Hindawi Publishing Corporation
Boundary Value Problems
Volume 2007, Article ID 56945, 10 pages
doi:10.1155/2007/56945
Research Article
A Note on the Relaxation-Time Limit of
the Isothermal Euler Equations
Jiang Xu and Daoyuan Fang
Received 3 July 2007; Accepted 30 August 2007
Recommended by Patrick J. Rabier
This work is concerned with the relaxation-time limit of the multidimensional isothermal
Euler equations with relaxation. We show that Coulombel-Goudon’s results (2007) can
hold in the weaker and more general Sobolev space of fractional order. The method of
proof used is the Littlewood-Paley decomposition.
Copyright © 2007 J. Xu and D. Fang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The multidimensional isothermal Euler equation with relaxation describing the perfect
gas flow is given by
nt + ∇ · (nu) = 0,
1
(nu)t + ∇ · (nu ⊗ u) + ∇ p(n) = − nu
τ
(1.1)
for (t,x) ∈ [0,+∞) × Rd , d ≥ 3, where n, u = (u1 ,u2 ,...,ud ) ( represents transpose)
denote the density and velocity of the flow, respectively, and the constant τ is the momentum relaxation time for some physical flow. Here, we assume that 0 < τ ≤ 1. The
pressure p(n) satisfies p(n) = An, and A > 0 is a physical constant. The symbols ∇, ⊗ are
the gradient operator and the symbol for the tensor products of two vectors, respectively.
The system is supplemented with the initial data
(n,u)(x,0) = n0 ,u0 (x),
x ∈ Rd .
(1.2)
2
Boundary Value Problems
To be concerned with the small relaxation-time analysis, we define the scaled variables
s
.
τ
nτ ,uτ (x,s) = (n,u) x,
(1.3)
Then the new variables satisfy the following equations:
nτs
τ
2
n τ uτ
τ
+τ
s
2
n τ uτ
+∇·
= 0,
τ
n τ uτ
n τ uτ ⊗ uτ
+
= −A∇nτ
2
τ
τ
(1.4)
with initial data
nτ ,uτ (x,0) = n0 ,u0 .
(1.5)
Let τ → 0, formally, we obtain the heat equation
ᏺs − AΔᏺ = 0,
(1.6)
ᏺ(x,0) = n0 .
The above formal derivation of heat equation has been justified by many authors, see
[1–3] and the references therein. In [2], Junca and Rascle studied the convergence of the
solutions to (1.1) towards those of (1.6) for arbitrary large initial data in BV (R) space.
Marcati and Milani [3] showed the derivation of the porous media equation as the limit of
the isentropic Euler equations in one space dimension. Recently, Coulombel and Goudon
[1] constructed the uniform smooth solutions to (1.1) in the multidimensional case and
proved this relaxation-time limit in some Sobolev space H k (Rd ) (k > 1 + d/2, k ∈ N). In
this paper, we weaken the regularity assumptions on the initial data and establish a similar
relaxation result in the more general Sobolev space of fractional order (H σ+ε (Rd ), σ =
1 + d/2, ε > 0) with the aid of Littlewood-Paley decomposition theory.
If fixed τ > 0, there are some efforts on the global existence of smooth solutions to the
system (1.1)-(1.2) for the isentropic gas or the general hyperbolic system, the interested
readers can refer to [4–7]. Now, we state main results as follows.
Theorem 1.1. Let n be a constant reference density. Suppose that n0 − n and u0 ∈ H σ+ε (Rd ),
there exist two positive constants δ0 and C0 independent of τ such that if
n0 − n,u0 2 σ+ε
H
(R d )
≤ δ0 ,
(1.7)
then the system (1.1)-(1.2) admits a unique global solution (n,u) satisfying
(n − n,u) ∈ Ꮿ [0, ∞),H σ+ε Rd .
(1.8)
J. Xu and D. Fang 3
Moreover, the uniform energy inequality holds:
(n − n,u)(·,t)2 σ+ε
H
1
(R d ) +
τ
t
0
u(·,σ)2 σ+ε
H
t
(Rd ) dσ + τ
0
(∇n, ∇u)(·,σ)2 σ −1+ε
H
2
≤ C0 n0 − n,u0 σ+ε
H
(R d ) ,
(Rd ) dσ
t ≥ 0.
