Initial Layer and Relaxation Limit of Non-Isentropic Compressible Euler Equations with Damping

Initial Layer and Relaxation Limit of Non-Isentropic Compressible Euler Equations with Damping

Fuzhou Wu^∗

Yau Mathematical Sciences Center, Tsinghua University Beijing 100084, China

Center of Mathematical Sciences and Applications, Harvard University Cambridge, Massachusetts 02138, USA

Abstract

In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions.

We prove that the velocity of the relaxing equations converges weakly to the velocity of the relaxed equations, while other variables of the relaxing equations converge strongly to the corresponding variables of the relaxed equations. We prove that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer;

while there is no initial layer for the well-prepared data, the convergence of the velocity is strong neart= 0. The strong convergence rates of all variables are also estimated.

Keywords: non-isentropic Euler equation with damping, relaxation limit, ill-prepared data, initial layer, strong convergence rate

1. Introduction

In this paper, we usep, u, S, %to denote the pressure, velocity, entropy and density of ideal gases respectively with the equation of gas state

%=%(p, S) := √γ¹

Ap^γ1exp{−^S_γ},

∗Corresponding author

Email address: [email protected]; [email protected](Fuzhou Wu)

whereA >0,γ= _C^Cp

V >1 are constants. Assume the constants ¯p,%,¯ S¯satisfy ¯p >

0,% >¯ 0, ¯p=A%¯^γe^S¯. Then we study the relaxation limit of the relaxing Cauchy problem for 3D non-isentropic compressible Euler equations with damping:























pt+u· ∇p+γp∇ ·u= 0, τ²ut+τ²u· ∇u+¹_%∇p+u= 0, S_t+u· ∇S= 0,

(p, u, S)(x,0) = (p₀(x, τ),^U0(x,τ)_τ , S₀(x, τ)),

(1.1)

where (p0(x, τ),U0(x, τ), S0(x, τ)) are small perturbations of (¯p,0,S). In (1.1),¯ (p, u, S, %) → (¯p,0,S,¯ %) as¯ |x| → +∞. τ is a small positive parameter repre- senting the relaxation time, letτ ∈(0,1]. The density%satisfies the equation

%t+u· ∇%+%∇ ·u= 0 by (1.1), and %0(x, τ) =%(p0(x, τ), S0(x, τ)) is small

5

perturbation of ¯%=%(¯p,S).¯

The equations (1.1) are derived from





















 ˆ

pt⁰+U · ∇pˆ+γp∇ · Uˆ = 0, Ut⁰+U · ∇U+¹_%ˆ∇pˆ+_τ¹U = 0, Sˆ_t⁰+U · ∇Sˆ= 0,

(ˆp,U,S)(x,ˆ 0) = (p₀(x, τ),U₀(x, τ), S₀(x, τ)),

(1.2)

with the time rescaling:

t=τ t⁰, u(x, t) =^U_τ(x, t⁰), p(x, t) = ˆp(x, t⁰), %(x, t) = ˆ%(x, t⁰), S(x, t) = ˆS(x, t⁰), (1.3) then (p(x, t), u(x, t), S(x, t), %(x, t)) satisfy the equations (1.1).

(1.2) has no explicit form of its relaxed system, so we study its equivalent systerm (1.1). Note that the initial velocity of (1.1) differs from that of (1.2). For both (1.1) and (1.2), the smallness of the initial data requireskU0(x, τ)kH⁴(R³)

10

to be small. Whileu(x,0) = ^U0(x,τ)_τ in (1.1) may be large whenτ is small.

Letτ = 0 in the relaxing equations (1.1), we formally obtain the following

relaxed equations























pt+u· ∇p+γp∇ ·u= 0,

1

%∇p+u= 0, St+u· ∇S= 0, (p, S)(x,0) = ( lim

τ→0p0(x, τ),lim

τ→0S0(x, τ)),

(1.4)

where%=%(p, S).

Due to its fundamental importance in both application and nonlinear PDE theory, the relaxation limit problems have been attracting much attention. We survey there some results closely related to this paper.

15

For the relaxing isothermal compressible Euler equations with damping:







%t+∇ ·(%u) = 0,

%ut+%u· ∇u+ ¯σ²∇%+¹_τ%u= 0,

(1.5)

where ¯σ² = Rθ∗ is constant. [1] proved the uniform bounds in the Sobolev spaceH^s(R^d), s∈N, s >1 +^d₂, after the time rescaling, the density%converges strongly to the solution of the heat equation in C([0, T], H^s0(B_r)), where 0 <

s⁰ < s, B_r is a ball with radius r. The results of [1] was extended in [2] to more general Sobolev space of fractional order. In [3], (1.5) was studied in one

20

dimension with BV large data away from vacuum, and it was proved in [3] that after the time rescaling,%converges strongly to the solution of the heat equation inL²(R×[0, T]) (global in space) by using the stream function.

For the relaxing isentropic compressible Euler equations with damping







%t+∇ ·(%u) = 0,

%ut+%u· ∇u+∇p+¹_τ%u= 0,

(1.6)

wherep(ρ) =Aρ^γ. [4] proved the uniform bounds in the Sobolev spaceH^s(R^d), s∈N, s >1 +^d₂, after the time rescaling, the density%converges strongly to the

25

solution of the porous media equation inC([0, T], H^s0(Br)), where 0< s⁰ < s.

