非線性波之交互作用及其應用

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行政院國家科學委員會專題研究計畫成果報告

非線性波之交互作用及其應用

計畫類別：個別型計畫

計畫編號： NSC92-2115-M-006-007-

執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日

執行單位：國立成功大學數學系暨應用數學所

計畫主持人：連文璟

報告類型：精簡報告

處理方式：本計畫可公開查詢

中華民國 93 年 8 月 13 日

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There are two directions in the current research. The first part of this report is a finished paper about the Boltzmann shock, which is already submitted. The second part is about the application of Green’s functions in some important physical models. Since it is in progress and not complete yet, this part will be reported in the future.

I

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PRESENCE OF BOUNDARY

WEN-CHING LIEN AND SHIH-HSIEN YU

Abstract. We study the time-asymptotic behavior of the Boltzmann shock layers with a given physical boundary of a half-space for the hard sphere collision model. As boundary conditions, we prescribe a Maxwellian at the far field and require a specular reflection at x = 0. When the macroscopic velocity at the far field is negative, we prove that if the initial data are suitably chosen, then a solution exists globally in time and tends toward the corresponding outgoing Boltzmann shock profile. The proof is essentially based on the macro-micro decomposition of solutions and the elementary energy methods.

Contents

1. Introduction 1

2. Preliminary 5

2.1. Macro-Micro Decomposition 5

2.2. Shock Profile 6

3. Construction of the Approximate Solution 7

3.1. The Macroscopic and Microscopic Equations 7

3.2. Reference Macro-Micro Decomposition 10

3.3. Local Existence in Time 10

4. Basic Estimates 11

4.1. Properties of the Macroscopic Variables 11

4.2. Matrix Representation 11

4.3. Estimates on the Eigenvalues for the Navier-Stokes Shock Profile 12

5. Stability Analysis 13

5.1. Lower Order Energy Estimates 14

5.2. Transversal Wave Estimates 19

5.3. Higher Order Energy Estimates 23

Appendix A. Estimates on Collision Operators 28

References 28

1. Introduction

In this paper we are interested in the time-asymptotic behaviors of the Boltzmann shock layers with a given physical boundary of a half space. We study the planar wave propagation on the half space x ∈ R⁺ with the following initial and boundary conditions:

Ft+ ξ1Fx= Q(F, F ), (x, t, ξ) ∈ R⁺× R⁺× R³ (1.1)

x→∞lim F (x, 0, ξ) = ρ+ e⁻

|ξ−u+|2

p 2T+

(2πT+)³ (1.2)

F (0, t, ξ) = F (0, t, −ξ).

(1.3)

ρ+(> 0), u+ = (u+1, 0, 0)(u+1< 0), and T+(> 0) are the macroscopic density, the macroscopic velocity, and the temperature respectively in the thermal equilibrium at the far field. Here the specific gas constant is assumed to be one. The collision operator, Q, is defined for hard spheres and the boundary condition is a specular reflection for this problem. Since u+1 < 0, the solution is expected to tend toward an outgoing Boltzmann shock profile.

Motivated by the boundary condition, we construct the approximate solution of (1.1)- (1.3) by superposing two travelling shock wave solutions moving in the opposite directions with the same speed. The aim of this paper is to discuss the time-asymptotic convergence of the solution to (1.1)-(1.3) toward the outgoing shock profile.

1

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There is extensive literature on the initial boundary value problems for the Boltzmann equation initiated by Cercignani. Existence, uniqueness and properties of asymptotic behavior are proved for solutions of the Milne and Kramers problems, which are to solve the linearized Boltzmann equation in a half space x > 0. Such studies on stationary solutions have been pursued analytically by [BCN], [CGS], [GPS], [UYY1]. The Milne problem has also been studied by asymptotic expansions for the condensation and evaporation by Sone et al [ASY],[S].

The result of this paper is new in two aspects. First, the present work treats the time-dependent solution of the initial boundary value problem and the solution tends to a non-stationary nonlinear wave pattern, in contrast to previous works on stationary solutions. Secondly, according to the construction of the approximate solution, this problem can also be viewed as the study of the interaction of two shocks. Such an approach is absent in the previous works, even in the field of viscous conservation laws.

The shock profile solutions of the Boltzmann equation are first constructed by [CN], where the existence and uniqueness of weak plane shocks are obtained by using a projection method similar to Lyapunov-Schmidt method, but the positivity property of the Boltzmann shock profile cannot be concluded from this approach. The time- asymptotic stability of the Maxwellian states has been shown by energy methods based on Fourier transform and spectral analysis [Ni, Uk1, Uk2, KMN]. Furthermore, the time-asymptotic stability and positivity of shock profiles are obtained in [LY] recently. In [LY], they introduce an elementary energy method based on a macro- micro decomposition of the equation into macroscopic and microscopic components to analyze the time-asymptotic stability of nonlinear waves. The decomposition effectively describes the Boltzamnn dynamics so that the methods of analyzing viscous conservation laws can be implemented with small modifications. The positivity of a Boltzmann shock layer is a consequence of the facts that the nonlinear collision operator Q is in a form suitable for the maximum principle and that a Boltzmann shock layer is time-asymptotically stable.

In the present paper, with the macro-micro decomposition we can regard the initial boundary value problem as a time-asymptotic stability problem, which solution tends toward the nonlinear wave pattern, a superposition of two Boltzmann shock layers.

In order to make this paper self-contained, we now consider the perturbation of a global Maxwellian state to illustrate the energy method by the macro-micro decomposition, though it repeats some contents in [LY].

