SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
Abstract. We prove a low Mach number limit for a dispersive fluid system [3] which contains thirdorder corrections to the compressible Navier-Stokes. We show that the classical solutions to this system
in the whole space Rn converge to classical solutions to ghost-effect systems [7]. Our analysis follows
the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and
Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy
structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and
to simplify the analysis in [4].
1. Introduction
In this paper we establish a low Mach number limit of a dispersive Navier-Stokes system (DNS)
derived from kinetic equations in [3]. The governing equations for this system are
∂t ρ + ∇x · (ρU ) = 0 ,
∂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) ,
(1.1)
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. The terms p, q, Σ represent the pressure, heat flux, and viscous stress
tensor (for a Newtonian fluid) respectively with
q(Θ) = −κ̂(Θ)∇x Θ ,
p = p (ρ, Θ) = ρΘ,
Σ = µ̂(Θ)(∇x U + (∇x U )T − 23 (∇x · U )I) .
The total energy density is given by
ρe = 21 ρ|U |2 + 32 ρΘ .
The quantities Σ̃ and q̃ denote dispersive corrections to the stress tensor and heat flux respectively.
They are given by
Σ̃ = τ̂1 (Θ)(∇x2 Θ − 13 (∆x Θ)I) + τ̂2 (Θ)(∇x Θ ⊗ ∇x Θ − 13 |∇x Θ|2 I)
+τ̂3 (ρ, Θ)(∇x U (∇x U )T − (∇x U )T ∇x U ) ,
q̃ = τ̂4 (Θ)(∆x U + 13 ∇x ∇x · U ) + τ̂5 (Θ)∇x Θ · (∇x U + (∇x U )T − 23 (∇x · U )I)
(
)
+τ̂6 (ρ, Θ) ∇x U − (∇x U )T · ∇x Θ .
1
2
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
Here we assume that the transport coefficients τ̂1 , · · · , τ̂6 are C ∞ functions of their variables 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.
One main feature of the DNS system is that it possesses an entropy structure: by the derivation in
[3], the transport coefficients in Σ̃ and q̃ satisfy
( )
Θ
τ̂2 2τ̂5
τ̂4
τ̂4 = τ̂1 ,
+ 2 = ∂Θ
,
(1.2)
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
Note that (1.2) implies that τ̂1 and τ̂4 have the same sign. Under assumption (1.2) one can show that
the DNS system dissipates the Euler entropy in the same way as the Navier-Stokes system.
This paper continues our previous work [4] on dispersive Navier-Stokes systems, where a low Mach
number limit is established for τ̂1 , τ̂4 > 0 without using the entropy structure. The positivity assumption of τ̂1 , τ̂4 is essential for the analysis in [4] because it guarantees that operators in the form
T = I − τ̂1 ∆x are invertible. It is the invertibility of T that provides the annihilation of the most
singular terms in the energy estimates. Thus this type of analysis does not generalize to the case
where τ̂1 , τ̂4 < 0. However, the original DNS system derived in [3] indeed has negative τ̂1 , τ̂4 and
the corresponding transport coefficient in the limiting ghost-effect system is also negative [7]. Our
goal in this paper is to show that the low Mach limit for the negative τ̂1 , τ̂4 can be established by
using the extra entropy structure (1.2). Moreover, our analysis here applies to the case with positive
τ̂1 , τ̂4 as well and it is more straightforward than that in [4]. The main reason is that the relation
between τ̂1 and τ̂4 in (1.2) shows that the DNS system has an antisymmetric structure which allows
us to perform direct energy estimates. Our result in this paper shows the DNS system indeed recovers
genuine ghost-effect systems instead of merely ghost-effect-type systems in [4].
We also want to point out that compared with the original system derived from kinetic equations
in [3], system (1.1) is simplified in the sense that terms involving second order derivatives of ρ are not
included within ∇x · Σ̃ and ∇x · q̃. This renders a second assumption that τ̂1 , τ̂2 , τ̂4 , τ̂5 are independent
of ρ since the entropy structure needs to be satisfied.
The limiting ghost effect system can be formally derived from kinetic equations using a Hilbert
expansion method (cf. Sone [7]). 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 [7]. 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,
∂t ϱ + ∇x · (ϱu) = 0,
∂t (ϱu) + ∇x · (ϱu ⊗ u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃,
3
5
2 ∂t (ρϑ) + 2 ρϑ∇x · u = −∇x · q,
(1.3)
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
)
(
q = −κ̄(ϑ)∇x ϑ,
Σ = µ̄(ϑ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
3
where µ̄(ϑ), κ̄(ϑ) > 0 are the viscosity and heat conductivity respectively. The thermal stress tensor
Σ̃ has the form
(
)
(
)
1
2
Σ̃ = τ̄1 (ϑ) ∇x ϑ − ∆x ϑ I + τ̄2 (ϑ) ∇x ϑ ⊗ ∇x ϑ − 31 |∇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), we introduce the same scaling as in [4]. Let ϵ be the Knudsen
number. We have
µ̂ = ϵµ̃,
κ̂ = ϵκ̃,
τ̂i = ϵ2 τ̃i , i = 1, · · · , 6 ,
p ∼ p0 + O(ϵ) ,
U ∼ O(ϵ) ,
∇x Θ ∼ O(1) ,
p = ρΘ ,
where p0 is a constant which is fixed as 1. Thus, we consider the case when p is around 1 and 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
ϵ
ϵ
ϵ
pϵ = eϵp ,
Uϵ = ϵuϵ ,
Θϵ = eθ ,
ρϵ = eρ ,
(1.4)
where the superscripts denote the fluctuation functions. We will consider a long-time scale such that
1
t = t̂ .
(1.5)
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
)
)
( ϵ 2 −ϵpϵ ϵ
) 2 −ϵpϵ ((
ϵ
ϵ 1
ϵ
ϵ
3
∇
·
u
−
(∂
+u
·∇
)p
+
e
κ(θ
)∇
θ
=
ϵe
Σ+
Σ̃
:
∇
u
+∇
·
q̃
x
x
x
x
x
5 t
5
5
ϵ
ϵ
ϵ
ϵ
2 −ϵpϵ
κ(θ )∇x p · ∇x θ ,
+ 5e
(
)
1
ϵ
ϵ
e−θ (∂t + uϵ · ∇x )uϵ + ∇x pϵ = e−ϵp ∇x · Σ + ∇x · Σ̃ ,
(1.6)
ϵ
((
)
)
ϵ
ϵ
ϵ
2 −ϵpϵ
3
2 (∂t +u ·∇x )θ +∇x ·u = ϵ e
Σ+ Σ̃ : ∇x uϵ +∇x · q̃
− e−ϵp ∇x · q ,
ϵ
in in
(pϵ , uϵ , θϵ )(x, 0) = (pin
ϵ , uϵ , θϵ )(x) ,
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 − 32 (∇x · uϵ )I ,
(
)
(
)
Σ̃ = τ1 (θϵ ) ∇x2 θϵ − 13 (∆x θϵ )I + τ2 (θϵ ) ∇x θϵ ⊗ ∇x θϵ − 13 |∇x θϵ |2 I
(
)
+ϵ2 τ3 (ϵpϵ , θϵ ) ∇x uϵ (∇x uϵ )T − (∇x uϵ )T ∇x uϵ ,
(
)
(
)
ϵ )T · ∇ θ ϵ
q̃ = τ4 (θϵ ) ∆x uϵ + 13 ∇x ∇x(· uϵ + τ6 (ϵpϵ , θϵ ) ∇x uϵ − (∇x u
x
)
+τ5 (θϵ )∇x θϵ · ∇x uϵ + (∇x uϵ )T − 32 (∇x · uϵ )I ,
(1.7)
while
κ(θϵ ) = κ̃(Θϵ ) Θϵ ,
µ(θϵ ) = µ̃(Θϵ ) ,
ρϵ = Pϵ /Θϵ = eϵp −θ ,
τ1 (θϵ ) = τ̃1 (Θϵ ) Θϵ ,
τ4 (θϵ ) = τ̃4 (Θϵ ) ,
ϵ
2
τ2 (θ ) = τ̃1 (Θϵ ) Θϵ + τ̃2 (Θϵ ) Θϵ , τ3 (ϵpϵ , θϵ ) = τ̃3 (ρϵ , Θϵ ) ,
τ5 (θϵ ) = τ̃5 (Θϵ )Θϵ , τ6 (ϵpϵ , θϵ ) = τ̃6 (ρϵ , Θϵ )Θϵ .