(1.9)
Based on Theorem 1.1, using the standard weak convergence method and compactness theorem [8], we can obtain the following relaxation-time limit immediately.
Corollary 1.2. Let (n,u) be the global solution of Theorem 1.1, then
nτ − n is uniformly bounded in Ꮿ [0, ∞),H σ+ε Rd ,
n τ uτ
is uniformly bounded in L2 [0, ∞),H σ+ε Rd .
τ
(1.10)
Furthermore, there exists some function ᏺ ∈ Ꮿ([0, ∞),n + H σ+ε (Rd )) which is a global
weak solution of (1.6). For any time T > 0, we have nτ (x,s) strongly converges to ᏺ(x,s)
in Ꮿ([0,T],(H σ +ε (Rd ))loc ) (σ < σ) as τ → 0.
2. Preliminary lemmas
On the Littlewood-Paley decomposition and the definitions of Besov space, for brevity,
we omit the details, see [9] or [7]. Here, we only present some useful lemmas.
Lemma 2.1 ([9, 7]). Let s > 0 and 1 ≤ p,r ≤ ∞. Then Bsp,r ∩ L∞ is an algebra and one has
f g Bsp,r f L∞ g Bsp,r + g L∞ f Bsp,r
if f ,g ∈ Bsp,r ∩ L∞ .
(2.1)
Lemma 2.2 [9, 7]. Let 1 ≤ p,r ≤ ∞, and I be open interval of R. Let s > 0 and be the smallest integer such that ≥ s. Let F : I → R satisfy F(0) = 0 and F ∈ W ,∞ (I; R). Assume that
v ∈ B sp,r takes values in J ⊂⊂ I. Then F(v) ∈ Bsp,r and there exists a constant C depending
only on s, I, J, and d such that
F(v)
B sp,r
≤ C 1 + v L∞ F W ,∞ (I) v Bsp,r .
(2.2)
Lemma 2.3 [7]. Let s > 0, 1 < p < ∞, the following inequalities hold.
(I) q ≥ −1:
2qs f ,Δq Ꮽg L p
⎧
⎪
⎪
⎪
Ccq f Bsp,2 g Bsp,2 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨
,
≤ Ccq f Bsp,2 g Bs+1
p,2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪Ccq f Bs+1
g Bsp,2 ,
⎩
p,2
d
+ ε (ε > 0),
p
d
f ∈ B sp,2 , g ∈ B s+1
+ ε (ε > 0),
p,2 , s =
p
d
s
f ∈ B s+1
+ ε (ε > 0).
p,2 , g ∈ B p,2 , s =
p
f ,g ∈ Bsp,2 , s = 1 +
(2.3)
4
Boundary Value Problems
If f = g, then
2qs f ,Δq Ꮽg L p ≤ Ccq ∇ f L∞ g Bsp,2 ,
s > 0.
(2.4)
(II) q = −1:
s g s ,
2−s f ,Δq Ꮽg L2d/(d+2) ≤ Cc−1 f B2,2
B2,2
s
f ,g ∈ B2,2
, s = 1+
d
+ ε (ε > 0),
2
(2.5)
where the operator Ꮽ = div or ∇, the commutator [ f ,h] = f h − h f , C is a harmless cons
≡
stant, and cq denotes a sequence such that (cq )l1 ≤ 1. (In particular, Besov space B2,2
s
H .)
3. Reformulation and local existence
Let us introduce the enthalpy Ᏼ(ρ) = Aln ρ (ρ > 0), and set
m(t,x) = A−1/2 Ᏼ n(t,x) − Ᏼ(n) .
(3.1)
Then (1.1) can be transformed into the symmetric hyperbolic form
d
1 0
,
∂t U + A j (u)∂x j U = −
u
τ
j =1
(3.2)
where
m
U=
,
u
uj
A j (u) = √
√
Aej
.
uj
Ae j
(3.3)
The initial data (1.2) become into
U0 =
√ A ln n0 − lnn ,u0
.