Similar results were obtained in the Besov spaceB^σ_2,1(R^d), σ = 1 +^d₂ (see [5]) and in the Chemin-Lerner space (see [6]).

Relaxation limit problem also appears in Euler-Poisson equations, see [7, 8, 9, 10] for weak solutions and [11, 12] for smooth solutions. It has been

30

proved that the current density, which is the product of the electron density and electron velocity, converges weakly to that of the drift-diffusion model. If the initial data are well-prepared, the current density converges strongly to that of the drift-diffusion model (see [13]). If the initial data are ill-prepared, the authors (see [14]) proved the difference between the current density of 1D

35

hydrodynamic model and that of the drift-diffusion model decays exponentially fast in the large time interval [0,¹_βlog(_τ¹_λ)] with λ ∈ (0,1), β > 0. The key of the proof in [14] is that the solutions of the relaxing and relaxed equations converge to the corresponding stationary solutions exponentially fast while both stationary solutions are close to each other. As to the relaxation limit of weak

40

solutions (see [15]) and classical solutions (see [16, 17, 18]) to non-isentropic Euler-Poisson equations, the current density converges weakly to that of the energy-transportation model or drift-diffusion model.

However, there have been no rigorous analysis of the initial layer and strong convergence of the velocity for the ill-prepared data in the above mentioned

45

papers. A main distinction of results in this paper is that we give results on the initial layer and strong convergence of the velocity, strong convergence rates of all variables. Our main concern is the non-isentropic flow (1.1), but our results are valid for the isentropic flow (1.6) and isothermal flow (1.5) (assuming no vacuum). We show that for the ill-prepared initial data, the strong convergence

50

of the velocity is not uniform neart= 0, there exists an initial layer whose thick- ness isO(τ²). Outside the initial layer, the velocity of the relaxing equations converge strongly to that of the relaxed equations. Only for the well-prepared initial data, there is no initial layer, the strong convergence of the velocity holds in [0, T]. The key of our analysis in this paper is uniform a priori estimates

55

with respect to τ and pointwise decay of the quantity u+ ¹_%∇p. Moreover, the methods in this paper can be applied to the relaxation limit problems for Euler-Poisson equations and Euler-Maxwell equations.

In this paper, all of our results are stated for the relaxing system (1.1) and

the relaxed system (1.4). The first result is the following convergence result:

60

Theorem 1.1. Fix an arbitraryT ∈(0,+∞), suppose for allτ≥0, the initial data for the relaxing Cauchy problem(1.1)satisfyk(p0(x, τ)−¯p,U0(x, τ), S0(x, τ)−

S)k¯ H⁴(R³) ≤₀,₀ is sufficiently small, depends on T but not on τ. Then the problem(1.1) admits a unique solution(p, u, S, %)in[0, T] satisfying

P

0≤`≤2

(∂_t^{`</sup>(p−p), τ ∂¯ <sup>`}_tu, ∂^{`</sup><sub>t</sub>(S−S), ∂¯ <sub>t</sub><sup>`}(%−%))¯

_{L∞([0,T],H4−`(}

R³))

+ P

0≤`≤2

u

_{H`([0,T],H4−`(}

R³))≤C(T, ₀),

(1.7)

such that asτ →0,

(p, S, %)→(˜p,S,˜ %)˜ in C([0, T], C^2+µ1(K)∩W^3,µ2(K)), u *u˜ in ∩

0≤`≤2H<sup>`</sup>([0, T], H^4−`(R³)),

(1.8)

whereµ₁∈[0,¹₂), µ₂∈[2,6),Kdenotes any compact subset ofR³,(˜p,u,˜ S,˜ %)˜ is the unique classical solution to the relaxed equations(1.4).

Assume k(p₀(x, τ)−lim

τ→0p₀(x, τ), S₀(x, τ)−lim

τ→0S₀(x, τ))k_H3(R³)≤O(τ^α1), then asτ→0,

k(p−p, S˜ −S, %˜ −%)k˜ _C([0,T],C1+µ1(R³)∩W^2,µ2(R³))≤O(τ^min{1,α1}), (1.9) whereµ1∈[0,¹₂], µ2∈[2,6].

The main results concerned with the initial layer and strong convergence of the velocity are stated in the following theorem:

65

Theorem 1.2. Let(p, u, S, %)and(˜p,u,˜ S,˜ %)˜ be the solutions obtained in Theo- rem1.1. For the ill-prepared data, i.e., lim

τ→0

1

τU₀(x, τ) +_% ¹

0(x,τ)∇p₀(x, τ) _∞6= 0, there exists an initial layer [0, t^∗] with t^∗ = Cτ^2−δ for the velocity u, where C >0,0< δ <2, such that asτ →0,|u(x, t^∗)−u(x,˜ 0)|_∞→0 and

ku−uk˜ _C([t∗,T], C^0+µ1(R³)∩W^1,µ2(R³))≤O(τ^min{1,α1}), µ1∈[0,¹₂], µ2∈[2,6].

(1.10) Ifδ= 0, for any constantC >0,u(x, Cτ²)does not converge to u(x,˜ 0).