For the sake of transparency, we express the perturbation as

F (x, t, ξ) = ω(ξ; u0, t0) + f (x, t, ξ), where

ω(ξ; u0, T0) ≡ e⁻^|ξ−u0|

2

p 2T0

(2πT0)³.

The macro-micro decomposition is made with respect to ω as follows. The collision invariants, χi, i = 0, · · · , 4, are normalized with respect to ω: 









χ0(ξ) ≡ 1 χi(ξ) ≡ ^ξ^√ⁱ^−u_T₀ⁱ⁰ χ4(ξ) ≡^√¹₆

|ξ−u0|² T0 − 3 Z

R³

ω(ξ; u0, T0)χi(ξ)χj(ξ)dξ = δij.

f is then decomposed into the macroscopic part f0 and the microscopic part f1. The macroscopic part is in the range of the projection operator P0, which is spanned by χiω; and the microscopic part is in the orthogonal complement, the range of the projection P1= I − P0. That is,





f (x, t, ξ) = f0(x, t, ξ) + f1(x, t, ξ), f0= P0f = ρ(x, t)χ0ω +P3

i=1mⁱ(x, t)χiω + e(x, t)χ4ω = ρ(x, t)χ0ω + h(x, t, ξ), f1= P1f

In particular, the Boltzmann equation is also decomposed into the macroscopic and microscopic equations as follows:

f0t+ P0ξ · ∇^x(f0+ f1) = 0

f1t+ P1ξ · ∇^x(f0+ f1) = (1 + ρ)Lf1+ N (f )

where L(f ) = Q(ω, f ) + Q(f, ω), N (f ) = Q(h + f1, h + f1). We then apply the basic energy method to both equations. For the microscopic equation, the negative definiteness of L restricted to the microscopic part yields the decaying of the microscopic component:

Z

R³

f1Lf1dξ < −ν0

Z

R³

(f1)²dξ.

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While in the energy estimates of the macroscopic equation, the following dissipation expression arises:

−P⁰ξ · ∇^xL⁻¹P1(ξ · ∇^xf0) = −P⁰ξ · ∇^xL⁻¹ P1

X3 i=1

(ξ · ∇^xmⁱ)χiω + P1(ξ · ∇^xe)χ4ω

! ,

which is exactly the dissipation of the momentum and the energy. Then the theory of conservation laws [Go, Ka, KM, Li1,MM] is applied to the macroscopic variables. The macroscopic component determines the wave propagation on one hand and the microscopic component, a faster decaying part, enjoys an equilibrating property on the other hand. And the coupling of the macroscopic and microscopic components gives rise to the dissipations.

In the present problem, we construct the approximate solution φ of (1.1)-(1.3) by superposing two Boltzmann shock profiles moving in the opposite directions with the same speed with some modifications. The Boltzmann shock profiles are obtained by [CN].

Let F denote a solution of (1.1) - (1.3), extended to the whole space R by setting F (x, t, ξ) = F (−x, t, −ξ), for x < 0.

Treat F as a perturbation of the approximate solution φ which is given explicitly in (3.1) later. Thus, we write F (x, t, ξ) ≡ φ(x, t, ξ) + J(x, t, ξ).

Therefore , J(x, t, ξ) satisfies

(1.4) Jt+ ξ1Jx= Q(φ + J, φ + J) − Q(φ, φ) − E(φ), where

E(φ) = φt+ ξ1φx− Q(φ, φ).

We choose the initial state J(x, 0, ξ) satisfying

(1.5)

Z ∞

−∞

Z

R³



 1

ξi

|ξ|²



 J(x, 0, ξ)dξdx = 0, for i = 1, 2, 3.

Due to the conservation laws for the macroscopic variables, it thus follows that Z ∞

−∞

Z

R³



 1

ξi

|ξ|²



 J(x, t, ξ)dξdx = 0, for i = 1, 2, 3.

Such property on macroscopic variables allows one to introduce an anti-derivative variable and the energy method can be applied to study the time-asymptotic stability problem. It is a well-known methodology in the field of viscous conservation laws. With regard to the Boltzmann equation we need to use the macroscopic component of the anti-derivative variable of the perturbation J(x, t, ξ).

Consider the anti-derivative:

W (x, t, ξ) ≡ Z x

−∞

J(y, t, ξ)dy.

We have

(1.6) Wt+ ξ1Wx=

Z x

−∞(Q(φ + J, φ + J) − Q(φ, φ) − E(φ))dy.

To solve equations (1.4) and (1.6), we make use of the shock profile of the Navier-Stokes equation. Let uN Sand TN S

denote the velocity and temperature for the corresponding shock profile of the Navier-Stokes equation constructed by the same conditions imposed for φ at the far field. We denote the corresponding local Maxwellians by

ωtr(x, t, ξ) = e⁻^(ξ1−uNS)

2 +ξ22 +ξ2 3

p 2TNS

(2πTN S)³ and the collision invariants ψi(x, t, ξ), i = 0, · · · , 4, are as follows:











ψ0(x, t, ξ) = 1 ψ1(x, t, ξ) = ^ξ¹^√^−u_T^{N S}

N S , ψi(x, t, ξ) = ^√_T^ξⁱ

N S, i = 2, 3.

ψ4(x, t, ξ) = ^√¹₆(^(ξ¹^−u^{N S}_T_{N S}⁾²^+ξ²²^+ξ³² − 3)

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We introduce the macroscopic and microscopic variables W0and W1 for W :

W0≡ P⁰W ≡ X4 i=0

Z

W ψidξ

ψiωtr, W1≡ P¹W ≡ W − W0,

where P0is the projection operator on the space spanned by ψiωtr, i = 0, · · · , 4 and P¹is the orthogonal projection P₁= I − P⁰. We also decompose J as

J = J0+ J1, J0≡ P0J, J1≡ P1J.