ϵ
ϵ
(1.8)
4
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
If we assume that for all indices α = (α1 , α2 , α3 ) such that |α| ≤ 3,
(∇x pϵ , ∇xα uϵ , ∇xα θϵ )(t, x) → (0, ∇xα u, ∇xα θ)(t, x)
as ϵ → 0,
then (1.6) 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 ,
3
2 (∂t + u · ∇x )θ + ∇x · u = −∇x · q ,
(1.9)
(1.10)
where P ∗ is the pressure which is an independent variable. By (1.9) we have
(
)
Σ = µ(θ) ∇x u + (∇x u)T − 23 (∇x · u)I ,
q = −κ(θ)∇x θ ,
( 2
)
(
)
Σ̃ = τ1 (θ) ∇x θ − 13 ∆x ΘI + τ2 (θ) ∇x θ ⊗ ∇x θ − 13 |∇x θ|2 I .
(1.11)
Here µ(θ) is the viscosity, κ(θ) is the heat conductivity, and τ1 (θ), τ2 (θ) ∈ C ∞ (R × R) are transport
coefficients with τ1 ≤ 0 as defined in (1.8).
We can write the limiting system (1.10) in the form of a ghost system as (1.3) by taking (ϱ, u, ϑ) =
(ϱ, u, eθ ) as the variables. System (1.10) then has the form
ϱϑ = 1 ,
∂t ϱ + ∇x · (ϱu) = 0 ,
(1.12)
∂t (ϱu) + ∇x · (ϱu ⊗ u) + ∇x P ∗ = ∇x · Σ + ∇x · Σ̃ ,
(
)
∇x · 52 u − κ(θ)∇x ϑ = 0 ,
where Σ and Σ̃ are defined in (1.11).
Our goal in this paper is to rigorously justify the above formal calculation of the low Mach number
limit from the dispersive system (1.6) to the ghost effect system (1.10) in the whole space.
2. Main Theorem and Outline of the Proof
In this section we will state the main theorem and give an outline of its proof. We adopt the same
notation as in [4].
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]
(∫
+ α0
0
t(
∥∇x (u, p)∥2H s
+
)
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1
ϵ
) 21
(τ )dτ ,
(2.1)
and
|||(p, u, θ−θ)|||ϵ,s,0 := ∥(pin , uin )∥H s + ∥Λ2s+1
(ϵpin , ϵuin , θin −θ)∥H s+1 ,
ϵ
(2.2)
where H s = H s (R3 ) is the usual Sobolev space such that derivatives up to order s of a function are
square-integrable. The invertible operator Λϵ is defined by
Λϵ := (I − ϵ2 ∆x )1/2
for any ϵ > 0.
(2.3)
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
5
Definition 2.2. Let (pϵ , uϵ , θϵ ) be the solution to (1.6). We call ψ = (ϵpϵ , ϵuϵ , θϵ ) the slow motion
because it varies on the time scale of order one. We call (pϵ , uϵ ) the fast motion because it varies on
the time scale of order ϵ.
The main theorem states
Main Theorem. Let s ≥ 6. Suppose that there exist constants M0 , θ, c0 , σ such that the initial data
in in
of the fluctuations (pin
ϵ , uϵ , θϵ ) satisfy
in
in
|||(pin
ϵ , uϵ , θϵ −θ)|||ϵ,s,0 ≤ M0 ,
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
(2.4)
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 (1.6) 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. Moreover, there exists a positive constant M depending only on T, M0 such that
|||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ≤ M .
(pϵ , uϵ , θϵ )
(2.5)
L∞ (0, T ; H s (R3 ))
Furthermore, the sequence of solutions
to system (1.6) converges weakly-∗ in
′
2
s
3
′
and strongly in L (0, T ; Hloc (R )) for all s < s to the limit (0, u, θ), where (u, θ) satisfies the ghost
effect system (1.10).
The low-Mach number limit of fluid equations has a long history. The interested reader is referred
to the references given in (for example) [1, 6]. Our proofs of the main theorem follows the same
framework as in [4], which is built upon the methodology developed by Métivier and Schochet [6] and
Alazard [1, 2]. In [6] Métivier and Schochet proved the incompressible limit for the non-isentropic
Euler equations for classical solutions with general initial data. In [1] Alazard proved the low Mach
number limit for the compressible Navier-Stokes for classical solutions with general initial data. In
[2] 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. The analysis in [1] considers different
types of estimates in different ranges of wave numbers. Here we will close the energy estimate directly
by studying the energy bounds of the slow motion and the fast motion in order without separating
the wave numbers.
The main difficulty in closing the energy estimate in this paper remains the same as in [4]: we
need to simultaneously remove the most singular terms in (1.6), which include the terms associated
with 1ϵ and the third-order terms in the dispersive part. Again 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
(1.6) and ψ ϵ = (ϵpϵ , ϵuϵ , θϵ ), then (1.6) 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 [4], when the coefficients τ1 , τ4 are unrelated, the term
A2 (ψ ϵ )U ϵ is not readily anti-symmetric. However, it is anti-symmetrizable by a certain symmetrizer
matrix composed of symmetric positive operators. As pointed out in the introduction, the positivity
of the operators in the symmetrizer matrix essentially depends on the fact that τ̂1 , τ̂4 are both positive.
6
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
Thus this symmetrizing method fails when τ̂1 , τ̂4 < 0. The key observation in resolving this difficulty
is that if one makes use of the intrinsic entropy structure of the DNS system, then τ̂1 and τ̂4 are
related by (1.2). It turns out this particular relation shows A2 (ψ ϵ )U ϵ is actually anti-symmetric. As
a consequence, energy estimates can then be carried out in a rather straightforward way because the
most singular term A2 (ψ ϵ )U ϵ naturally disappears in the L2 -type estimates. We want to mention that
this argument applies to the case where τ1 , τ4 are both positive as well.
Proof of the main theorem consists of two main steps: uniform bounds of the solution (pϵ , uϵ , θϵ ) in
an appropriate norm and local energy decay of the fast motion (pϵ , uϵ ). We will only show the details
for deriving the uniform bounds because the local energy decay does not depend on the signs of τ̂1 , τ̂4
and is proved in exactly the same way as in [4]. The proposition for the uniform bounds states
Proposition 2.3. For each fixed ϵ > 0, let (pϵ , uϵ , θϵ ) ∈ C([0, T ]; H s (R3 )) be the solution to the scaled
DNS system (1.6). Let
Ω = |||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ,
in in
Ω0 = |||(pin
ϵ , uϵ , θϵ −θ)|||ϵ,s,0 .