For the well-prepared data, i.e., lim

τ→0

1

τU0(x, τ) +_% ¹

0(x,τ)∇p0(x, τ) _H2(

R³)

= 0, assuming

1

τU0(x, τ) +_% ¹

0(x,τ)∇p0(x, τ) _H2(

R³)≤O(τ^α2), asτ →0, ku−uk˜ _C([0,T],C0+µ1(R³)∩W^1,µ2(R³))≤O(τ^{min{1,α1,α2}}), µ1∈[0,¹₂], µ2∈[2,6].

(1.11) Remark 1.3. (i) In the above theorems,(p, u, S, %)depend onτ, while(˜p,u,˜ S,˜ %)˜ do not. Compare (1.8) with (1.9),(1.10),(1.11), the strong convergence with such higher regularities in (1.8) must be restricted to arbitrary compact subset K. The strong convergence in (1.9),(1.10),(1.11) holds in the whole space R³,

70

their proofs do not need compact embedding.

(ii) In Theorem 1.2 for ill-prepared data, δ 6= 0 implies that the thickness of the initial layer isO(τ²). The strong convergence ofu is not uniform near t= 0, u|t=0 6= ˜u|t=0, u(x, Ct²) does not converge to u(x,˜ 0), while u(x, Ct^2−δ) converges to u(x,˜ 0) when δ >0. Simply, for any fixed small number t_∗ >0,

75

u(x, t)→u(x, t)˜ inC([t_∗, T], C^0+µ1(R³)∩W^1,µ2(R³)),µ₁∈[0,¹₂], µ₂∈[2,6].

(iii)(p, S, %)converge strongly to (˜p,S,˜ %)˜ in[0, T], but for ill-prepared data, (pt, St, %t) may not converge strongly to (˜pt,S˜t,%˜t) near t = 0, they may have an initial layer (see (5.3)). In this paper, we do not have enough regularity of

∇ ·(v−˜v)to prove the initial layer and strong convergence ofpt, St, %t.

80

(iv) If one replaces the initial data(p0(x, τ),¹_τU0(x, τ), S0(x, τ))in(1.1)with (p0(x),_τ¹U0(x), S0(x)), where(p0(x),U0(x), S0(x))are independent ofτ, then for the well-prepared data,p0(x)≡constand U0(x)≡0 (equilibrium states); while for the ill-prepared data,p0(x)6=constor U0(x)6= 0 (non-equilibrium states).

In the following, we give more comments on Theorem 1.1 and Theorem 1.2.

85

If τ > 0 is fixed, the global existence of classical solutions to the equations (1.2) is proved in [19, 20, 21]. However, for the relaxation limit problem in this paper,τ >0 is variant and approaches 0, so we need the uniform existence of the solutions and uniform regularities (1.7) which are different from [19, 20, 21]. The uniform a priori estimates with respect to τ produce the convergence results.

90

In order to have enough regularities to treat the initial layer of the velocity,

(p0(x, τ)−p,¯ U0(x, τ), S0(x, τ)−S) are required to be in¯ H⁴(R³) for all τ≥0.

The uniform a priori estimates for the equations (1.1) imply the uniform regularities (1.7). Passing to the limit, we have the convergence results (1.8).

Let us give some comments and remarks on the initial layer as follows. The asymptotic expansions of the solutions to (1.1) give us some indication of the initial layer. We illustrate this as follows: assume that the initial data have asymptotic expansion

(p(x,0), u(x,0), S(x,0), %(x,0)) = P

m≥0

τ^2m(p^m₀ , u^m₀, S₀^m, %^m₀), and solutions of the equations (1.1) have the asymptotic expansion

(p(x, t), u(x, t), S(x, t), %(x, t)) = P

m≥0

τ^2m(p^m(x, t), u^m(x, t), S^m(x, t), %^m(x, t)), then the leading order profiles satisfy the equations





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















∂tp⁰+u⁰· ∇p⁰+γp⁰∇ ·u⁰= 0, u⁰+_%¹

0∇p⁰= 0,

∂_tS⁰+u⁰· ∇S⁰= 0, (p⁰, S⁰)(x,0) = (p⁰₀, S₀⁰),

(1.12)

if the initial velocity is well-prepared, i.e.,u⁰₀=−_%¹0

0∇p⁰₀,where %⁰=%(p⁰, S⁰).

95

Note that in this case,u⁰+_%¹₀∇p⁰= 0 matchesu⁰₀+_%¹0 0

∇p⁰₀= 0.

However, if the initial velocity is ill-prepared, i. e., u⁰₀ 6=−_%¹0 0

∇p⁰₀, we may assume that the velocity has the asymptotic expansion

u(x, t) = X

m≥0

τ^2m(u^m(x, t) + ˆu^m(x, z)), z= t τ²,

then the leading order profile of the initial layer correction ˆusatisfies the equa-

tion 





∂_zuˆ⁰+ ˆu⁰= 0, ˆ

u⁰(x,0) =u⁰₀+_%¹0 0

∇p⁰₀.

(1.13)

Then ˆu(x, z) = (u⁰₀+_%¹0 0

∇p⁰₀)e^−z.Thus, the difference betweenu⁰and−_%¹0∇p⁰ decays exponentially within the initial layer.