Note that the projection P0to equation (1.4) and (1.6) can be regarded as a linearization around the Navier-Stokes shock profile in the sense that the physical quantities u and T in the background Maxwellian are chosen from the shock profile of the Navier-Stokes equation.

In this paper, we share several technical difficulties with those in [LY]. For the readability of this paper, we address them again. First, we need to apply the transversal wave estimate. When we apply the theory of hyperbolic conservation laws to the macroscopic part, we recover the 3 families of elementary waves. In the present problem, the compressibility of the Navier-Stokes shock profile can be used in the estimate involving the first and third families, but not for the second family which produces transversal terms. Originated from the energy estimate for the stability analysis of a viscous shock profile[Go], we refine the transversal wave estimate in the current situation.

Secondly, the Boltzmann equation is nonlinear due to the collision operator. We split the collision operator Q into the linear part L and the nonlinear part N . The negative definiteness of L yields the decaying of the microscopic component. But we are left with the nonlinear part N , which produces terms like k(1 + |ξ|)¹²J1kL²_ξ. It leads us to make the right a priori assumption and establish the higher order energy estimate to resolve the nonlinear terms.

Finally, we have error terms caused by the approximate solution and the drifting Maxwellians. The facts that the projection P0is determined by the physical quantities of the Navier-Stokes shock profile and the local Maxwellians ωtrvary along the same shock profile certainly result in several error terms. When the shock strength is sufficiently small, the Boltzmann shock and the Navier-Stokes shock are close enough, which allows us to control all those errors. In addition, Kawashima’s method[Ka] is applied to control the density term, which is absent in considering the perturbation of a global Maxwellian.

We state the main theorem as follows:

Theorem 1.1. Consider the hard sphere model of equations (1.1)-(1.3). Suppose that the shock strength of φ is sufficiently small. Under the condition (1.5), there exists a constant δ0> 0 such that if the initial data are sufficient small:

X

|α|≤4

k∂x^αW0kL^∞_x,t(L²_ξ)+ k∂x^α∂tW0kL^∞_x,t(L²_ξ)

+ X

|α|≤3

k∂x^α{(1 + |ξ|)¹²J1}kref,L^∞_x,t(L²_ξ)+ k∂x^α∂t{(1 + |ξ|)¹²J1}kref,L^∞_x,t(L²_ξ)

≤ δ0

then the Cauchy problem (1.4) and (1.6) has a global solution in time. And the solution J(x, t, ξ) decays to zero in the k · kref,L^∞_x(L²_ξ)norm ast → ∞.

That is, the solutionF (x, t, ξ) to (1.1)-(1.3) tends toward the outgoing shock profile φ(x, t, ξ) in the k·kref,L^∞_x(L²_ξ)norm ast → ∞.

Remark. Although both the strength of shocks and the magnitude of perturbations of the initial data are small, these two parameters are chosen independently. When the strength of the shock is sufficiently small, the existence of the Boltzmann shock profile can be guaranteed and the shocks of the Boltzmann equation and the Navier-Stokes equation are close enough. Therefore, we can make use of the compressibility of the Navier-Stokes shock profiles.

As for δ0, the smallness assumption is to close the higher order estimate due to the nonlinearity of the Boltzmann equation. This is also a common short point of the energy method for viscous conservation laws.

This paper is organized as follows. Basic facts about the Boltzmann equation and the shock profiles are summa- rized in Section 2. In Section 3, the approximate solution for the present problem is constructed and the related equations are derived. Section 4 collects several technical estimates developed to analyze the macroscopic and microscopic equations. And the stability analysis is finally shown in Section 5. The properties of the collision operator for hard spheres is put in the Appendix.

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2. Preliminary

2.1. Macro-Micro Decomposition. We first recall several important properties of the Boltzmann equation Ft+ ξ · ∇xF = Q(F, F ), (x, t, ξ) ∈ R³× R⁺× R³.

In the present paper we consider the hard sphere as our model and the collision operator can be written as follows ([G], [H]):

Q(g, h) ≡ Z

R³

Z

S²

[g(ξ⁰)h(ξ_?⁰) − g(ξ)h(ξ^?)]C(Ω, ξ − ξ^?)dΩdξ?, where

ξ⁰ = ξ + (Ω · (ξ^?− ξ))Ω, ξ_?⁰ = ξ?− (Ω · (ξ?− ξ))Ω,

Ω ∈ S². The function C(Ω, ξ − ξ^?) for a hard sphere is

C(Ω, ξ − ξ^?) ≡ |Ω · (ξ − ξ^?)|.

The local equilibrium distributions are the distributions F with Q(F, F ) = 0, for which the only solutions are the Maxwellians

F (ξ) = ρ0ω(ξ; u0, T0), where

ω(ξ; u0, T0) = e⁻^|ξ−u0|

2

p 2T0

(2πT0)³.

The macroscopic density ρ0, velocity u0= (u01, u02, u03), and the temperature T0of the local thermal equilibrium state may vary. In the collision process, mass, momentum, and energy are conserved, i.e. for any distributions F and G,

Z

R³

Q(F, G)dξ = 0 Z

R³

ξiQ(F, G)dξ = 0, i = 1, 2, 3, (2.1)

Z

R³|ξ|²Q(F, G)dξ = 0.

Set the collision invariants χi(ξ), i = 0, · · · , 4, as follows:







χ0(ξ) = 1 χi(ξ) =^ξⁱ^√^−u_T⁰ⁱ

0 , i = 1, 2, 3.