Then there exists an increasing function C(·) such that
√
|||(pϵ , uϵ , θϵ −θ)|||ϵ,s,T ≤ C(Ω0 )e(
T +ϵ)C(Ω)
,
which further indicates [6] 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 rest of the paper is devoted to the proof of Proposition 2.3. The outline of the proof is as follows.
In Section 3 we derive the a priori estimate (Proposition 3.1) for the slow motion ψ ϵ = (ϵpϵ , ϵuϵ , θϵ ).
This estimate implies addition bounds on various time derivatives of the fast motion, the slow motion
ϵ
ψ ϵ , and another slowly varying quantity curl(eθ uϵ ). Estimates of these additional bounds only depend
on the order of the system. In particular, they are not affected by the signs of τ̂1 , τ̂4 . Thus these
estimates are identical to those in [4] and are omitted in this paper. In Section 4 we derive the a priori
estimate for the fast motion (pϵ , uϵ ). This is done in two steps: first we show the bounds for certain
time derivatives of the fast motion (Proposition 4.1). Then we prove the desired H s -bounds for the
fast motion using Proposition 4.1. Again since the second step does not depend on the signs of τ̂1 , τ̂4 ,
we will omit its details in this paper. Overall, in this paper we will focus on details for the a priori
estimate of the slow motion (Proposition 3.1) and the bounds for time derivatives of the fast motion
(Proposition 4.1).
3. A Priori Estimates – Slow Motion
In this section we derive a priori estimates for slow motions which vary on the time scale of order
one. Estimates in the next section for fast motions depend heavily on those for slow motions.
Assume there exist two constants α1 , α2 > 0 such that
0 < α1 < h(ϵp0 ), g(θ0 ), µ(θ0 ), κ(θ0 ) < α2 < ∞ .
(3.1)
For t > 0 and s > 5, recall the norm ||| · |||ϵ,s,t defined by
(
)
|||(p, u, θ)|||ϵ,s,t := sup ∥(p, u)∥H s + ∥Λ2s+1
(ϵp, ϵu, θ)∥H s+1
ϵ
[0,t]
(∫
+ α0
0
t(
∥∇x (u, p)∥2H s
+
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1
ϵ
)
)1
(3.2)
2
(τ ) dτ
,
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
7
We also define
R = R(t) = ∥(p, u)∥H s + ∥Λ2s+1
(ϵp, ϵu, θ)∥H s+1 ,
ϵ
(3.3)
(
)1/2
R′ = R′ (t) = ∥(∇x u, ∇x p)∥2H s + ∥∇x Λ2s+1
(ϵu, θ)∥2H s+1
.
ϵ
(∫
)1/2
t
Thus |||(p, u, θ)|||ϵ,s,t = sup[0,t] R(s) + 0 (R′ (s))2 ds
. We will repeatedly use the following basic
calculus inequalities (see for example [?]): for |k1 | + |k2 | + · + |kn | = s with s > d + 2 = 5 ,
∥∇xk1 u1 ∇xk2 u2 · · · ∇xkn un ∥L2 ≤ ∥u1 ∥H s ∥u2 ∥H s · · · ∥un ∥H s ,
∀n ∈ N ,
(
)
∥[f, ∇xm ] g∥L2 (R3 ) ≤ cm ∥∇x f ∥L∞ ∥∇xm−1 g∥L2 + ∥∇xm f ∥L2 ∥g∥L∞
(3.4)
≤ cm (∥∇x f ∥L∞ ∥g∥H s−1 + ∥f ∥H s ∥g∥L∞ ) ,
∥∇xm (f g)∥L2 (R3 )
≤ cm (∥f ∥L∞
∥∇xm g∥L2
+
∥∇xm f ∥L2
(3.5)
∥g∥L∞ )
≤ cm (∥∇x f ∥L∞ ∥g∥H s + ∥f ∥H s ∥g∥L∞ ) ,
(3.6)
for any 1 ≤ |m| ≤ s.
3.1. A Priori Estimate of the Slow Motion. The a priori estimate for the slow motion (ϵp, ϵu, θ)
is stated in the following proposition.
Proposition 3.1. Let (p, u, θ) be a solution to the system (1.6). Suppose s > 5. There exists a
constant α0 > 0 such that if we define the norm ||| · |||♭ϵ,s,t as
(∫
|||(p, u, θ)|||♭ϵ,s,t : = sup ∥Λϵ2s+1 (ϵp, ϵu, θ)∥H s+1 + α0
0
[0,t]
t
)1/2
∥∇x Λ2s+1
(ϵu, θ)∥2H s+1 (τ ) dτ
ϵ
,
(3.7)
then for all T ∈ [0, 1], we have
|||(p, u, θ)|||♭ϵ,s,T ≤ C(Ω0 ) e(
√
T +ϵ)C(Ω)
,
(3.8)
where
Ω = |||(p, u, θ)|||ϵ,s,T ,
Ω0 = |||(p, u, θ)|||ϵ,s,0 .
(3.9)
Proof. Define
(p̂, û, θ̂) = (ϵp − θ, ϵu, θ) .
(3.10)
Algebraic calculation using (1.6) shows that the system for (p̂, û, θ̂) has the form
1
(∂t + u · ∇x )p̂ + ∇x · û = 0 ,
ϵ
1
1
g(θ)(∂t + u · ∇x ) û + ∇x p̂ + ∇x θ̂ = h(p̂)∇x · Σ + ϵh(p̂)∇x · Σ̃ ,
ϵ
ϵ
1
3
2 (∂t + u · ∇x ) θ̂ + ϵ ∇x · û = ϵh(p̂)∇x · q̃ + ϵh(p̂) Σ̃ : ∇x û
+ h(p̂)Σ : ∇x û − h(p̂)∇x · q ,
(p, u, θ)(x, 0) = (pin , uin , θin )(x) .
(3.11)
8
where
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
(
)
q = −κ(θ)∇x θ̂ ,
Σ = µ(θ) ∇x û + (∇x û)T − 23 (∇x · û)I ,
(
)
(
)
Σ̃ = τ1 (θ) ∇x2 θ̂ − 13 (∆x θ̂)I + τ2 (θ) ∇x θ ⊗ ∇x θ̂ − 31 (∇x θ · ∇x θ̂)I
(
)
+ τ3 (ϵp, θ) (∇x û)(∇x û)T − (∇x û)T (∇x û) ,
(
)
(
)
q̃ = τ4 (θ) ∆x û + 31 ∇x ∇x · û + τ6 (ϵp, θ) ∇x û − (∇x û)T · ∇x θ .
(
)
+ τ5 (θ)∇x θ · ∇x û + (∇x û)T − 32 (∇x · û)I .