The above arguments of (1.12) and (1.13) indicate the relationship between the existence of initial layer and a class of initial data on a formal level. We are

100

not concerned with the asymptotic expansions in this paper (as to asymptotic expansion analysis in relaxation limit problem for Euler-Poisson equations, see [22, 23, 24]; for Euler-Maxwell equations, see [25, 26]) and only focus on rigorous analysis of the initial layer and relaxation limit of the relaxing equations (1.1).

The relaxation limit is a singular limit, since (p,−¹_%∇p, S, %) converge strong- ly to (˜p,u,˜ S,˜ %), instead of˜ u→u. In order to measure the difference between˜ uand−¹_%∇p, we introduce a quantity:

η=u+¹_%∇p . (1.14)

The pointwise decay of η outside the initial layer is the key to the strong convergence of the velocity. η satisfies the following transport equations with damping and forcing

η_t+u· ∇η+ 1

τ²η=forcing terms, (1.15) whereτ·[forcing terms] is bounded uniformly with respect to τ, the damping

105

effect becomes stronger asτ decreases.

Also,η satisfies another equation:

η=τ²(u_t+u· ∇u). (1.16) Then the equations (1.15) and (1.16) produce the following estimates respec- tively:











|η|²_∞≤Ckηk²_H2(R³)≤Ckη|t=0k²_H2(R³)exp{−_τ^t2}+Cτ²,

T

R

0

kηtk²_H1(R³)dx≤Cτ².

(1.17)

Therefore, for the well-prepared data, lim

τ→0η(x, t) = 0 for anyt∈[0, T], there is no discrepancy between η(x, t)|t>0 and η(x)|t=0 = u0+_%¹

0∇p0, thus there is no initial layer, u → u˜ strongly in C([0, T], C^0+µ1(R³)∩W^1,µ2(R³)), µ1 ∈ [0,¹₂], µ2∈[2,6].

110

While for the ill-prepared data, lim

τ→0η(x)|t=0 6= 0, while lim

τ→0η(x, t_∗) → 0 pointwisely for any fixed small numbert∗ >0, thus the discrepancy between η|t=0 and η|t=t∗ make the initial layer of the velocity exist. Within [0, t∗], η must decreases rapidly, ¹_%∇pis uniform bounded, thenuchanges dramatically.

In [t_∗, T], η →0 strongly, u→u˜ strongly. Near t= 0, the behavior ofη is not

115

uniform with respect toτ,η(x, Cτ²) does not converge to 0, while η(x, Cτ^2−δ) converges to 0 whenδ >0. Correspondingly,u|_t=06= ˜u|_t=0,u(x, Cτ²) does not converge to ˜u(x,0), whileu(x, Cτ^2−δ) converges to ˜u(x,0) whenδ >0.

Finally, in order to prove that the thickness of the initial layer isO(τ²), we have the following equation ofη(x, z) by rescaling the time variablez= _τ^t2:

∂zη(x, z) +u(x, z)· ∇η(x, z) +η(x, z) =new forcing terms, (1.18) where ¹_τ·u(x, z) and¹_τ·[new forcing terms] are bounded uniformly with respect toτ. (1.18) produces the estimate kη(x, z)kL²(R³)>0 when τ is small.

120

The uniform a priori estimates for the relaxing equations are hard to obtain for the bounded domain Ω with fixed boundary∂Ω, due to the characteristic boundary conditionu·n|_∂Ω= 0. However, our results can be extended with- out difficulties to the periodic domains T³ due to the convenience of periodic boundary conditions. The results are the same except that R³ and K in the

125

theorems need to be replaced byT³, there is nothing new in methodology.

The rest of this paper is organized as follows: In Section 2, we reformulate the equations into appropriate forms and derive the equations ofη. In Section 3, we prove uniform a priori estimates for the relaxing equations. In Section 4, we prove the uniqueness of the relaxed equations and the relaxation limit of the

130

the relaxing equations. In Section 5, we estimate the strong convergence rates of the pressure, entropy and density. In Section 6, we study the initial layer and strong convergence of the velocity, and then we prove the thickness of the initial layer for the ill-prepared data.

2. Preliminaries

135

In this section, we will reformulate the equations (1.1) into appropriate form- s, define the energy functionals and derive the equations of the quantityη.

For the relaxing equations (1.1) together with their initial data (p0, u0, S0, %0) and constants ¯p,S,¯ %, we introduce the constants:¯

k₁=q

1

γ%¯¯p, k₂=q_γp¯

¯

%, define the variables:

ξ=p−p,¯ v= _k¹

1u, φ=S−S,¯ ζ=%−%,¯ where (ξ, v, φ, ζ)→(0,0,0,0) as|x| →+∞.

In order to prove the uniform a priori estimates, similar to the symmetriza- tion used in [27, 20], we symmetrize the equations (1.1) into the following form:

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



















ξ_t+k₂∇ ·v=−γk₁ξ∇ ·v−k₁v· ∇ξ, τ²vt+k2∇ξ+v=−k1τ²v· ∇v+_k¹

1(¹_%¯−¹_%)∇ξ, φt=−k1v· ∇φ,

(ξ, v, φ)(x,0) = (p0(x, τ)−p,¯ _k¹

1τU0(x, τ), S0(x, τ)−S),¯

(2.1)

where%=ζ+ ¯%=%(ξ+ ¯p, φ+ ¯S) and ζ satisfies the equationζt+k1v· ∇ζ+ k₁%∇ ·v= 0 by the equation of gas state.

