Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24, 22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.


Introduction and Main Result
We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation which takes the form with the initial data where the time variable t ∈ R + , spatial variables x = (x 1 , x 2 , x 3 ) ∈ R × T 2 := D with T 2 := (R/Z) 2 being the two-dimensional unit flat torus, the particle velocity v = (v 1 , v 2 , v 3 ) ∈ R 3 and f = f (t, x, v) represents the distributional density of particles at time-space (t, x) with velocity v. In the Boltzmann equation (1.1), the physical parameter κ called the Knudsen number is proportional to the mean free path of the interacting particles. Since we are concerned with the time-asymptotic behavior of the solution to the Boltzmann equation (1.1), the Knudsen number κ will be fixed to be 1 in the following.
For the hard sphere model, the collision operator Q(f, f ) takes the following bilinear and symmetric form where the unit vector Ω ∈ S 2 + = {Ω ∈ S 2 : (v − v * ) · Ω ≥ 0}, and (v, v * ) and (v ′ , v ′ * ) are the pair velocities of the two particles before and after a binary elastic collision respectively, which together with the conservation laws of momentum and energy during the collisions, imply the following relations It is well-known that Boltzmann equation is closely related to the system of classical fluid mechanics, in particular, the systems of compressible Euler and Navier-Stokes equations, as described by the famous Hilbert expansion and Chapman-Enskog expansion, respectively. Either the Hilbert expansion or the Chapman-Enskog expansion yields the compressible Euler equations in the leading order with respect to the Knudsen number κ. The system of compressible Euler equations is a typical example of hyperbolic conservation laws system, which has the distinguished feature that the solution may blow up in finitetime, i. e., the formation of the shock wave, no matter how smooth or small the initial values are. In fact, there are three basic wave patterns to the system of hyperbolic conservation laws, that is, two nonlinear waves, shock and rarefaction waves, in the genuinely nonlinear characteristic fields and a linear wave, contact discontinuity, in the linearly degenerate field. These dilation invariant solutions, and their linear superposition in the increasing order of characteristic speed, called Riemann solutions, govern both the local and large time asymptotic behavior of general solutions to the inviscid Euler system. Hence, one can expect the wave phenomena to Boltzmann equation as for the macroscopic fluid dynamics and it is interesting and important both mathematically and physically to prove the dynamic stability of these basic wave patterns for the Boltzmann equation.
For the Boltzmann equation with slab symmetry, i. e., the spatial one-dimensional case, the stability of the these three basic wave patterns are well-understood until now. For instance, the pioneering study on the stability of viscous shock wave was first proved by Liu-Yu [18] with the zero total macroscopic mass condition by the energy method based on the micro-macro decomposition while the existence of viscous shock profile to the Boltzmann equations is given by Caflish-Nicolaenko [2] and Liu-Yu [19]. Then stability of rarefaction wave fan is proved by Liu-Yang-Yu-Zhao [17] and the stability of viscous contact wave, which is the viscous version of contact discontinuity, by Huang-Yang [12] with the zero mass condition and Huang-Xin-Yang [11] without the zero mass condition. Furthermore, Yu [24] proved the stability of single viscous shock profile without zero mass condition by the elegant point-wise method based on the Green function around the shock profile. Recently, Wang-Wang [23] proved the stability of the superposition of two viscous shock profiles to the Boltzmann equation without the zero mass condition by the weighted characteristic energy method.
However, for the three-dimensional Boltzmann equation (1.1), there is no any result on the stability of these three basic wave patterns, as far as we know. The main purpose of the present paper is to establish the first result on the time-asymptotic stability of planar rarefaction wave to 3D Boltzmann equation in the infinite long flat nozzle domain D := R × T 2 . In our stability proof, we first use the macromicro decomposition invented by Liu-Yu [18] and Liu-Yang-Yu [16] to rewrite Boltzmann equation (1.1) by a fluid-type system, which is a compressible Navier-Stokes equations with temperature-dependent viscosities and heat-conductivities coupled with the microscopic terms, and a non-fluid part equation where the linearized collision operator around the local Maxwellian are strongly dissipative. Then we construct a smooth profile to the 1D self-similar rarefaction wave fan and we look for the solution to the 3D Boltzmann equation around this profile and prove the stability of this planar rarefaction wave.