Note that the equation for p̂ = ϵp − θ is simply the linearized density equation. Let Ψk denote any
k-th order differential operator. The general form of Ψk w is
Ψk w = a(x) ∇xα w ,
where |α| = k. Use this notation and apply ∇xα with |α| ≤ s + 1 to (3.11). We have
1
(∂t + u·∇x )p̂α + ∇x · ûα = R1,α ,
ϵ
1
1
g(θ)(∂t + u·∇x ) ûα + ∇x p̂α + ∇x θ̂α = h(p̂)µ(θ)(∆x ûα + 31 ∇x ∇x ûα )
ϵ
ϵ
+ 23 ϵh(p̂)τ1 (θ)∇x ∆x θ̂α + R2,α
+ ϵΨ2 (ûβ , θ̂β ) + Ψ1 (p̂β , ûβ , θ̂β )
+ Ψ0 (p̂β , ûβ , θ̂β ) + F1 (p̂ν , ûν , θ̂ν )
3
2 (∂t
(3.12)
1
+ u·∇x ) θ̂α + ∇x · ûα = 43 ϵh(p̂)τ4 ∇x ∇x · û + h(p̂)κ(θ̂)∆x θ̂
ϵ
+ R3,α + ϵΨ2 (ûβ , θ̂β ) + Ψ1 (p̂β , ûβ , θ̂β )
+ Ψ0 (p̂β , ûβ , θ̂β ) + F2 (p̂ν , ûν , θ̂ν ) ,
in in
(p̂α , ûα , θ̂α )(x, 0) = (p̂in
α , ûα , θ̂α )(x) ,
where |β| = |α| and |ν| ≤ max{|α| − 1, 0} and (p̂α , ûα , θ̂α ) = (∇xα p̂, ∇xα û, ∇xα θ̂). The coefficients of
Ψk depend only on (p̂, û, θ̂) and ∇xγ (p̂, û, θ̂) with |γ| ≤ 3. The commutator terms are
R1,α = −[∇xα , u] · ∇x p̂ ,
R3,α = −[∇xα , 32 u] · ∇x θ̂ .
R2,α = −[∇xα , g(θ)]∂t û − [∇xα , g(θ)u] · ∇x û ,
(3.13)
Note that by (1.2) and (1.8), we have
2
3 ϵh(p̂)τ1 (θ̂)
= 43 ϵh(p̂)τ4 (θ̂) .
Thus the leading-orders in the dispersive terms form an antisymmetric structure, which will vanish in
L2 -type energy estimates. Now we perform the L2 energy estimate by multiplying (3.12) by (p̂α , ûα , θ̂α )
and integrating over R3 . The estimates for each equation are as follows. By (3.4), we only need to
check the leading order terms. For the p̂-equation, we have
2
1 d
2 dt ∥p̂α ∥L2
1
+ ⟨p̂α , ∇x · ûα ⟩ ≤ 12 ∥∇x u∥L∞ ∥p̂α ∥2L2 + ∥p̂α ∥L2 ∥R1,α ∥L2 .
ϵ
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
9
Note that R1,α = 0 if α = 0. For 1 ≤ |α| ≤ s + 1 we have
)
(
∥R1,α ∥L2 ≤ Cα ∥∇x u∥L∞ ∥∇x|α| p̂∥L2 + ∥∇x|α| u∥L2 ∥∇x p̂∥L2
≤ C(∥u∥H s+1 ∥∇x p̂∥W 1,∞ + ∥u∥W 1,∞ ∥p̂∥H s+1 ) ,
by (3.5). Thus,
2
1 d
2 dt ∥p̂α ∥L2
Similarly, for any m ∈ N,
1
+ ⟨p̂α , ∇x · ûα ⟩ ≤ C(R)(1 + R′ ) .
ϵ
d
1
∥(ϵ∇x )m p̂α ∥2L2 + ⟨(ϵ∇x )m p̂α , ∇x · (ϵ∇x )m ûα ⟩
dt
ϵ
≤ 21 (∥∇x u∥L∞ + 1) ∥(ϵ∇x )m p̂α ∥2L2 + 12 ∥ϵm R1,m+|α| ∥2L2 ,
1
2
(3.14)
(3.15)
where
∥ϵm R1,m+|α| ∥L2 ≤ C(∥(ϵ∇x )m ∇xα u∥L2 ∥∇x p̂∥W 1,∞ + ∥u∥W 1,∞ ∥(ϵ∇x )m ∇xα p̂∥L2 )
= C(∥(ϵ∇x )m−1 ∇x ∇xα û∥L2 ∥∇x p̂∥W 1,∞ + ∥u∥W 1,∞ ∥(ϵ∇x )m p̂∥H s+1 ) .
(3.16)
If we choose m ≤ 2s + 1, then
2s+1
∥ϵm R1,m+α ∥L2 ≤ C(∥Λ2s
p̂∥H s+1 ) .
ϵ ∇x û∥H s+1 ∥∇x p̂∥W 1,∞ + ∥u∥W 1,∞ ∥Λϵ
≤ C(R)(1 + R′ ) .
Hence,
m
2
1 d
2 dt ∥(ϵ∇x ) p̂α ∥L2
1
+ ⟨(ϵ∇x )m p̂α , ∇x · (ϵ∇x )m ûα ⟩ ≤ C(R)(1 + R′ ) ,
ϵ
(3.17)
for any m ≤ 2s + 1.
Next, we check the û-equation. The L2 estimate gives
1
1
2
1 d
2 dt ⟨ûα , g(θ̂)ûα ⟩ + ϵ ⟨ûα , ∇x p̂α ⟩ + ϵ ⟨ûα , ∇x θ̂α ⟩ + α0 ∥∇x ûα ∥L2
≤ 12 (∥∂t g(θ)∥L∞ + ∥∇x (g(θ)u)∥L∞ )∥ûα ∥2L2 + ⟨ûα , 32 ϵh(p̂)τ1 (θ)∇x ∆x θ̂α ⟩
+ ϵC(R)(R′ )2 + C(R)(1 + R′ ) + ⟨ûα , R2,α ⟩ ,
where α0 is determined by the (positive) lower bound of µ(θ) and κ(θ). By (3.13),
R2,α = [∇xα , g(θ)](u · ∇x û) + [∇xα , g(θ)](g(θ)−1 ∇x p) + [∇xα , g(θ)](Ψ2 + ϵΨ3 )(û, θ̂)
+ [∇xα , g(θ)](Ψ1 + Ψ0 )(p̂, û, θ̂) − [∇xα , g(θ)u] · ∇x û .
Note that R2,α = 0 for α = 0. Thus, we need only to consider |α| ≥ 2. We show the details for
estimates of the leading order terms, which are [∇xα , g(θ)](g(θ)−1 ∇x p), [∇xα , g(θ)](Ψ2 + ϵΨ3 )(û, θ̂), and
[∇xα , g(θ)u] · ∇x û. First,
∥[∇xα , g(θ)](g(θ)−1 ∇x p)∥L2 + ∥[∇xα , g(θ)]Ψ2 (û, θ̂))∥L2 + ∥[∇xα , g(θ)u] · ∇x û∥L2
(
)
≤C ∥g(θ) − g(0)∥W 1,∞ ∥g(θ)−1 ∇x p∥H s + ∥g(θ) − g(0)∥H s+1 ∥g(θ)−1 ∇x p∥W 1,∞
(
)
+ C ∥g(θ) − g(0)∥H s+1 ∥Ψ2 (û, θ̂)∥W 1,∞ + ∥g(θ) − g(0)∥W 1,∞ ∥Ψ2 (û, θ̂)∥H s
(
)
s
+ C ∥g(θ) − g(0)∥H s+1 ∥Ψ2 (û, θ̂)∥W 1,∞ + ∥g(θ) − g(0)∥W 1,∞ ∥Ψ2 (û, θ̂)∥H
≤C(R)(1 + R′ ) .