χ4(ξ) = ^√¹₆(^|ξ−u_T₀⁰^|² − 3) which are normalized with respect to the Maxwellian state:

Z

χiχjω(ξ)dξ = δij, i, j = 0, · · · , 4.

The linearized collision operator

L(h) ≡ Q(ω, h) + Q(h, ω) is self-adjoint and non-positive, i.e.,

hLg, hi = hg, Lhi, hLg, gi ≤ 0.

Here h, i is the inner product on the space L²(R³) with respect to the variable ξ:

hg, hi ≡ Z

ghdξ.

In fact, χiω, i = 0, · · · , 4, span the null space of L. We denote by P⁰ the projection operator on the space spanned by χiω, i = 0, · · · , 4, and by P¹the orthogonal projection: P1= I − P⁰. F can be decomposed into the macroscopic part F0 and the microscopic part F1:

F0= P0F = X4 i=0

( Z

χiF dξ)χiω,

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F1= P1F = F − F⁰. It thus follows that

L(F0) = 0 and L(F ) = L(F1).

We introduce new norms for F (x, t, ξ), which will be used in the energy estimates:

kF kL²_ξ(x, t) ≡ hF, F i¹², kF kL^∞_x,t(L²_ξ) ≡ sup

(x,t)∈R×R⁺kF kL²_ξ(x, t)

We also note that for hard spheres [C] and Grad’s cutoff potentials [G], L is the sum of a multiplication operator and a compact operator K:

Lh(ξ) = −ν(ξ)h(ξ) + K(h)(ξ)

The collision frequency ν(ξ) has a positive lower bound. As a result, L is negative definite on the microscopic part:

Z

R³

F1L(F1)dξ < −ν0

Z

R³

F₁²dξ

for some positive constant ν0. Let P0≡ ker(L) and its orthogonal complement be denoted by P1. We will write the negative operator restricted to the space P1 as

L ≡ L|¯ ^P1≤ −ν0

Lemma 2.1. For anygi(x, t, ξ) satisfying P0gi ≡ 0,

|hg1, Lg2i| ≤ −1

2{γhg1, Lg1i + γ⁻¹hg2, Lg2i}

for any constantγ > 0.

Proof: By the self-adjoint and non-positive property of L.

2.2. Shock Profile. Let φ(x − st, ξ) be a travelling wave solution of the Boltzmann equation (2.2) Ft+ ξ1Fx= Q(F, F ), (x, t, ξ) ∈ R × R⁺× R³.

φ thus satisfies

(2.3) −sφ⁰+ ξ1φ⁰ = Q(φ, φ).

Let (ρ, u, T ) denote the macroscopic variables of the travelling wave solution:

ρ(x − st) ≡ Z

R³φ(x − st, ξ)dξ m(x − st) ≡

Z

R³

ξ1φ(x − st, ξ)dξ, u ≡ m ρ E(x − st) ≡

Z

R³

|ξ|²

2 φ(x − st, ξ)dξ (m²

2ρ + ρT ) ≡ E ≡ ρ(u² 2 + e)

Then the states (ρ_±, m_±, E_±) ≡ limx→±∞(ρ, m, E)(x) satisfy the Rankine-Hugoniot condition:

s(ρ₋− ρ⁺) = m₋− m⁺,

s(m₋− m+) = (u₋m₋+ p₋) − (u+m++ p+), (2.4)

s(E₋− E+) = (u₋(E₋+ p₋) − u+(E++ p+)), and the entropy condition

p₋− p⁺ > 0.

Here p ≡ ρRT, R ≡ 1. For the existence of the shock profile φ, see [CN].

Denote by φT(x − st, ξ) the corresponding local thermal equilibrium distribution:

φ^T(x − st, ξ) ≡ ρ(x − st)e⁻(ξ1−u(x−st))2 +ξ22+ξ2 3 2T (x−st)

p(2πT (x − st))³ . By direct calculations, φT satisfies the following lemma.

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Lemma 2.2.

Z

R³(φ − φ^T)dξ = 0, Z

R³

ξi− ui

√T (φ − φ^T)dξ = 0, i = 1, 2, 3, Z

R³

(|ξ − u|²

T − 3)(φ − φ^T)dξ = 0.

Here for the shock profileφ of (2.2), u1= u, u2= u3= 0.

Let (ρN S, uN S, EN S)(x − st) be the approximate shock profile of the Navier-Stokes equation obtained by the Chapman-Enskog expansion, which connects the same end states (ρ_±, u_±, E_±). The corresponding local Maxwellians are denoted by

ωtr(x − st, ξ) = e⁻^(ξ1−uNS)

2 +ξ22 +ξ2 3

p 2TNS

(2πTN S)³ ,

φtr(x − st, ξ) = ρN S

e⁻^(ξ1−uNS)

2 +ξ22 +ξ2 3

p 2TNS

(2πTN S)³ .

We consider a weak shock φ with strength ≡ |ρ−−ρ⁺| 1. The rate of the profile φ converging to the Maxwellian equilibrium states is given in the following theorem.

Theorem 2.3. On the profileφ(x − st, ξ), there exist C¹, C2> 1 and C3∈ (0, 1) such that

|φ(x, ξ) − φtr(x, ξ)| ≤ C1²e^−C³^|x|ρ(x)e⁻^(ξ1−u(x))

2 +ξ22+ξ2 3 2C2T (x)

p(T (x))³ (2.5)

|∂x^kφ(x, ξ)| ≤ C1^1+ke^−C³^|x|ρ(x)e⁻^(ξ1−u(x))^{2 +ξ2}^{2 +ξ}

23 2C2T (x)

p(T (x))³ , k = 1, · · · , 10.