140

Letτ = 0 in the relaxing equations (2.1), we formally obtained the following relaxed equations, which are equivalent to the relaxed equations (1.4).























ξ_t+k₂∇ ·v=−γk₁ξ∇ ·v−k₁v· ∇ξ, k2∇ξ+v=_k¹

1(¹_%¯−¹_%)∇ξ, φt=−k1v· ∇φ,

(ξ, φ)(x,0) = ( lim

τ→0p0(x, τ)−p,¯ lim

τ→0S0(x, τ)−S),¯

(2.2)

where%=ζ+ ¯%=%(ξ+ ¯p, φ+ ¯S).

In order to use the energy method to derive uniform a priori estimates with respect toτ, we define the following generic energy functionalE[ξ](t) and three specific energy functionalsEX[ξ](t),E1[ξ](t),E2[v](t):

Definition 2.1. Define E[ξ](t) := P

0≤`≤2,0≤`+|α|≤4

k∂_t^`D^αξ(t)k²_L2(R³),

E_X[ξ](t) := P

0≤`≤2,0<`+|α|≤4

k∂_t^`D^αξk²_L2(R³),

E1[ξ](t) :=E[ξ](t)− P

0≤`≤2,`+|α|=4

R

R³ ξ

p(∂^`_tD^αξ)²dx, E2[v](t) :=E[v](t) + P

0≤`≤2,`+|α|=4

R

R³

(^%_%¯−1)|∂_t^`D^αv|²dx.

(2.3)

Moreover, we use the following two notations: E[ξ, v](t) :=E[ξ](t) +E[v](t),

145

E[ξ, v, φ, ζ](t) :=E[ξ, v](t) +E[φ](t) +E[ζ](t).

It is easy to get the following lemma states that E[ξ](t) and E1[ξ](t) are equivalent,E[v](t) andE2[v](t) are equivalent.

Lemma 2.2. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where0< 1, then there existc1>0, c2>0such that for anyt∈[0, T], c₁E[ξ](t)≤ E1[ξ](t)≤c₂E[ξ](t), c₁E[v](t)≤ E2[v](t)≤c₂E[v](t). (2.4) Remark 2.3. Different from the symmetrization(2.1), there is an straight way to symmetrize(1.1), which is















ξt+γk1p∇ ·v=−k1v· ∇ξ, τ²vt+v+_k¹

1%∇ξ=−τ²k1v· ∇v, φ_t+k₁v· ∇φ= 0,

(2.5)

with the weighted energy functionals:

E_ξ[ξ] :=P

`,α

R

R³

¯ p

p(∂_t^`∂_x^αξ)², E_v[v] :=P

`,α

R

R³ ρ

¯

ρ(∂_t^`∂_x^αv)².

However, if we use (2.5) to prove uniform a priori estimates, the uniqueness and convergence rates, we have to introduce many different weighted energy

150

functionals into this paper. So we still use(2.1) because (2.1) makes the proofs in this paper a little easier.

Finally, we define the quantity ηprecisely and derive its equations. Define η(x, t) =v(x, t) +_k ¹

1%(x,t)∇ξ(x, t). (2.6)

Note that hereη = _k¹

1(u+¹_%∇p), which differs fromη introduced in the intro- duction (see (1.14)) up to a constant coefficient _k¹

1. Actually, the coefficient can be any positive number,ηdefined in (2.6) makes our calculation easier, whileη

155

introduced in (1.14) is convenient to represent our ideas.

Differentiate (2.6) wit respect tot, we have ηt=vt+_k¹

1%∇ξt−_k^ζt

1%²∇ξ,

then insert the evolution equations ofv, ξ, ζinto the above equation, we get the following transport equation with damping and forcing forη:

ηt+k1v· ∇η+_τ¹2η=−¹_%(∇v)∇ξ−^γ−1_% ∇ξ∇ ·v−^γp_%∇(∇ ·v), (2.7) where (∇v) is a matrix,τ·[R.H.S. of(2.7)] is bounded uniformly with respect toτ, ’R.H.S.’ is the abbreviation for ’right hand side’.

Besides the equation (2.7), η satisfies another equation:

η=τ²(vt+k1v· ∇v), (2.8) In the rest of this paper, we will use the following notations: X .Y denotes the estimateX ≤CY for some implied constantC >0 which may be different

160

line by line. [A, B] is the commutator ofAandB.

3. Uniform A Priori Estimates for the Relaxing Equations

In this section, we derive unform a priori estimates for the relaxing Cauchy problem (2.1). In order to discuss the initial layer and strong convergence of the velocity, we need to estimate the higher order time derivatives.

165

The following lemma gives uniform a priori estimate forE1[ξ](t) +τ²E2[v](t), which is equivalent toE[ξ](t) +τ²E[v](t).