10
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
We apply integration by part for the estimate related to [∇xα , g(θ)](ϵΨ3 (û, θ̂)). Specifically, suppose
′′
′
∇xα = ∂xi ∇xα = ∂xi xj ∇xα for some i, j ∈ {1, 2, 3} and α′ , α′′ ∈ N3 such that |α′ | = |α| − 1 and
|α′′ | = |α| − 2. Then
⟨
⟩
ûα , [∇xα , g(θ)](ϵΨ3 (û, θ̂)) ⟨
⟩ ⟨
⟩
′
′′
= ∂xi ûα , [∇xα , g(θ)](ϵΨ3 (û, θ̂)) + ∂xj ((∂xi g(θ)) ûα ) , ∇xα (ϵΨ3 (û, θ̂)) ≤ C∥∇x û∥H s+1 ∥g(θ) − g(0)∥H s ∥ϵΨ3 (û, θ̂)∥W 1,∞ + C∥∇x g(θ)∥L∞ ∥∇x û∥H s+1 ∥ϵΨ3 (û, θ̂)∥H s−1
+ C∥∇x û∥H s+1 ∥g(θ) − g(0)∥W 1,∞ ∥ϵΨ3 (û, θ̂)∥H s−1 + C∥∇x ∇x g(θ)∥L∞ ∥û∥H s+1 ∥ϵΨ3 (û, θ̂)∥H s−1
≤C(R)(1 + R′ ) + ϵC(R)(R′ )2 .
Thus,
1 d
2 dt ⟨ûα ,
1
1
g(θ̂)ûα ⟩ + ⟨ûα , ∇x p̂α ⟩ + ⟨ûα , ∇x θ̂α ⟩ + α0 ∥∇x ûα ∥2L2
ϵ
ϵ
2
≤ ⟨ûα , 3 ϵh(p̂)τ1 (θ)∇x ∆x θ̂α ⟩ + ϵC(R)(R′ )2 + C(R)(1 + R′ ) .
(3.18)
Applying (ϵ∇x )m will only affect the terms containing the fast motion in R2,α . We show the calculation
for the leading-order terms involving the fast motion in R2,α :
∥[ϵm ∇xm+|α| , g(θ)](u · ∇x û)∥L2 + ∥[ϵm ∇xm+|α| , g(θ)u] · ∇x û∥L2
≤ ∥ϵm ∇xm+|α| g(θ)∥L2 ∥u · ∇x û∥W 1,∞ + ∥g(θ)∥W 1,∞ ∥ϵm ∇xm+|α|−1 (u · ∇x û)∥L2
+ ∥ϵm ∇xm+|α| (g(θ)u)∥L2 ∥∇x û∥W 1,∞ + ∥g(θ)u∥W 1,∞ ∥ϵm ∇xm+|α|−1 (∇x û)∥L2
(3.19)
m−1
∇x û∥H |α| ∥∇x û∥W 1,∞
≤ C(∥θ∥L∞ ) ∥u∥W 1,∞ ∥∇x û∥W 1,∞ ∥Λm
ϵ θ∥H |α| + C(∥θ∥L∞ ) ∥Λϵ
)
(
m−1
m
+ ∥g(θ)∥W 1,∞ ∥u∥W 1,∞ ∥Λϵ û∥H |α| + ∥∇x û∥L∞ ∥Λϵ û∥H |α| .
(
)
m+|α|
m+|α|
The leading term (in the fast motion) in [ϵm ∇x
, g(θ)](g(θ)−1 ∇x p) is g(θ)−1 ∇x g(θ) ϵm ∇x
p.
In order to control this term, we apply integration by parts:
) m m+|α| m m+|α| (
−1
û, g(θ) ∇x g(θ) ϵ ∇x
p⟩
⟨ϵ ∇x
(3.20)
) m−1
(
−1
m
∞ ∥Λ ∇x û∥ |α| ∥Λ
.
p̂∥
+
∥g(θ)
∇
g(θ)∥
≤ ∥∇x (g(θ)−1 ∇x g(θ))∥L∞ ∥Λm
û∥
|α|
|α|
x
L
ϵ
ϵ
ϵ
H
H
H
Thus, if we restrict to 0 ≤ m ≤ 2s + 1, then
m
1 d
2 dt ⟨(ϵ∇x ) ûα ,
1
g(θ̂)(ϵ∇x )m ûα ⟩ + ⟨(ϵ∇x )m ûα , ∇x (ϵ∇x )m p̂α ⟩
ϵ
1
+ ⟨(ϵ∇x )m ûα , ∇x (ϵ∇x )m θ̂α ⟩ + α0 ∥∇x (ϵ∇x )m ûα ∥2L2
ϵ
≤ ⟨(ϵ∇x )m ûα , 23 ϵh(p̂)τ1 (θ)∇x ∆x (ϵ∇x )m θ̂α ⟩ + ϵC(R)(R′ )2 + C(R)(1 + R′ ) .
(3.21)
The structure of the θ̂ equation is similar to that of the û equation. Thus, the L2 estimate for θ̂
gives
1
2
1 d
3
2 dt ⟨θ̂α , 2 θ̂α ⟩ + ϵ ⟨θ̂α , ∇x · ûα ⟩ + α0 ∥∇x θ̂α ∥L2
(3.22)
′ 2
′
4
≤ ⟨θ̂α , 3 ϵh(p̂)τ4 (θ)∇x ∆x · ûα ⟩ + ϵC(R)(R ) + C(R)(1 + R ) .
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
11
and
1
d
⟨(ϵ∇x )m θ̂α , 32 (ϵ∇x )m θ̂α ⟩ + ⟨(ϵ∇x )m θ̂α , ∇x · (ϵ∇x )m ûα ⟩ + α0 ∥∇x (ϵ∇x )m θ̂α ∥2L2
dt
ϵ
≤ ⟨(ϵ∇x )m θ̂α , 34 ϵh(p̂)τ4 (θ)∇x ∆x · (ϵ∇x )m ûα ⟩ + ϵC(R)(R′ )2 + C(R)(1 + R′ ) .
1
2
(3.23)
for any m ∈ N.
Adding (3.14), (3.17), (3.18), (3.17), (3.22), and (3.23), we obtain
d m
d
2
m
m
∥Λϵ (p̂, θ̂)∥2H s+1 + 12 ⟨Λm
û, g(θ̂)Λm
ϵ û⟩ + α0 ∥∇x (Λϵ û, Λϵ θ̂)∥H s+1
dt
dt ϵ
≤ ϵC(R)(R′ )2 + C(R)(1 + R′ ) .
1
2
By the lower bound (3.1) for g(θ), we have that
∫ T
√
m
2
m
2
( T +ϵ)C(Ω)
∥Λϵ (p̂, û, θ̂)∥H s+1 + α0
∥∇x (Λm
.
ϵ û, Λϵ θ̂)∥H s+1 ≤ C(Ω0 ) e
(3.24)
0
Remark 3.1. Notice that (3.16), (3.19), and (3.20) indicate that the dependence of (ϵ∇x )m (p̂, û, θ̂)
on the fast motion does not change when m increases. In particular, the estimate for (ϵ∇x )m (p̂, û, θ̂)
depends only on ∥u∥W 1,∞ for any m ∈ N. This shows we can choose any m ∈ N in order to close the
estimate in Proposition 3.1 (with R, R′ , Ω0 , Ω redefined with the corresponding m).