(2.6)

Remark. The above theorem is proved in [CN] and the second inequality is the consequence of the scalings.

3. Construction of the Approximate Solution

3.1. The Macroscopic and Microscopic Equations. In this section we construct the approximate solution of (1.1) - (1.3) by superposing two travelling wave solutions moving in the opposite directions with the same speed.

The state (ρ+, m+, E+) is given by (1.2) at x = ∞. Since (1.3) implies that the macroscopic velocity and momentum are zero at x = 0, we first construct a Boltzmann shock profile with these two given conditions. By solving the Rankine-Hugoniot condition (2.4) and equation (2.3), there exist ρ0, E0 and a wave speed s > 0 such that (ρ+, m+, E+) and (ρ0, 0, E0) can be connected by a travelling wave solution φ+(x − st, ξ) on the whole space R = (−∞, ∞). Let (ρ+(x − st), m+(x − st), E+(x − st)) denote the corresponding macroscopic variables of φ+(x − st, ξ). According to the construction,

x→∞lim(ρ+(x), m+(x), E+(x)) = (ρ+, m+, E+),

x→−∞lim (ρ+(x), m+(x), E+(x)) = (ρ0, 0, E0).

We can also construct the other travelling wave solution φ₋(x + st, ξ) in the same way, satisfying

x→∞lim(ρ₋(x), m₋(x), E₋(x)) = (ρ0, 0, E0),

x→−∞lim (ρ₋(x), m₋(x), E₋(x)) = (ρ+, −m+, E+).

In fact, we can construct φ₋(x, ξ) = φ+(−x, −ξ). It thus follows that ρ+(x) = ρ₋(−x) m+(x) = −m−(−x)

E+(x) = E₋(−x) T+(x) = T₋(−x)

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We now choose the approximate solution φ(x, t, ξ) to be

(3.1) φ(x, t, ξ) ≡ φ⁺(x − s(t + t⁰), ξ) + φ₋(x + s(t + t0), ξ) − ρ⁰ω(ξ; 0, T0).

Here t0≡ ⁻³, ≡ |ρ⁺− ρ⁰| << 1. It follows from the above construction that φ(x, t, ξ) = φ(−x, t, −ξ).

Let F denote a solution of (1.1) - (1.3), extended to the whole space R by setting F (x, t, ξ) = F (−x, t, −ξ), for x < 0.

We now write

F (x, t, ξ) ≡ φ(x, t, ξ) + J(x, t, ξ) Then by (1.1), (2.3) and (3.1), J(x, t, ξ) satisfies

Jt+ ξ1Jx= Q(φ + J, φ + J) − Q(φ, φ) − E(φ), where

E(φ) ≡ φ^t+ ξ1φx− Q(φ, φ).

Also,

J(x, t, ξ) = J(−x, t, −ξ).

We choose the initial state J(x, 0, ξ) satisfying Z ∞

−∞

Z

R³



 1

ξi

|ξ|²



 J(x, 0, ξ)dξdx = 0, for i = 1, 2, 3.

Due to the conservation laws for the macroscopic variables, it thus follows that Z ∞

−∞

Z

R³



 1

ξi

|ξ|²



 J(x, t, ξ)dξdx = 0, for i = 1, 2, 3.

We consider the anti-derivative:

W (x, t, ξ) ≡ Z x

−∞

J(y, t, ξ)dy.

We thus have

Wt+ ξ1Wx= Z x

−∞(Q(φ + J, φ + J) − Q(φ, φ) − E(φ))dy.

Let (ρ^±_{N S}, u^±_{N S}, E_{N S}^± )(x ∓ st) be approximate shock profiles of the Navier-Stokes equation connecting the same end states as those of φ_±(x ∓ st, ξ). Define

(3.2)



 ρN S

uN S

EN S



 (x, t) =



 ρ⁺_{N S} u⁺_{N S} E_{N S}⁺



 (x − s(t + t⁰)) +



 ρ⁻_{N S} u⁻_{N S} E_{N S}⁻



 (x + s(t + t0)) −



 ρ0

0 E0



 .

We denote the corresponding local Maxwellians by

2 +ξ22+ξ2 3

p 2TNS

(2πTN S)³ , (3.3)

φtr(x, t, ξ) = ρN Se⁻^{(ξ1−uNS )}

2+ξ22 +ξ2 3

p 2TNS

(2πTN S)³ , (3.4)

where EN S= ρN S(¹₂u²_{N S}+ TN S).

In the following we use φtr and ωtr for the macro-micro decomposition. Let L and L be the linearized collision operator around φtr and φ respectively:

L(J) ≡ Q(φ^tr, J) + Q(J, φtr) L(J) ≡ Q(φ, J) + Q(J, φ) We introduce the following deviations:

D(J) ≡ L J − LJ N(J) ≡ Q(J, J)

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Set the collision invariants ψi(x, t, ξ), i = 0, · · · , 4, as follows:











ψ0(x, t, ξ) = 1 ψ1(x, t, ξ) = ^ξ¹√^−u^{N S}

TN S , ψi(x, t, ξ) = √^ξⁱ

TN S, i = 2, 3.

ψ4(x, t, ξ) = √¹

6(^(ξ¹^−u^{N S}_T⁾²^+ξ²²^+ξ³²

N S − 3)

which are normalized with respect to the Maxwellian state ωtr

Z

ψiψjωtrdξ = δij, i, j = 0, · · · , 4.

Note that the functions ψiωtr, i = 0, · · · , 4, span the kernel of L:

L(ψiωtr) = 0, for i = 0, · · · , 4.