Lemma 3.1. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where0< 1, then there exists a constantC1>0such that for anyt∈[0, T], d

dtE1[ξ](t) +τ²d

dtE2[v](t) + 2E2[v](t)≤C1

√(EX[ξ](t) +E[v](t)). (3.1)

Proof. Let ξ·(2.1)1+v·(2.1)2, integrate in R³, note that R

R³

∇ ·(ξv) dx= 0, then we get

d dt

R

R³

|ξ|²+τ²|v|²dx+ 2R

R³

|v|²dx

= R

R³

2γk1v· ∇(ξ²)−2k1ξv· ∇ξ−2k1τ²v· ∇v·v+_k²

1(¹_%¯−¹_%)∇ξ·vdx .√

k∇ξkL²(R³)kvkL²(R³)+√

kvk²_L2(R³).√

(EX[ξ](t) +E[v](t)).

(3.2) Let∂_t^{`</sup>D<sup>α</sup>ξ·∂<sub>t</sub><sup>`}D^α(2.1)₁+∂^{`</sup><sub>t</sub>D<sup>α</sup>v·∂<sub>t</sub><sup>`}D^α(2.1)₂, where 0≤`≤2,1≤`+|α| ≤4, integrate inR³, note that R

R³

∇ ·(∂_t^{`</sup>D<sup>α</sup>ξ∂<sub>t</sub><sup>`}D^αv) dx= 0, then we get

d dt

R

R³

|∂_t^{`</sup>D<sup>α</sup>ξ|<sup>2</sup>+τ<sup>2</sup>|∂<sub>t</sub><sup>`}D^αv|²dx+ 2R

R³

|∂_t^`D^αv|²dx

= R

R³

−2γk1(∂_t^{`</sup>D<sup>α</sup>ξ)∂<sub>t</sub><sup>`}D^α(ξ∇ ·v)−2k1(∂^{`</sup><sub>t</sub>D<sup>α</sup>ξ)∂<sup>`}_tD^α(v· ∇ξ)

−2k1τ²(∂_t^{`</sup>D<sup>α</sup>v)·∂<sub>t</sub><sup>`}D^α(v· ∇v) +_k²

1(∂_t^{`</sup>D<sup>α</sup>v)·∂<sub>t</sub><sup>`}D^α[(¹_%¯−¹_%)∇ξ] dx:=I₁. (3.3) When 1≤`+|α| ≤3, it is easy to check that I1.√

(EX[ξ](t) +E[v](t)).

When ` +|α| = 4, we estimate the quantity I1 − _dt^d R

R³ ξ

p(∂_t^`D^αξ)²dx+

τ^{2 d}_dtR

R³

(^%_%¯−1)|∂_t^`D^αv|²dx, then

I₁−_dt^d R

R³ ξ

p(∂_t^`D^αξ)²dx+τ^{2 d}_dtR

R³

(^%_%¯−1)|∂_t^`D^αv|²dx

=−2γk1

R

R³

(∂^{`</sup><sub>t</sub>D<sup>α</sup>ξ)ξ∇ ·(∂<sub>t</sub><sup>`}D^αv) dx−2k1

R

R³

(∂_t^{`</sup>D<sup>α</sup>ξ)v· ∇(∂<sup>`}_tD^αξ) dx

−2k1τ²R

R³

v· ∇(∂_t^{`</sup>D<sup>α</sup>v)·(∂<sub>t</sub><sup>`}D^αv) dx+_k²

1

R

R³

(¹_%¯−_%¹)(∂_t^{`</sup>D<sup>α</sup>v)· ∇(∂<sup>`}_tD^αξ) dx +2τ²R

R³

(^%_%¯−1)(∂_t^{`</sup>D<sup>α</sup>v)·(∂<sub>t</sub><sup>`}D^αv_t) dx+τ²R

R³ ζt

¯

%|∂_t^`D^αv|²dx

−2R

R³ ξ

p(∂_t^{`</sup>D<sup>α</sup>ξ)(∂<sub>t</sub><sup>`}D^αξ_t) dx−R

R³

∂_t(^ξ_p)(∂_t^`D^αξ)²dx .√

(E_X[ξ](t) +E[v](t))−2R

R³ ξ

p(∂_t^{`</sup>D<sup>α</sup>ξ)[∂<sub>t</sub><sup>`}D^αξ_t+k₁γp∇ ·(∂_t^`D^αv)] dx +_k²

1

R

R³

(¹_%¯−¹_%)(∂^{`</sup><sub>t</sub>D<sup>α</sup>v)·[∇(∂<sub>t</sub><sup>`}D^αξ) +k1τ²%(∂^`_tD^αvt)] dx.

(3.4) Apply∂_t<sup>`</sup>D<sup>α</sup> to (2.1)1, where 0≤`≤2, `+|α|= 4, we get

∂_t^{`</sup>D<sup>α</sup>ξt+k1γp∇ ·(∂<sup>`}_tD^αv) =−k1∂_t^`D^α(v· ∇ξ)

−k1γ P

`₁+|α₁|>0

∂_t^{`1</sup>D<sup>α1</sup>ξ∇ ·(∂<sub>t</sub><sup>`2}D^α2v).

(3.5)

Plug (3.5) into the following integral, we get R

R³ ξ

p(∂_t^{`</sup>D<sup>α</sup>ξ)(∂<sub>t</sub><sup>`}D^αξt+k1γp∇ ·(∂_t^`D^αv)) dx=R

R³ ξ

p(∂_t^`D^αξ)[R.H.S. of (3.5)] dx .√

(EX[ξ](t) +E[v](t)) +^k₂¹ R

R³

|∂_t^`D^αξ|²∇ ·(^ξ_pv) dx .√

(EX[ξ](t) +E[v](t)).