3.2. Additional Bounds. In this part we list out additional bounds based on Proposition 3.1. The
details for deriving these bounds are omitted because they are very similar to those in [4]. This is
because these bounds do not depend on the signs of τ̂1 , τ̂4 . Instead they only depend on the order of
the equations. First, the slow motion curl(e−θ u) satisfies the equation:
(
(
))
(
(
))
(∂t + u · ∇x ) curl e−θ u = curl e−ϵp ∇x · Σ + ∇x · Σ̃ + curl(e−θ u ∂t θ)
(3.25)
+ [curl, u] · ∇x (e−θ u) + curl(e−θ u(u · ∇x θ)) .
It has the following bound:
Lemma 3.1. Let s > 5. Let (p, u, θ) be a solution to the fluctuation equations (1.6). Then there exists
an increasing function such that
∫ T
√
−θ
2
∥∇x curl(e−θ u)∥2H s−1 (τ ) dτ ≤ C(Ω0 )e T C(Ω) .
sup ∥curl(e u))∥H s−1 +α0
(3.26)
[0,T ]
0
2(s−k)
Next we show some bounds of terms ∥(ϵ∂t )k (u, p)∥L2 and ∥Λϵ
(ϵ∂t )k (ϵp, ϵu, θ)∥H s+1−k for 1 ≤ k ≤ s.
These bounds will be used in the next section. They are immediate consequences of Proposition 3.1.
Lemma 3.2. Let (p, u, θ) be the solution to (1.6). Let 1 ≤ k ≤ s. Then
∥(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′ ) .
(3.27)
12
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
Lemma 3.3. Let (p, u, θ) be the solution to (1.6) and ψ = (ϵp, ϵu, θ). Then for all 1 ≤ k ≤ s,
∥Λ2(s−k)+1
(ϵ∂t )k ψ∥H s+1−k ≤ C(R) ,
ϵ
∥∇x Λ2(s−k)+1
(ϵ∂t )k ψ∥H s+1−k ≤ C(R)(1 + R′ ) ,
ϵ
)
√
k
k
−θ
( T +ϵ)C(Ω)
sup ∥Λ2(s−k)+1
(ϵ∂
)
ψ∥
+
∥(ϵ∂
)
curl(e
u)∥
≤
C(Ω
)e
,
s+1−k
s−k−1
t
t
0
H
H
ϵ
(
∫
(3.28)
[0,T ]
T
(
0
∥∇x Λ2(s−k)+1
(ϵ∂t )k ψ∥2H s+1−k + +∥(ϵ∂t )k ∇x curl(e−θ u)∥H s−k−1
ϵ
)
dτ ≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
,
where R, R′ , Ω0 , Ω are defined in (3.3) and (3.9).
We also recall the following bound in [1] (Remark 6.21):
Lemma 3.4 ([1]). Let F ∈ C ∞ (R1 × R3 × R1 ) such that F (0) = 0. Let ψ = (ϵp, ϵu, θ) be the solution
to (1.6). Then
√
√
(3.29)
∥F (ψ)∥L∞ (0,T ;H s ) ≤ C(Ω0 ) + T C(Ω) ≤ C(Ω0 )e( T +ϵ)C(Ω) .
4. A priori estimate – fast motion
In this section we establish estimates for p and the acoustic part of u. Using the various bounds
obtained for the slow motion in Section 3, we can show Sobolev bounds of (∇x ·u, ∇x p) by studying
estimates for (ϵ∂t )k (p, u) with 1 ≤ k ≤ s which is summarized in the following proposition:
Proposition 4.1. Let (p, u, θ) be a solution to (1.6). Let s > 5. Define
(pγ , uγ , θγ ) := (ϵ∂t )γ (p, u, θ) ,
where 1 ≤ γ ≤ s . Then there exists an increasing function C(·) such that
∫ T
(
)
√
2
2
sup ∥pγ ∥L2 + ∥uγ ∥Hϵ1 + α0
∥∇x (pγ , uγ )∥2L2 dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) .
[0,T ]
(4.2)
0
Proof. Applying the operator (ϵ∂t )γ to the system (1.6), we have
(
((
)
))
)
(
γ 2
3
1
2
=
(ϵ∂
)
(∂
+
u·∇
)p
+
∇
·
u
−
h(ϵp)κ(θ)
∇
θ
ϵh(ϵp)
Σ
+
Σ̃
:
∇
u
+
∇
·
q̃
t
x
γ
x
γ
x
γ
t
x
x
5
ϵ
5
5
)
(
− (ϵ∂t )γ 52 h(ϵp)κ(θ)∇x p · ∇x θ + g1 ,
(
)
g(θ)(∂t + u·∇x ) uγ + 1ϵ ∇x pγ = h(ϵp)∇x · Σγ + (ϵ∂t )γ h(ϵp)∇x · Σ̃ + g2 ,
(
((
)
))
γ
2
3
(∂
θ
+
u·∇
θ
)
+
∇
·
u
=
(ϵ∂
)
ϵ
h(ϵp)
Σ+
Σ̃
:
∇
u+∇
·
q̃
x γ
x
γ
t
x
x
2 t γ
+ (ϵ∂t )γ (h(ϵp)∇x · (κ(θ)∇x θ)) + g3 ,
where
(4.1)
(
)
Σγ = µ(θ) ∇x uγ + (∇x uγ )T − 23 (∇x · uγ )I ,
(
)
(
)
Σ̃γ = τ1 (θ) ∇x2 θγ − 31 (∆x θγ )I + τ2 (θ) ∇x θ ⊗ ∇x θγ − 13 ∇x θ · ∇x θγ I ,
(
)
+ τ3 (ϵp, θ) (∇x (ϵu))(∇x (ϵuγ ))T − (∇x (ϵu))T ∇x (ϵuγ ) .
(4.3)
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
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
g2 = [g(θ), (ϵ∂t )γ ]∂t u + [g(θ)u, (ϵ∂t )γ ] · ∇x u + [Ψ2 + Ψ1 , (ϵ∂t )γ ]u ,
13
(4.4)
g3 = [u, 23 (ϵ∂t )γ ]∇x θ ,
where Ψ2 and Ψ1 denote second and first-order homogeneous differential operators respectively. The
pγ -equation indicates that we need to consider the system for
(pγ , ũγ ) = (pγ , uγ − 52 h(ϵp)κ(θ)∇x θγ ) .