We now introduce the macroscopic and microscopic variables W0 and W1 for W : W0≡ P⁰W ≡

X4 i=0

Z

W ψidξ

ψiωtr, W1≡ P1W ≡ W − W0,

where P0is the projection operator on the space spanned by ψiωtr, i = 0, · · · , 4 and P¹is the orthogonal projection P₁= I − P⁰. We also decompose J as

J = J0+ J1, J0≡ P⁰J, J1≡ P¹J.

Applying P0, P1, we obtain from (1.4) and (1.6),

P0∂tW0+ P0ξ1P0∂xW0+ P0ξ1J1= P0(−

Z x

−∞

E(φ)(y, t, ξ)dy).

(3.5)

P1∂tJ0+ P1∂tJ1+ P1ξ1∂xJ0+ P1ξ1∂xJ1− L(J1) = D(J) + N (J) − P1(E(φ)).

(3.6)

Since φ is constructed by superposing the travelling waves φ+and φ₋, E(φ) = φt+ ξ1φx− Q(φ, φ)

= (∂tφ++ ξ1∂xφ+− Q(φ+, φ+)) + (∂tφ₋+ ξ1∂xφ₋− Q(φ−, φ₋)) +Q(φ+, φ+) + Q(φ₋, φ₋) − Q(φ, φ)

= Q(φ+, φ+) + Q(φ₋, φ₋) − Q(φ, φ) By (2.1),

Z

R³

E(φ)dξ = 0 Z

R³

ξiE(φ)dξ = 0, i = 1, 2, 3, Z

R³|ξ|²E(φ)dξ = 0.

It thus follows that

P0( Z x

−∞

E(φ)(y, t, ξ)dy) = 0, P1(E(φ)) = E(φ).

From (3.6), we have

J1 = L⁻¹(P1∂tJ0+ P1∂tJ1+ P1ξ1∂xJ0+ P1ξ1∂xJ1)

−L⁻¹D(J) − L⁻¹N (J) − L⁻¹P1(E(φ)).

(3.7)

By substituting (3.7) for J1, we obtain from (3.5)

P0∂tW0+ P0ξ1P0∂xW0+ P0ξ1L⁻¹(P1∂tJ0+ P1∂tJ1+ P1ξ1∂xJ0+ P1ξ1∂xJ1)

−P0ξ1L⁻¹[D(J) + N (J) + P1(E(φ))] = 0.

(3.8)

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3.2. Reference Macro-Micro Decomposition. We introduce a reference macro-micro decomposition to estimate the derivatives of the microscopic part, which will be used in the higher order energy estimates and especially the estimates for J1.

To be sufficiently close to the background Maxwellian state ωtr, we actually choose the local Maxwellian state at x = 0 for the approximate solution φ. Let M0(ξ) denote such a Maxwellian state:

M0(ξ) ≡ e⁻^|ξ|2^2T0 p(2πT0)³.

The reference macro-micro decomposition is made with respect to M0(ξ) as follows:

hh1, h2iref ≡ Z

R³

h1h2

M0

dξ,

P0ref

h ≡ X4 i=0

hh, χⁱM0i^ref χiM0, P1ref

h ≡ h − P0ref,

where χiis defined in Section 2.1 with u0i= 0 for i = 1, 2, 3. The norms in the reference decomposition are defined as follows:

khkref,L²_ξ≡ hh, hi

1 2

ref

khkref,L²_x(L²_ξ)≡ Z

Rkhk²ref,L²_ξdx khkref,L^∞_x(L²_ξ)(t) ≡ sup

x∈Rkhkref,L²_ξ(x, t) khkref,L^∞_x,t(L²_ξ)≡ sup

(x,t)∈R×R⁺khkref,L²_ξ(x, t)

Remark: For the macroscopic component P0h, kP0hkL²_ξ and kP0hkref,L²_ξ are equivalent; that is, there exists K > 1 such that for any h(x, t, ξ),

K⁻¹kP⁰hkL²_ξ ≤ kP⁰hkref,L²_ξ≤ KkP⁰hkL²_ξ. 3.3. Local Existence in Time. We rewrite equation (1.4) as follows:

Jt+ ξ1Jx = L0J + {(L − L⁰)J + Q(J, J) − E(φ)}

= L0J + R(J) where

L0J ≡ Q(M0, J) + Q(J, M0).

Since R(J) is small, we can apply the Picard iteration to obtain the local existence of J.

We have equation (3.5) for W0:

P0∂tW0+ P0ξ1P0∂xW0+ P0ξ1J1= 0.

By calculating the inner product of the above equation with χiseparately, i = 0, 1, 4, we obtain a hyperbolic system with a given source term as follows:

∂thχ0, W0i + ∂xhχ0, ξ1W0i + hχ0, P0ξ1J1i = 0,

∂thχ¹, W0i + ∂^xhχ¹, ξ1W0i + hχ¹, P0ξ1J1i = 0,

∂thχ4, W0i + ∂xhχ4, ξ1W0i + hχ4, P0ξ1J1i = 0, where χi are the collision invariants normalized with respect to M0:







χ0(ξ) = 1 χi(ξ) =^ξⁱ^√^−u_T⁰ⁱ

0 , i = 1, 2, 3.

χ4(ξ) = √¹

6(^|ξ−u_T₀⁰^|² − 3)

Here u0= 0. Note that the inner product with χ2 or χ3is zero due to the planar wave assumption. Therefore, the local existence of W0follows directly from the local existence of J in k · kref,L²_x(L²_ξ).

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4. Basic Estimates

In this section we state several basic estimates, which will be used in Section 5 to analyze equation (3.8). We first introduce the following notations.