(3.6) Apply∂<sup>`</sup><sub>t</sub>D<sup>α</sup>tok1τ<sup>2</sup>vt+k<sup>2</sup><sub>1</sub>τ<sup>2</sup>v·∇v+k1v+<sup>1</sup><sub>%</sub>∇ξ= 0, where 0≤`≤2, `+|α|= 4, we get

1

%∇(∂_t^{`</sup>D<sup>α</sup>ξ) +k1τ<sup>2</sup>(∂<sub>t</sub><sup>`}D^αvt)

=−k1∂_t^{`</sup>D<sup>α</sup>v−k<sup>2</sup><sub>1</sub>τ<sup>2</sup>∂<sub>t</sub><sup>`}D^α(v· ∇v)− P

`1+|α1|>0

∂_t^{`1</sup>D<sup>α1</sup>(<sup>1</sup><sub>%</sub>)∂<sub>t</sub><sup>`2}D^α2∇ξ.

(3.7)

Plug (3.7) into the following integral, we get R

R³

(¹_%¯−_%¹)(∂_t^{`</sup>D<sup>α</sup>v)·[∇(∂<sub>t</sub><sup>`}D^αξ) +k₁τ²%∂_t^`D^αv_t] dx

= R

R³

(¹_%¯−¹_%)(∂_t^`D^αv)·%[R.H.S. of (3.7)] dx . ^k122^τ

R

R³

|∂_t^`D^αv|²∇ ·[(^%_%¯−1)τ v] dx+√

(E_X[ξ](t) +E[v](t)) .√

(EX[ξ](t) +E[v](t)).

(3.8)

Plug (3.6) and (3.8) into (3.4), we get I₁−_dt^d R

R³ ξ

p(∂_t^`D^αξ)²dx+τ^{2 d}_dtR

R³

(^%_%¯−1)|∂_t^`D^αv|²dx.√

(EX[ξ](t) +E[v](t)).

(3.9) After Summing `andα, we have

d dt

P

0≤`≤2,1≤`+|α|≤4

R

R³

|∂_t^`D^αξ|²dx− P

0≤`≤2,`+|α|=4

R

R³ ξ

p(∂_t^`D^αξ)²dx +τ^{2 d}_dt

P

0≤`≤2,1≤`+|α|≤4

R

R³

|∂_t^`D^αv|²dx+ P

0≤`≤2,`+|α|=4

R

R³

(^%_%¯−1)|∂tD^αv|²dx +2

P

0≤`≤2,1≤`+|α|≤4

R

R³

|∂_t^`D^αv|²dx+ P

0≤`≤2,`+|α|=4

R

R³

(^%_%¯−1)|∂^`_tD^αv|²dx

.√

(EX[ξ](t) +E[v](t)).

(3.10) By (3.2) + (3.10), we proved that there exists a constantC₁>0 such that

d

dtE1[ξ](t) +τ²d

dtE2[v](t) + 2E2[v](t)≤C1

√(EX[ξ](t) +E[v](t)). (3.11)

Thus, Lemma 3.1 is proved.

The structure of the equations (2.1) implies EX[ξ](t) can be estimates by E[v](t), as the following lemma stated:

170

Lemma 3.2. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where0< 1, then there exists a constantc₃>0such that for anyt∈[0, T],

EX[ξ](t)≤c3E[v](t). (3.12)

Proof. Apply D^αto ∇ξ=−k1τ²%vt−k²₁τ²%v· ∇v−k1%v, where 0≤ |α| ≤3, we get

kD^α∇ξk²_L2(R³).τ⁴kD^α(%v_t)k²_L2(R³)+τ⁴kD^α(%v· ∇v)k²_L2(R³)+kD^α(%v)k²_L2(R³)

.kD^αvtk²_L2(R³)+kD^αvk²_L2(R³)+E[ζ]E[v](t) +E[τ v](t)E[v](t) +E[τ v](t)E[ζ]E[v](t).

(3.13) ApplyD^αto ξt=−k1v· ∇ξ−k1γp∇ ·v, where 0≤ |α| ≤3, we get

kD^αξtk²_L2(R³).kD^α(v· ∇ξ)k²_L2(R³)+kD^α(p∇ ·v)k²_L2(R³)

.kD^α∇ ·vk²_L2(R³)+EX[ξ]E[v](t).

(3.14) Similar to the above estimate, apply∂tD^α toξt, where 0≤ |α| ≤2, we get

kD^αξttk²_L2(R³).kD^α∇ ·vtk²_L2(R³)+EX[ξ]E[v](t). (3.15) By (3.13) + (3.14) + (3.15), we have

EX[ξ](t)≤C2E[v](t) +C2E[ζ]E[v](t) +C2E[τ v](t)E[v](t) +C2E[τ v](t)E[ζ]E[v](t) +C2EX[ξ]E[v](t)

≤C₂E_X[ξ]E[v](t) +C₂(1 + 2+²)E[v](t),

(3.16)

for someC₂>0.