(3.4)
Remark 1. The variable change is from the open set {(n,u) ∈ (0,+∞) × Rd } to the whole
space {(m,u) ∈ Rd × Rd }. It is easy to show that the system (1.1)-(1.2) is equivalent to
(3.2)–(3.4) for classical solutions (n,u) away from vacuum.
First, we recall a local existence and uniqueness result of classical solutions to (3.2)–
(3.4) which has been obtained in [7].
σ
, then there
Proposition 3.1. For any fixed relaxation time τ > 0, assume that U0 ∈ B2,1
exist a time T0 > 0 (only depending on the initial data U0 ) and a unique solution U(t,x) to
σ
σ −1
) ∩ Ꮿ1 ([0,T0 ],B2,1
).
(3.2)–(3.4) such that U ∈ Ꮿ1 ([0,T0 ] × Rd ) and U ∈ Ꮿ([0,T0 ],B2,1
J. Xu and D. Fang 5
4. A priori estimate and global existence
In this section, we will establish a uniform a priori estimate, which is used to derive the
global existence of classical solutions to (3.2)–(3.4). Defining the energy function
2
1
Eτ (T)2 := sup U(t)H σ+ε +
τ
0≤t ≤T
T
u(t)2 σ+ε dt + τ
T
H
0
∇x U(t)2 σ −1+ε dt,
(4.1)
H
0
then we have the following a priori estimate.
Proposition 4.1. For any given time T > 0, if U ∈ Ꮿ([0,T],H σ+ε ) is a solution to the
system (3.2)–(3.4), then the following inequality holds:
Eτ (T)2 ≤ C S(T) Eτ (0)2 + Eτ (T)2 + Eτ (T)4 ,
(4.2)
where S(T) = sup0≤t≤T U(·,t)H σ+ε , C(S(T)) denotes an increasing function from R+ to
R+ , which is independent of τ,T,U.
Proof. The proof of Proposition 4.1 is divided into two steps. First, we estimate the
L∞ ([0,T],H σ+ε ) norm of U, and the L2 ([0,T],H σ+ε ) one of u. Then, we estimate the
L2 ([0,T],H σ −1+ε ) norm of ∇U.
Step 1. Applying the operator Δq to (3.2), multiplying the resulting equations by Δq m
and Δq u, respectively, and then integrating them over Rd , we get
t 1
1 Δq m2 2 + Δq u2 2 L
L 0 +
2
τ
=
1
2
t
Rd
0
Rd
+
0
Δq u(σ)2 2 dσ
L
0
t
t
2 2
div u Δq m + Δq u dx dσ
(4.3)
u,Δq · ∇mΔq m + u,Δq · ∇uΔq u dx dσ.
In what follows, we first deal with the low-frequency case. By performing integration by
parts, then using Hölder- and Gagliardo-Nirenberg-Sobolev inequality, we have (d ≥ 3)
t 2 t Δ−1 u(σ)2 2 dσ
Δ−1 m2 2 + Δ−1 u2 2 +
L
L
L
τ
0
≤
t
0
t
t
+2
2 2uLd Δ−1 mL2d/(d−2) Δ−1 ∇mL2 + ∇uL∞ Δ−1 uL2 dσ
+2
≤
0
0
u,Δ−1 · ∇m
2
L2d/(d+2)
Δ−1 m
L2d/(d−2)
+ u,Δ−1 · ∇uL2 Δ−1 uL2 dσ
2 2uLd Δ−1 ∇mL2 + ∇uL∞ Δ−1 uL2 dσ
0
t
0
u,Δ−1 · ∇m
L2d/(d+2)
Δ−1 ∇m 2 + u,Δ−1 · ∇u 2 Δ−1 u 2 dσ.