Definition 4.1. Leth(x, t, ξ) and g(x, t, ξ) be functions defined on R × R⁺× R³, andA be an operator on L²(R × R⁺× R³). Then

hh|gi(x, t) ≡ Z

R³

hg ωtrdξ, hh|A|gi(x, t) ≡

Z

R³

hAg ωtr dξ, whereωtr is defined in (3.3) as

2 +ξ22+ξ2 3

p 2TNS

(2πTN S)³ .

4.1. Properties of the Macroscopic Variables. In the study of the Boltzmann and Navier-Stokes equations the macroscopic dissipation occurs only for the momentum and the energy. The following lemma is used in Section 5 to analyze the dissipation of the macroscopic part.

Lemma 4.2. There existsC > 0, which depends on the Navier-Stokes shock profiles such that for the macroscopic function f0≡ ρ(x, t)ψ⁰ωtr+ m(x, t)ψ1ωtr+ e(x, t)ψ4ωtr,

(4.1) hP¹ξ1f0|P¹ξ1f0i(x, t) = (7

3TN S)m²(x, t) + (5

3TN S)e²(x, t) (4.2) C⁻¹(m²+ e²) ≤ |h ¯L⁻¹P1ξ1f0| ¯L⁻¹P1ξ1f0i| ≤ C(m²+ e²),

Proof: Write

hP¹ξ1f0|P¹ξ1f0i = hξ¹f0|ξ¹f0i − hP⁰ξ1f0|P⁰ξ1f0i

The first equality can be obtained by straightforward computations and the following integrals:





Rξ₁²ωtrdξ = u²_{N S}+ TN S, R

ξ₁²ψ²₁ωtrdξ = u²_{N S}+ 3TN S

Rξ₁²ψ1ωtrdξ = 2uN S√

TN S, R

ξ₁²ψ1ψ4ωtrdξ = √⁴ 6uN S√

TN S

Rξ₁²ψ4ωtrdξ = ^√²₆TN S, R

ξ₁²ψ²₄ωtrdξ = u²_{N S}+⁷₃TN S

The second inequality follows from the fact that P1ξ1(ρψ0ωtr) = 0 and L|^P1 is invertible. By straightforward computations and Schwartz’s inequality, we can obtain the following

Lemma 4.3. [LY ] There exists C > 0 which depends on the Navier-Stokes shock profiles such that the following holds for all P1-valued L² function f1:

Z

R³hP⁰ξ1f1|P⁰ξ1f1i dx ≤ C Z

R³hf¹|f¹i dx.

4.2. Matrix Representation. We now consider the 3 × 3 matrix P⁰ξ1P0, where the entries are calculated as follows:



 hψ⁰ωtr|ξ¹|ψ⁰ωtri hψ⁰ωtr|ξ¹|ψ¹ωtri hψ⁰ωtr|ξ¹|ψ⁴ωtri hψ¹ωtr|ξ¹|ψ⁰ωtri hψ¹ωtr|ξ¹|ψ¹ωtri hψ¹ωtr|ξ¹|ψ⁴ωtri hψ⁴ωtr|ξ¹|ψ⁰ωtri hψ⁴ωtr|ξ¹|ψ¹ωtri hψ⁴ωtr|ξ¹|ψ⁴ωtri



 =



 u1

√T 0

√T u1 √2 6

√T 0 ^√²₆√

T u1





By straightforward computations, its eigenvalues λi are λ1= u1−

r5T

3 , λ2= u1, λ3= u1+ r5T

3 .

Here u1 = uN S, T = TN S defined in (3.2). Note that ωtr is a local Maxwellian defined around the Navier-Stokes shock profile; therefore, these eigenvalues λi, i = 1, 2, 3, are defined only for the Navier-Stokes shock profile in this paper. Let rj= (rj1, rj2, rj3)^tand lj= (lj1, lj2, lj3) denote the corresponding right and left eigenvectors normalized by lj· ri= δij and |rj| = 1, i, j = 1, 2, 3. We define rj and lj as follows:

r_j ≡ (rj1ψ0+ rj2ψ1+ rj3ψ4)ωtr

lj≡ (l^j1ψ0+ lj2ψ1+ lj3ψ4)ωtr

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Note that r^t_j= lj since P0ξ1P0 is symmetric. Then for any function V (x, t, ξ), we have P0ξ1rj = λjrj,

hljξ1|P0V i = λjhl^j|V i

4.3. Estimates on the Eigenvalues for the Navier-Stokes Shock Profile. Recall that φ+(x − st, ξ) is the Boltzmann shock profile connecting (ρ+, m+, E+) at x = ∞ and (ρ0, 0, E0) at x = −∞. And φ−(x + st, ξ) is the Boltzmann shock profile connecting (ρ+, −m+, E+) at x = −∞ and (ρ0, 0, E0) at x = ∞. λ⁺i and λ⁻_i , i = 1, 2, 3, are the eigenvalues of the corresponding Navier-Stokes shock profiles of φ+and φ₋ respectively, which are given in Section 4.2 when φ is replaced by φ+ and φ₋ respectively. That is,

(4.3) λ⁺₁ = u⁺_{N S}−

s 5T_{N S}⁺

3 , λ⁺₂ = u⁺_{N S}, λ⁺₃ = u⁺_{N S}+ s

5T_{N S}⁺ 3 ,

(4.4) λ⁻₁ = u⁻_{N S}−

s 5T_{N S}⁻

3 , λ⁻₂ = u⁻_{N S}, λ⁻₃ = u⁻_{N S}+ s

5T_{N S}⁻ 3 . We have the following theorem:

Theorem 4.4. There existC4∈ (0, 1), C³> 0 and C5> 1 such that the eigenvalues of the corresponding Navier- Stokes shock profiles ofφ+ andφ₋ given in (4.3) and (4.4) satisfy

(4.5) C4²e^−2C³^|x|≤ −∂^xλ⁺₃ ≤ ²

C4e^−C⁵^|x|

(4.6) C4²e^−2C³^|x|≤ −∂xλ⁻₁ ≤ ² C4

e^−C⁵^|x|

Here λ⁻₁ = u⁻_{N S}− q5T_{N S}⁻

3 , λ⁺₃ = u⁺_{N S}+ q5T_{N S}⁺

3 .