By the uγ and θγ equations, we derive that the system for (pγ , ũγ ) has the form
(
((
)
))
γ 2
3
1
(∂
+
u·∇
)p
+
∇
·p
=
(ϵ∂
)
ϵh(ϵp)
Σ
+
Σ̃
:
∇
u
+
∇
·
q̃
t
x
γ
x
γ
t
x
x
5
ϵ
5
(
)
− (ϵ∂t )γ 25 h(ϵp)κ(θ)∇x p · ∇x θ + g1 ,
g(θ)(∂t + u·∇x ) ũγ + 1ϵ ∇x pγ =
where
4
15 g(θ)h(ϵp)κ(θ)∇x ∇x
(4.5)
· ũγ + h(ϵp)∇x · Σũγ + g4 ,
(
)
Σũγ = µ(θ) ∇x ũγ + (∇x ũγ )T − 32 (∇x · ũγ )I ,
4
g(θ)h(ϵp)κ(θ)∇x ∇x · ( 25 h(ϵp)κ(θ)∇x θγ )
g4 = h(ϵp)∇x · Σ̃γ + g2 + 15
(2
)
4
− (g(θ)∇x θγ )∂t 5 h(ϵp)κ(θ) − 15
g(θ)h(ϵp)κ(θ)∇x (RHSθγ )
+ g(θ)[ 52 h(ϵp)κ(θ)∇x , u · ∇x ]θγ ,
where RHSθγ is the right-hand side of the θγ -equation in (4.3). In the L2 -estimate we will only show the
details for the leading-order terms for the slow motion and the fast motion. Denoting ψ = (ϵu, ϵp, θ) as
the slow motion, the leading-orders on the right-hand side of the pγ -equation expressed in the abstract
form in terms of the homogeneous differential operators are
Γ1 = Ψ2 θγ + Ψ3 ûγ + Ψ1 (uγ , pγ ) + Ψ1 ((ϵ∂t )γ−1 ∂t θ) ,
(4.6)
where the coefficients of Ψk ’s depend up to the first-order spatial derivatives of ψ, u, p. In particular,
the coefficient of Ψ3 depends only on the slow motion. The leading-order terms on the right-hand side
of the ũγ -equation are
Γ2 = Ψ3 θγ + Ψ0 ((ϵ∂t )γ−1 ∂t θ) + Ψ1 (uγ ) + ϵΨ4 ûγ ,
(4.7)
where the coefficients of the operators depend up to the fourth order spatial derivatives of ψ and up
to the second-order derivatives of u. In particular, the coefficients of Ψ3 and ϵΨ4 depend only on the
slow motion.
Now we compute the contribution of Γ1 , Γ2 in the L2 -estimate. To this end, we multiply (4.15) by
pγ and ũγ and integrate over Rn . The estimates for ⟨Γ1 , pγ ⟩ and ⟨Γ2 , ũγ ⟩ are as follows.
|⟨Γ1 , pγ ⟩| ≤ ∥pγ ∥L2 ∥Ψ2 θγ ∥L2 + |⟨pγ , Ψ3 ûγ ⟩| + ∥pγ ∥L2 ∥Ψ1 uγ ∥L2 + C(R)∥pγ ∥2L2
(
)
+ ∥pγ ∥L2 Ψ1 (ϵ∂t )γ−1 ∂t θ L2
(
)
≤ C(R) 1 + R′ + |⟨pγ , Ψ3 ûγ ⟩| ,
where by integration by parts
|⟨pγ , Ψ3 ûγ ⟩| ≤ ∥δ1 (ψ)∥L∞ ∥∇x pγ ∥L2 ∥∇x2 ûγ ∥L2 + C(R)(1 + R′ ) .
14
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
Here we have used δ1 (ψ) to denote the coefficient of the operator Ψ3 . Note that by Proposition 3.1,
we have
√
∥δ1 (ψ)∥L∞ ≤ C(Ω0 )e(
T +ϵ)C(Ω)
.
Overall, we have
√
|⟨Γ1 , pγ ⟩| ≤ C(R)(1 + R′ ) + C(Ω0 )e(
T +ϵ)C(Ω)
∥∇x pγ ∥L2 ∥∇x2 ûγ ∥L2 .
Similarly, the contribution from Γ2 satisfies
√
|⟨Γ2 , ũγ ⟩| ≤ C(R)(1 + R′ ) + C(Ω0 )e(
T +ϵ)C(Ω)
∥Λϵ ∇x2 ûγ ∥2L2 .
The L2 -estimate of (pγ , ũγ ) thus has the form
)
1 d (
∥pγ ∥2L2 + ∥ũγ ∥2L2 + α0 ∥∇x ũγ ∥2L2 (τ )
2 dt
√
(
)
≤ C(R)(1 + R′ ) + C(Ω0 )e( T +ϵ)C(Ω) ∥∇x pγ ∥L2 ∥∇x2 ûγ ∥L2 + ∥Λϵ ∇x2 ûγ ∥2L2 .
This implies
sup
[0,T ]
(
∥pγ ∥2L2
+
∥ũγ ∥2L2
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
)
∫
+ α0
(
∫
0
T
1+
∫
0
T
+ ϵ0
0
T
∥∇x ũγ ∥2L2 dτ
∥Λϵ ∇x2 ûγ ∥2L2
)
dτ
∫
T
+ ϵ0
0
∥∇x pγ ∥2L2 dτ
(4.8)
∥∇x pγ ∥2L2 dτ ,
1
8
where 0 < ϵ0 < is small enough and is to be determined. Note that the last inequality follows from
the last inequality in (3.28).
∫T
To conclude, we need the bound for 0 ∥∇x pγ ∥2L2 dτ . The proof is similar to the corresponding part
in [4] since again the estimates involved here do not depend on the sign of τ̂1 , τ̂4 . However for the
completeness of the proof of the theorem, we include the details here. To this end, we multiply the
uγ -equation in (4.3) 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
(4.9)
∫ T∫
+
0
R3
Γ · (∇x pγ )(x, τ ) dx dτ ,
where
Γ = −g(θ)(ϵu)·∇x uγ + ϵh(ϵp)∇x · Σγ + ϵh(ϵp)(ϵ∂t )γ (∇x · Σ̃) + ϵg2 .
(4.10)
Note that by Lemma 3.2, the first term in Γ satisfies
(∫ T
)1/2
′ 2
∥g(θ)(ϵu)·∇x uγ ∥L2 (R3 ×[0,T ]) ≤ ϵ
(C(R)(1 + R )) (τ ) dτ
≤ ϵ C(Ω) .
0
By Lemma 3.3, the second term in Γ satisfies
√
∥ϵh(ϵp)∇x · Σγ + ϵh(ϵp)(ϵ∂t )γ (∇x · Σ̃)∥L2 (R3 ×[0,T ]) ≤ C(Ω0 )e(
T +ϵ)C(Ω)
.
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
WITH AN ENTROPY STRUCTURE
15
By checking the leading-orders in g2 , we deduce that g2 satisfies
(∫ T
)1/2
≤ C(Ω) .
∥g2 ∥L2 (R3 ×[0,T ]) ≤
(C(R)(1 + R′ ))2 (τ ) dτ
0
Here by Lemma 3.2 and Lemma 3.3 the lower-order terms in g2 satisfy the same bound . Thus the
last term in Γ satisfies
(∫ T
)1/2
′ 2
∥ϵg2 ∥L2 (R3 ×[0,T ]) ≤ ϵ
(C(R)(1 + R )) (τ ) dτ
≤ ϵ C(Ω) ,
0
where R, R′ , Ω are defined in (3.3) and (3.9). 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
R
0
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
+
(4.11)
2
1
4 ∥∇x pγ ∥L2 (R3 ×[0,T ]) .
To estimate the first term on the right-hand side of (4.9), we integrate by parts in both t and x. Then
∫ T∫
g(θ) ∂t ((ϵ∂t )γ (ϵu)) · ∇x pγ (x, τ ) dx dτ
3
∫0 R
∫
= (g(θ)uγ ) · ((ϵ∂t )γ (∇x (ϵp)))(x, T ) dx − (g(θ)uγ ) · ((ϵ∂t )γ (∇x (ϵp)))(x, 0) dx
R3
R3
(4.12)
∫ T∫
∫ T∫
+
(ϵ∂t g(θ))(∇x · uγ ) pγ dx dτ +
((ϵ∂t ∇x g(θ)) · uγ ) pγ dx dτ
0
R3
0
R3
∫ T∫
+
R3
0
(g(θ)(∇x · uγ )) (ϵ∂t pγ ) dx dτ +
∫ T∫
0
R3
((∇x g(θ)) · uγ ) (ϵ∂t pγ ) dx dτ .