L
L
L
(4.4)
6
Boundary Value Problems
Multiplying the factor 2−2(σ+ε) on both sides of (4.4), from Lemma 2.3 and Young inequality, we obtain
t
2 2 2
t 2
2−2(σ+ε) Δ−1 u(σ)L2 dσ
2−2(σ+ε) Δ−1 mL2 + Δ−1 uL2 +
0
τ 0
t
2
2
1
≤
uLd 2−2(σ −1+ε) Δ−1 ∇mL2 + ∇uL∞ 2−2(σ+ε) Δ−1 uL2 dσ
0 2
+C
t
t
0
c−1 uH σ+ε mH σ+ε 2−(σ −1+ε) Δ−1 ∇mL2 + c−1 u2H σ+ε 2−(σ+ε) Δ−1 uL2 dσ
2
1
≤
uLd 2−2(σ −1+ε) Δ−1 ∇mL2 + ∇uL∞ 2−2(σ+ε) Δ−1 u2L2
2
0
t
2
1
+ C mH σ+ε c−2 1 u2H σ+ε + τ2−2(σ −1+ε) Δ−1 ∇mL2 dσ
τ
0
t
2
1
1 2
τ≤
+ C uH σ+ε c−1 u2H σ+ε + 2−2(σ+ε) Δ−1 uL2 dσ
τ
τ
0
dσ
1
,
τ
(4.5)
where C is some positive constant independent of τ. For the high-frequency case, we can
also achieve the similar inequality:
2 2 t 2
22q(σ+ε) Δq mL2 + Δq uL2 +
0
τ
t
0
2
22q(σ+ε) Δq u(σ)L2 dσ
2
2 ∇uL∞ 2
Δq ∇mL2 + 22q(σ+ε) Δq uL2 dσ
0
t
2
1
+ C mH σ+ε cq2 u2H σ+ε + τ22q(σ −1+ε) Δq ∇mL2 dσ
τ
0
t
2
1 2q(σ+ε) 1 2
1
2
σ+ε
+ C u H
Δq u L2 dσ
τ≤
,
cq uH σ+ε + 2
≤C
t
2q(σ −1+ε) τ
0
τ
(4.6)
τ
where we have taken the advantage of the fact Δq ∇mL2 ≈ 2q Δq mL2 (q ≥ 0).
By summing (4.6) on q ∈ N ∪ {0} and adding (4.5) together, then according to the
imbedding property in Sobolev space, we have
t 2 t
u2H σ+ε dσ
m2H σ+ε + u2H σ+ε 0 +
τ 0
t
t
1
1
≤ C mH σ+ε
u2H σ+ε + τ ∇m2H σ −1+ε dσ + C uH σ+ε u2H σ+ε dσ
τ
τ
0
0
t
1
+ C mH σ+ε u2H σ+ε + τ ∇m2H σ −1+ε dσ
τ
0
t
1
1
+ C uH σ+ε u2H σ+ε + u2H σ+ε dσ.
0
τ
τ
(4.7)
J. Xu and D. Fang 7
Therefore, for any t ∈ [0,T], the following inequality holds:
U(t)2 σ+ε + 2
H
t
τ
0
u2H σ+ε dσ ≤ C S(t) Eτ (0)2 + Eτ (t)2 .
(4.8)
Step 2. Thanks to the important skew-symmetric lemma developed in [1, 6, 10], we are
going to estimate the L2 ([0,T],H σ −1+ε ) norm of ∇U.
Lemma 4.2 (Shizuta-Kawashima). For all ξ ∈ Rd , ξ = 0, the system (3.2) admits a real
skew-symmetric smooth matrix K(ξ) which is defined in the unit sphere Sd−1 :
⎛
⎞
⎜ 0
K(ξ) = ⎜
⎝ ξ
−
|ξ |
then
d
K(ξ)
⎛√
⎜
ξ j A j (0) = ⎝
j =1
A|ξ |
0
ξ
|ξ | ⎟
⎟,
0
(4.9)
⎠
⎞
0
√ ξ ⊗ξ⎟
⎠.
− A
|ξ |
(4.10)
The system (3.2) can be written as the linearized form
∂t U +
d
A j (0)∂x j U =
j =1
d
j =1
1 0
A j (0) − A j (u) ∂x j U −
.