Assume is so small that C₂E[v](t)≤ ¹₂. Let c₃= 2C₂(1 + 2+²), we get E_X[ξ](t)≤c₃E[v](t). Thus, Lemma 3.2 is proved.

Based on the above a priori estimates, we prove not only the uniformL^∞ bound ofE[ξ, τ v](t), but also the uniform bound of

T

R

0

E[v](s) ds.

175

Lemma 3.3. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where0< 1, then there exists a constantc4>0such that for anyt∈[0, T], E[ξ, τ v](t)≤c4k(ξ0,U0)k²_H4(R³),

T

R

0

E[v](s) ds≤c₄k(ξ0,U0)k²_H4(R³).

(3.17)

Proof. Since Lemmas 3.1 and 3.2 are proved, plug (3.12) into (3.1), then we get

d

dtE1[ξ](t) +τ^{2 d}_dtE2[v](t) + 2E2[v](t)≤C1(1 +c3)√ E[v](t)

≤ ^C1(1+c_c ³⁾

1

√E2[v](t).

(3.18)

Sinceis sufficiently small, we assume≤ _C2 ^c21

1(1+c3)², then,

d

dtE1[ξ](t) +τ^{2 d}_dtE2[v](t) +E2[v](t)≤0. (3.19) By using Lemma 2.2, it is easy to get

E[ξ, τ v](t) +

t

R

0

E[v](s) ds≤c4k(ξ0,U0)k²_H4(R³), (3.20) wherec₄= ^C_c³

1. Thus, Lemma 3.3 is proved.

The following lemma concerns the uniform bound ofE[φ](t). Here, the finite- ness ofT plays a key role in the proof.

Lemma 3.4. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where 0 < 1, then there are constants c5 > 0, c6 > 0 such that for any t∈[0, T],

E[φ](t)≤c5kφ0k²_H4(R³)exp{c6Tk(ξ0,U0)k²_H4(R³)}. (3.21) Proof. Let∂_t^{`</sup>D<sup>α</sup>φ·∂<sub>t</sub><sup>`}D^α(2.1)3, where 0≤`≤2,0≤`+|α| ≤4, we get

(|∂_t^{`</sup>D<sup>α</sup>φ|<sup>2</sup>)<sub>t</sub>=−v· ∇|∂<sub>t</sub><sup>`}D^αφ|²−2∂_t^{`</sup>D<sup>α</sup>φ[∂<sub>t</sub><sup>`}D^α, v· ∇]φ. (3.22) Integrate (3.22) in R³, we have

d dt

R

R³

|∂_t^`D^αφ|²dx= R

R³

|∂_t^`D^αφ|²∇ ·vdx−2R

R³

∂_t^{`</sup>D<sup>α</sup>φ[∂<sub>t</sub><sup>`}D^α, v· ∇]φdx:=I₂. (3.23) When`+|α| ≤4, it is easy to check I2.E[v](t)<sup>12</sup>E[φ](t) by using commu- tator estimates. Sum`, α, we have

d

dtE[φ](t)≤C₄E[v](t)¹²E[φ](t). (3.24)

Integrate (3.24) in (0, t), where t ∈ [0, T], we obtain the uniform a priori estimate forE[φ](t) which is independent ofτ.

E[φ](t)≤c₅kφ0k²_H4(R³)exp{

t

R

0

C₄E[v](s)¹²ds}

≤c5kφ0k²_H4(R³)exp{C4T

T

R

0

E[v](s) ds}

≤c₅kφ0k²_H4(R³)exp{c6Tk(ξ0,U0)k²_H4(R³)},

(3.25)

wherec5>0, c6=C4c4>0. Thus, Lemma 3.4 is proved.

Due to ζ = %(ξ+ ¯p, φ+ ¯S)−%, we can estimate¯ E[ζ](t) in the following

180

lemma:

Lemma 3.5. For any fixedT ∈(0,+∞), τ ∈[0,1], if sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤,

where 0 < 1, then there are constants c7 > 0, c8 > 0 such that for any t∈[0, T],

E[ζ](t)≤c7kφ0k²_H4(R³)exp{c6Tk(ξ0,U0)k²_H4(R³)}+c8k(ξ0,U0)k²_H4(R³). (3.26) Proof. Sinceζ is expressed in terms ofξand φ, namely,

ζ= √γ¹

Aexp{−^S_γ^¯}h

(ξ+ ¯p)^1γexp{−^φ_γ} −p¯^1γi

, (3.27)

it is easy to get the tame estimate of composed functions (see [28]):

E[ζ](t)≤C5E[ξ](t) +C5E[φ](t)

≤c7kφ0k²_H4(R³)exp{c6Tk(ξ0,U0)k²_H4(R³)}+c8k(ξ0,U0)k²_H4(R³),

(3.28)

wherec7=c5C5>0, c8=c4C5>0. Thus, Lemma 3.5 is proved.

Lemmas 3.3,3.4,3.5 imply that E[ξ, τ v, φ, ζ](t) is bounded by fixed T and the initial data, i.e. k(ξ0,U0)kH⁴(R³),kφ0kH⁴(R³). Thus, as long as₀, which is the bound of the initial data and depends onT, is chosen to be small enough,

185

the a priori assumption sup

0≤t≤T

E[ξ, τ v, φ, ζ](t)≤will be valid.