wϵ , ∇x · uϵ − 25 e−ϵp κ(θϵ )∇x θϵ .
Then the wave equation for wϵ is
( ϵ
)
ϵ2 ∂tt wϵ + ∇x · eθ ∇x wϵ = ϵF̃ (pϵ , uϵ , θϵ ) ,
where F̃ is a smooth function in its variables with F̃ (0) = 0. Therefore, the same
argument guarantees that
(
)
ϵ
s′ −1
∇x · uϵ − 25 e−ϵp κ(θϵ )∇x θϵ → 0 strongly in L2 ([0, T ]; Hloc
(R3 ))
for all 1 ≤ s′ < s.
7. Passing to the limit. Combining the uniform bound given by Theorem 2.4
and the local strong convergence of the fast motion (6.3), we can prove the following
convergence result.
Theorem 7.1. Using the same assumptions and notations in Theorem 6.1, the
family {(uϵ , θϵ )| ϵ ∈ (0, 1]} converges weakly in L∞ ([0, T ]; H s (R3 )) and strongly in
s′
L2 ([0, T ]; Hloc
(R3 )) for all s′ < s to a limit (u, θ) satisfying the ghost effect system:
(
)
∇x · 52 u − κ(θ)∇x θ = 0 ,
(7.1)
e−θ (∂t u + u · ∇x u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃ ,
3
2 (∂t
+ u · ∇x )θ + ∇x · u = −∇x · q ,
(u, θ)(x, 0) = (v in , θin )(x) ,
for some P ∗ satisfying ∇x P ∗ ∈ C([0, T ]; H s−2 (R3 )), and ϑ, Σ, Σ̃ be defined as
)
(
q = −κ(θ)∇x θ ,
Σ = µ(θ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
( 2
)
(
)
1
Σ̃ = τ̄1 (θ) ∇x θ − 3 ∆x ΘI + τ̄2 (θ) ∇x θ ⊗ ∇x θ − 31 |∇x θ|2 I .
A Low Mach Number Limit of a Dispersive Navier-Stokes System
43
where τ̄1 , τ̄2 are the transport coefficients such that τ̄1 (θ) = τ1 (0, θ) and τ̄2 (θ) = τ2 (0, θ)
where τ1 , τ2 are defined in (1.9). The initial data (v in , θin ) satisfies that
Π(e−θ v in ) = uin ,
in
∇x · (v in − 52 κ(θin )∇x θin ) = 0 ,
where Π is the projection onto the divergence free of e−θ v in and uin , θin are defined
in (2.4).
Proof. [Sketch of proof] First the uniform bound of the norm defined by (4.33)
gives that
in
sup (∥(pϵ , uϵ )∥H s + ∥θϵ ∥H s+1 ) < ∞ .
[0,T ]
for s ≥ 6. The equation for θϵ then implies that ∂t θϵ is uniformly bounded in
C([0, T ]; H s−3 (R3 ). Extracting a subsequence, we can assume that for all s′ < s,
θϵ → θ
′
s +1
C 0 ([0, T ]; Hloc
(R3 )) ,
strongly in
′
s +1
where the limit θ belongs to C([0, T ]; Hloc
(R3 )) ∩ L∞ (0, T ; H s+1 (R3 )). Meanwhile,
one can check that the decay conditions
(6.2) are
(
) satisfied by θ by checking the
equations for |x|1+γ (θ − θ) and ∇x |x|1+γ (θ − θ) . Therefore, by Theorem 6.1, we
(
)
ϵ
have the local strong convergence of pϵ and ∇x · uϵ − 25 e−ϵp κ(θϵ )∇x θϵ as in (6.3).
By the uniform boundedness of (pϵ , uϵ ) in C([0, T ]; H s (R3 )), we can assume that
weak∗ in L∞ (0, T ; H s (R3 )) ,
(pϵ , uϵ ) → (p, u)
(7.2)
By the limits
θϵ → θ
(7.3)
pϵ → 0
(
)
ϵ
∇x · uϵ − 25 e−ϵp κ(θϵ )∇x θϵ → 0
′
s +1
in C 0 ([0, T ]; Hloc
(R3 )) ,
′
s
in L2 ([0, T ]; Hloc
(R3 )) ,
′
s −1
in L2 ([0, T ]; Hloc
(R3 )) ,
we obtain
(7.4)
′
s −1
in L2 ([0, T ]; Hloc
(R3 )) ,
∇x · uϵ → ∇x · u
Thus by (7.3) the limit (u, θ) satisfies the first equation in (7.1). By the equation for
ϵ
ϵ
curl (e−θ uϵ ), there exists a subsequence of curl (e−θ uϵ ) such that for all s′ < s,
(7.5)
′
s −1
curl (e−θ uϵ ) → curl (e−θ u) strongly in C([0, T ]; Hloc
(R3 )) ,
ϵ
ϵ
Combining (7.4) and (7.5), we have up to a subsequence,
′
s
uϵ → u strongly in L2 ([0, T ]; Hloc
(R3 )) ,
for all s′ < s. The third equations in (7.1) is shown by direct applications of the
uniform bounds of (pϵ , uϵ ) in L∞ (0, T ; H s (R3 )), local strong convergence of (pϵ , uϵ )
s′ +1
s′
in L2 ([0, T ]; Hloc
(R3 ) and strong convergence of θϵ in C 0 ([0, T ]; Hloc
(R3 )). For the
second equation in (7.1), first we have the limiting projected equation using various
convergence of (pϵ , uϵ , θϵ )
(
)
(
)
(7.6)
Π ∂t (e−θ u) + ∇x · (e−θ u ⊗ u) = Π ∇x · Σ + ∇x · Σ̃ ,
44
A Low Mach Number Limit of a Dispersive Navier-Stokes System
where Π is the projection operator onto the divergence free part of e−θ u. Moreover,
by the first and third equations in (7.1), we have ∂t u ∈ H s−2 (R3 ) for each t ∈ [0, T ].
Therefore there exists P ∗ such that the second equation in (7.1) holds. Furthermore,
by Theorem 3.2, the solution to the ghost effect system (7.1) is unique. Therefore,
the above convergence in fact holds for the full sequence of (pϵ , uϵ , θϵ ), which thereby
completes the proof.
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Anal. 180, 1-73 (2006).
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