τ u
(4.11)
Let
Ᏻ=
d
A j (0) − A j (u) ∂x j U.
(4.12)
j =1
From Lemma 2.1, we have
ᏳH σ −1+ε ≤ C uH σ −1+ε ∇U H σ −1+ε .
(4.13)
Apply the operator Δq to the system (4.11) to get
d
1 0
.
∂t Δq U + A j (0)∂x j Δq U = Δq Ᏻ −
τ Δq u
j =1
(4.14)
By performing the Fourier transform with respect to the space variable x for (4.14) and
∗
multiplying the resulting equation by −iτ(Δ
q U) K(ξ), “∗” represents transpose and
conjugator, then taking the real part of each term in the equality, we can obtain
τ Im
Δ
qU
∗
= − Im
K(ξ)
d
∗
d
Δq U + τ Δ
U
K(ξ)
ξ j A j (0) Δ
q
qU
dt
j =1
Δ
qm
∗ ξ |ξ |
Δ
q u + τ Im Δ
qU
∗
K(ξ) Δ
qᏳ .
(4.15)
8
Boundary Value Problems
Using the skew-symmetry of K(ξ), we have
Im
Δ
qU
∗
K(ξ)
∗
d
1 d
Δq U =
K(ξ)Δ
Im Δ
qU
qU .
dt
2 dt
(4.16)
Substituting (4.10) into the second term on the left-hand side of (4.15), it is not difficult
to get
τ Im
Δ
qU
∗
d
∗
d
K(ξ) Δ
K(ξ)
ξ j A j (0) Δ
qU + τ Δ
qU
qU
dt
j =1
√
√
2
2
∗
τ d
− 2 A|ξ |Δ
≥
K(ξ)Δ
Im Δ
qU
q U + τ A|ξ | Δ
qU
qu .
2 dt
(4.17)
With the help of Young inequality, the right-hand side of (4.15) can be estimated as
− Im
∗ ξ ∗
K(ξ) Δ
qᏳ
|ξ |
√
2
2 Cτ 2
A + C Δ
+
Δ
,
|ξ |Δ
≤τ
qU
qu
qᏳ
2
τ |ξ |
|ξ |
Δ
qm
Δ
q u + τ Im Δ
qU
(4.18)
where the positive constant C is independent of τ. Combining with the equality (4.15)
and the inequalities (4.17)-(4.18), we deduce
√
2 C
2 Cτ 2 τ d
∗
1 A Δ
Δ
+
−
|ξ | Δq U ≤
K(ξ)Δ
|ξ | +
τ
Im Δ
qu
qᏳ
qU
qU .
2
τ
|ξ |
|ξ |
2 dt
(4.19)
Multiplying (4.19) by |ξ | and integrating it over [0,t] × Rd , from Plancherel’s theorem,
we reach
t
τ
0
Δq ∇U 2 2 dσ ≤ C
L
τ
t
0
Δq u2 2 + Δq ∇u2 2 dσ + Cτ
L
t
L
0
∗
τ
t
− Im
|ξ | Δ
K(ξ)Δ
qU
q U dξ 0
2
Rd
t
2
2
C t 2q ≤
2 Δq u 2 dσ + Cτ Δq Ᏻ 2 dσ
τ
0
L
0
Δq Ᏻ2 2 dσ
L
(4.20)
L
2 2 + Cτ22q Δq U(t) 2 + Δq U(0) 2 ,
L
L
where we have used the uniform boundedness of the matrix K(ξ) (ξ = 0).
Multiplying the factor 22q(σ −1+ε) (q ≥ −1) on both sides of (4.20) and summing it on
q, we have
t
τ
0
t
2
2 C t
u2H σ+ε dσ + Cτ Ᏻ2H σ −1+ε dσ + Cτ U(t)H σ+ε + U(0)H σ+ε
τ 0
0
≤ C S(t) Eτ (0)2 + Eτ (t)2 + Eτ (t)4 .
(4.21)
∇U 2H σ −1+ε dσ ≤
J. Xu and D. Fang 9
Together with the inequalities (4.8) and (4.21), (4.2) follows immediately, which com
pletes the proof of Proposition 4.1.