Remark. The above inequalities are consequences of a well-known fact that the sound speed across a weak compressible Navier-Stokes shock profile is strictly monotone. One can obtain it easily by a two dimensional phase diagram. We omit the proof.

We now estimate ∂xλ1(x, t) and ∂xλ3(x, t) for later use. Recall that λ1= uN S−

r5TN S

3 , λ3= uN S+

r5TN S

3 ,

where 

 ρN S

uN S

EN S



 (x, t) =



 ρ⁺_{N S} u⁺_{N S} E_{N S}⁺



 (x − s(t + t⁰)) +



 ρ⁻_{N S} u⁻_{N S} E_{N S}⁻



 (x + s(t + t0)) −



 ρ0

0 E0



 .

And

TN S(x, t) = EN S

ρN S −u²_{N S} 2 . Due to the construction (3.2), we actually have

∂xλ⁻₁(x) = ∂xλ⁺₃(−x).

Since ρN S, uN S, TN Sare the corresponding density, velocity and temperature profiles of a traveling wave solution of the compressible Navier-Stokes equations, we can measure the decay of φtr:

∂xφtr(x, t, ξ) = O(1)²E(x, t)

1 + |ξ¹− u^{N S}| + |ξ¹− u^{N S}|²+ ξ²₂+ ξ²₃ φtr,

∂tφtr(x, t, ξ) = O(1)²E(x, t)

1 + |ξ¹− u^{N S}| + |ξ¹− u^{N S}|²+ ξ²₂+ ξ²₃ φtr, where

(4.7) E(x, t) ≡ e^−C³^|x−s(t+t⁰^)|+ e^−C³^|x+s(t+t⁰^)|. Moreover, by straightforward computations, we can obtain

∂xλ3(x, t) = ∂xλ⁺₃(x − s(t + t0)) + O(1)²E

−(x, t) (4.8)

∂xλ1(x, t) = ∂xλ⁻₁(x + s(t + t0)) + O(1)²E₊(x, t) (4.9)

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where

E₊(x, t) ≡ e^−C³^|x−s(t+t⁰^)|

E₋(x, t) ≡ e^−C³^|x+s(t+t⁰^)|. 5. Stability Analysis

We first derive some technical lemmas for later use. Lower order energy estimates are mainly shown in Subsection 5.1. Transversal wave estimates follow in Subsection 5.2. And the final higher order energy estimates are concluded in Subsection 5.3.

We decompose W and J as follows:

W ≡ W0+ W1, J ≡ J0+ J1

W0≡ X3 j=1

wjrj

h≡ J0− hωtr|J0iωtr≡ (h1ψ1(ξ) + h2ψ4(ξ))ωtr

Due to the decomposition, we have

J0= P0∂xW0. We introduce the following notation to analyze nonlinear terms:

N(J) ≡ Q(h + J¹, h + J1).

Thus,

N (J) = ρJ

ρN S

L(h + J1) + N (J) = ρJ

ρN S

L(J1) + N (J) where ρJ≡ hω^tr|J⁰i.

We rewrite equations (3.5) and (3.6) again.

P0∂tW0+ P0ξ1P0∂xW0+ P0ξ1J1 = 0 (5.1)

P1∂tJ0+ P1∂tJ1+ P1ξ1∂xJ0+ P1ξ1∂xJ1 = (1 + ρJ

ρN S

)L(J1)

+D(J) + N (J) − P¹(E(φ)).

(5.2) Write

W0≡ (R(x, t)ψ⁰+ M (x, t)ψ1+ E(x, t)ψ4) ωtr,

and 





Rx= ρ, Mx= m, Ex= e.

We note that ρJ(x, t) = ρ(x, t) + O(1)²E(x, t)kW0k∞, where E (x, t) is defined in (4.7) and kW⁰k∞≡ sup

(x,t)∈R×R⁺(|R| + |Mⁱ| + |E|)(x, t).

Motivated by the method in [Ka], we can prove the following estimate forRτ 0

R_∞

−∞ρ²dxdt:

Lemma 5.1.

Z τ 0

Z _∞

−∞

ρ²dxdt = O(1) Z

ρM dx

t=τ

t=0+ O(1) Z τ

0

Z _∞

−∞

m²+ e²+ hJ¹|J¹idxdt +O(1)

Z τ 0

Z ∞

−∞

⁴E²(x, t)hW⁰|W⁰idxdt Proof: By straightforward calculations, we have

P0(∂tW0) = Rtψ0ω + Mtψ1ω + Etψ4ω + O(1)²E(x, t)kW⁰k∞(X ψiω), P0(∂xW0) = Rxψ0ω + Mxψ1ω + Exψ4ω + O(1)²E(x, t)kW⁰k∞(X

ψiω).

Substituted by the above expressions, (5.1) leads to the following equations:

Rt+ uN SRx+p

TN SMx+ O(1)²E(x, t)kW0k∞= 0 (5.3)

Mt+p

TN SRx+ uN SMx+ 2

√6

pTN SEx+ hξ¹J1, ψ1i + O(1)²E(x, t)kW⁰k∞= 0 (5.4)

非線性波之交互作用及其應用

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