Estimates of each term are as follows. First, for each t ∈ [0, T ], by Lemma 3.3, Lemma 3.4, and (4.8),
∥((ϵ∂t )γ ∇x (ϵp))(·, t)∥L2 (R3 ) ≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
√
( T +ϵ)C(Ω)
∥g(θ)(·, t)∥L∞ (R3 ) ≤ C(Ω0 )e
,
.
∥uγ (·, t)∥L2 (R3 ) ≤ ∥ũγ ∥L2 (R3 ) + ∥ 52 h(ϵp)κ(θ)∇x θγ ∥L2 (R3 )
√
≤ C(Ω0 )e(
T +ϵ)C(Ω)
+ 18 ∥∇x pγ ∥L2 (R3 ×[0,T ]) .
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) + 1 ∥∇x pγ ∥2 2 3
L (R ×[0,T ]) .
8
3
(4.13)
R
By Lemma 3.3,
∥pγ ∥L∞ (0,T ;L2 (R3 )) + ∥∇x · uγ ∥L2 (R3 ×[0,T ]) ≤ C(R)(1 + R′ ) ,
∥(ϵ∂t )∇x g(θ)∥L∞ (R3 ×[0,T ]) ≤ C(Ω0 )e(
Therefore,
√
T +ϵ)C(Ω)
.
∫ T∫
√
√
( T +ϵ)C(Ω)
≤
(ϵ∂
g(θ))(∇
·
u
)
p
dx
dτ
T
C(Ω)
≤
C(Ω
)e
,
t
x
γ
γ
0
0 R3
∫ T∫
√
√
≤ T C(Ω) ≤ C(Ω0 )e( T +ϵ)C(Ω) .
((ϵ∂
∇
g(θ))
·
u
)
p
dx
dτ
t
x
γ
γ
3
0
R
(4.14)
16
C. DAVID LEVERMORE, WEIRAN SUN AND KONSTANTINA TRIVISA
To estimate the last two terms in (4.12) we need to use the pγ -equation in (4.3), which is
ϵ∂t pγ = − 53 (ϵu)·∇x pγ − 53 ∇x ·uγ − 32 (ϵ∂t )γ (h(ϵp)∇x · (κ(θ)∇x θ)) .
(
((
)
))
+ 32 (ϵ∂t )γ ϵ2 h(ϵp) Σ + Σ̃ : ∇x u + ∇x · q̃ + ϵg1 .
(4.15)
Using (4.15) we show bounds for the second last term in (4.12). First, by Lemma 3.3 and (??), the
contribution from the first term in ϵ∂t pγ is
∫ T∫
5
(g(θ)(∇
·
u
))
(
(ϵu)·∇
p
)
dx
dτ
x
γ
x
γ
3
0 R3
√
≤ 53 T ∥g(θ)u∥L∞ (R3 ×[0,T ]) ∥∇x · ûγ ∥L∞ ([0,T ];L2 (R3 )) ∥∇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 ])
+ 18 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) .
The contribution from the term 35 ∇x · uγ in ϵ∂t pγ is
∫ T∫
5
(g(θ)(∇x · uγ )) ( 3 ∇x · uγ ) dx dτ ≤ 35 ∥g(θ)∥L∞ (R3 ×[0,T ]) ∥∇x · uγ ∥2L2 (R3 ×[0,T ])
0
R3
≤ C(Ω0 )e(
√
T +ϵ)C(Ω)
(1 + ϵ0 ∥∇x pγ ∥L2 (R3 ×[0,T ]) ) .
By Lemma 3.2 and Lemma 3.3, the contribution from the term ϵg1 in ϵ∂t pγ satisfies
∫ T∫
(g(θ)(∇x · uγ )) (ϵg1 ) dx dτ 3
0
R
≤ ∥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)
(ϵC(Ω))
≤ C(Ω0 )e
.
The rest of the terms in ϵ∂t pγ are all for the slow motion (ϵ∂t )γ ψ = (ϵ∂t )γ (ϵp, ϵu, θ). The highest order
spatial derivatives for these
terms are (ϵ∇x )∇x2 . Therefore by Lemma 3.3, their L2 (R3 × [0, T ])-norms
√
( T +ϵ)C(Ω) . Hence, their contribution to the second last term of (4.12) is also
are bounded by C(Ω√
0 )e
(
T
+ϵ)C(Ω)
bounded by C(Ω0 )e
. Altogether we have
∫ T∫
(
)
√
( T +ϵ)C(Ω)
2
(g(θ)(∇
·
u
))
(ϵ∂
p
)
dx
dτ
≤
C(Ω
)e
1
+
ϵ
∥∇
p
∥
2
3
x
γ
t γ
0
x γ L (R ×[0,T ]) .
0
0
R3
Hence we choose ϵ0 small enough such that
√
1
.
8
Note that this can be done because C(·) is a fixed function depending only
on the particular form of
√
(
T
+ϵ)C(Ω)
g(·) and we can consider the time period such that Ω < 1. Thus C(Ω0 )e
is a fixed quantity.
The last term in (4.12) is bounded in a similar way. Then we have
∫ T
√
∥∇x pγ ∥2L2 (τ ) dτ ≤ C(Ω0 )e( T +ϵ)C(Ω) + 34 ∥∇x pγ ∥2L2 (R3 ×[0,T ]) ,
ϵ0 C(Ω0 )e(
0
T +ϵ)C(Ω)
≤
SINGULAR LIMIT OF A DISPERSIVE NAVIER-STOKES SYSTEM
which gives
∫
0
√
T
∥∇x pγ ∥2L2 (τ ) dτ ≤ C(Ω0 )e(
WITH AN ENTROPY STRUCTURE
T +ϵ)C(Ω)
.
Combining (4.16) with (4.8), we thereby finish the proof of Proposition 4.1.
17
(4.16)
The derivation of the a priori estimate for ∥(p, u)∥H s from Proposition 4.1 follows the same way as
in [4]. Again one uses the relation
∇x · u = − 35 (ϵ∂t )p − 35 (ϵu) · ∇x p + 25 e−ϵp ∇x · (κ(θ)∇x θ) + 52 ϵ e−ϵp Σ : ∇x (ϵu)
+ 25 ϵ e−ϵp Σ̃ : ∇x (ϵu) + 52 ϵ2 e−ϵp ∇x · q̃ ,
(
)
∇x p = −e−θ (ϵ∂t )u − e−θ (ϵu) · ∇x u + ϵe−ϵp ∇x · Σ + ∇x · Σ̃ ,
(4.17)
and apply it iteratively to transfer the time derivatives into spatial derivatives. Since the signs of τ̂1 , τ̂4
do not contribute and the proof is rather identical to [4], we simply state the result without providing
a detailed proof:
Proposition 4.2. Let (p, u, θ) be a solution to the fluctuation equations (1.6). Then
∫ T
(
)
2
2
sup ∥p∥H s + ∥∇x · u∥H s−1 +
(∥∇x ∇x · u∥2H s−1 + ∥∇x p∥2H s )(τ ) dτ
[0,T ]
√
( T +ϵ)C(Ω)
≤ C(Ω0 )e
0
.
Conclusion. Combining Proposition 3.1, Lemma 3.1, and Proposition 4.2, we complete the proof of
Theorem 2.3.
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[2]
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[4]
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[6]
[7]
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