Proof of Theorem 1.1. In fact, Proposition 3.1 also holds on the framework of the funcσ+ε
). There exists a sufficiently small number 0 independent of τ
tional space H σ+ε (≡ B2,2
such that Eτ (T) ≤ 0 ≤ 1 from (4.1), we have
Eτ (T)2 ≤ C Eτ (0)2 + Eτ (T)3 ,
(4.22)
where the constant C is independent of τ. Without loss of generality, we may assume
C ≥ 1. Similar to that in [1], we achieve that
Eτ (t) ≤ min 0 ,
1
τ (0)
, 2CE
2C
!
(4.23)
for any t ≥ 0 if
U0 H σ+ε
≤
1
.
3/2
2(2C)
(4.24)
Note that the density
n − n = n exp A−1/2 m − 1 ;
(4.25)
from Lemma 2.2, the definition of Eτ (t), and the standard continuity argument, we can
obtain the following result: there exist two positive constants δ0 , C0 independent of τ if
the initial data satisfy
n0 − n2 σ+ε + u0 2 σ+ε ≤ δ0 ,
H
H
(4.26)
then the system (1.1)-(1.2) exists as a unique global solution (n,u). Moreover, the uniform energy estimate holds:
(n − n,u)(·,t)2 σ+ε + 1
H
τ
t
0
u(·,σ)2 σ+ε dσ + τ
2
≤ C0 n0 − n,u0 σ+ε ,
H
H
t
0
(∇n, ∇u)(·,σ)2 σ −1+ε dσ
H
(4.27)
t ≥ 0,
which completes the proof of Theorem 1.1.
The proof of Corollary 1.2 is similar to that in [1]; here, we omit the details, the interested readers can refer to [1].
Acknowledgments
This work was supported by NUAA’s Scientic Fund for the Introduction of Qualified Personnel (S0762-082), NSFC 10571158, and Zhejiang Provincial NSF of China (Y605076).
10
Boundary Value Problems
References
[1] J.-F. Coulombel and T. Goudon, “The strong relaxation limit of the multidimensional isothermal Euler equations,” Transactions of the American Mathematical Society, vol. 359, no. 2, pp.
637–648, 2007.
[2] S. Junca and M. Rascle, “Strong relaxation of the isothermal Euler system to the heat equation,”
Zeitschrift für Angewandte Mathematik und Physik, vol. 53, no. 2, pp. 239–264, 2002.
[3] P. Marcati and A. Milani, “The one-dimensional Darcy’s law as the limit of a compressible Euler
flow,” Journal of Differential Equations, vol. 84, no. 1, pp. 129–147, 1990.
[4] B. Hanouzet and R. Natalini, “Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,” Archive for Rational Mechanics and Analysis, vol. 169,
no. 2, pp. 89–117, 2003.
[5] T. C. Sideris, B. Thomases, and D. Wang, “Long time behavior of solutions to the 3D compressible Euler with damping,” Communications in Partial Differential Equations, vol. 28, no. 3-4, pp.
795–816, 2003.
[6] W.-A. Yong, “Entropy and global existence for hyperbolic balance laws,” Archive for Rational
Mechanics and Analysis, vol. 172, no. 2, pp. 247–266, 2004.
[7] D. Y. Fang and J. Xu, “Existence and asymptotic behavior of C 1 solutions to the multidimensional compressible Euler equations with damping,” http://arxiv.org/abs/math.AP/0703621.
[8] J. Simon, “Compact sets in the space L p (0,T;B),” Annali di Matematica Pura ed Applicata,
vol. 146, no. 1, pp. 65–96, 1987.
[9] J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and
Its Applications, The Clarendon Press, Oxford University Press, New York, NY, USA, 1998.
[10] Y. Shizuta and S. Kawashima, “Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,” Hokkaido Mathematical Journal, vol. 14, no. 2, pp.
249–275, 1985.
Jiang Xu: Department of Mathematics, Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
Email address: jiangxu 79@nuaa.edu.cn
Daoyuan Fang: Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Email address: dyf@zju.edu.cn