Compared with the stability problem in one-dimensional case, the main difficulties in the proof of the stability of planar rarefaction wave lie in the wave propagations along the transverse directions (x 2 , x 3 ) ∈ T 2 and their interactions with the planar rarefaction wave in the direction x 1 ∈ R. Motivated by the previous work of the second author and his collaborator in [14] for the stability of planar rarefaction wave of the two-dimensional compressible Navier-Stokes equations, we use some underlying wave structures to overcome the difficulties mentioned above. However, different from [14], here we need to consider the case that both viscosities and heat conductivity depend on the temperature and moreover, we need to cope with the microscopic terms and their interactions with the fluid part. Furthermore, we need to make full use of the advantage of the different forms of the fluid system in the micro-macro decomposition for Boltzmann equation when we estimate the fluid parts. Now we formulate our problem. Since we are concerned with the time-asymptotic stability of planar rarefaction wave to 3D Boltzmann equation (1.1), it is assumed that the far fields conditions of initial data on the x 1 −direction with ρ ± > 0, u ± = (u 1± , 0, 0) t , θ ± > 0 being prescribed constant states. Moreover, the periodic boundary conditions are imposed on (x 2 , x 3 ) ∈ T 2 for the solution f (t, x, v). Here the two end states (ρ ± , u ± , θ ± ) are connected by the rarefaction wave solution to the Riemann problem of the corresponding 1D compressible Euler system with the Riemann initial data It could be expected that the large-time behavior of the solution to the 3D Boltzmann equation (1.1)-(1.3) is closely related to the Riemann problem to the corresponding 3D compressible Euler equations with the Riemann initial data There are essential differences between the one-dimensional Riemann problem (1.4)-(1.5) and the multi-dimensional Riemann problem (1.6)-(1.7). In the 2D isentropic regime, that is, the system (1.6) with the constant entropy and then the energy equation (1.6) 3 being satisfied trivially, it is first proved by Chiodaroli-DeLellis-Kreml [5] and Chiodaroli-Kreml [6] that there are infinitely many bounded admissible weak solutions to (1.6)-(1.7) satisfying the natural entropy condition for shock Riemann initial data by using the convex integration methods in DeLellis-Székelyhid [7] while the construction of infinitely many admissible weak solutions in [5,6] seems essential to the multi-dimensional system and could not be applied to one-dimensional problem (1.4)-(1.5). Then Klingenberg-Markfelder [13] and Brezina-Chiodaroli-Kreml [1] extend the results in [5,6] to the case when the corresponding Riemann initial data contain shock or contact discontinuity. On the other hand, similar to the one-dimensional case, for the Riemann solution only containing rarefaction waves to (1.6)-(1.7), Chen-Chen [4] and Feireisl-Kreml [8], Feireisl-Kreml-Vasseur [9] independently proved the uniqueness of the uniformly bounded admissible weak solution even the rarefaction waves are connected with vacuum states (cf. [4]).
As mentioned before, we can expect that the large-time behavior of the solution to the Boltzmann equation (1.1)-(1.3) is determined by the Riemann problem to the corresponding inviscid Euler system (1.6) or (1.4), which contains planar shock wave and rarefaction wave in the genuinely nonlinear characteristic fields and contact discontinuity in the linearly degenerate field. Our goal in the paper is to prove the above expectations in mathematics rigor and to investigate the dynamic stability of planar rarefaction wave for the 3D Boltzmann equation (1.1) at a first step. Now we first carry out the micro-macro decomposition around the local Maxwellian to the Boltzmann equation (1.1) as introduced by Liu-Yu [18] and Liu-Yang-Yu [16]. In fact, for any solution f (t, x, v) to equation (1.1), there are five macroscopic (fluid) quantities: the mass density ρ(t, x), the momentum m(t, x) = ρu(t, x), and the total energy E(t, x) = ρ e + 1 2 |u| 2 (t, x) defined by where ξ i (v) (i = 0, 1, 2, 3, 4) are the collision invariants given by and satisfy For a solution f (t, x, v) to the Boltzmann equation (1.1), we decompose it into the macroscopic (fluid) component, i.e., the local Maxwellian M = M [ρ,u,θ] (t, x, v), and the microscopic (non-fluid) component, i.e., G = G(t, x, v) as follows (cf. [16]) Here, the local Maxwellian M is associated to the solution f (t, x, v) of the equation (1.1) in terms of the five fluid quantities (ρ, u, θ)(t, x) defined by where θ(t, x) is the temperature which is related to the internal energy e(t, x) by e = 3 2 Rθ with R > 0 being the gas constant, and u(t, x) = u 1 (t, x), u 2 (t, x), u 3 (t, x) t is the fluid velocity.
For some given global or local Maxwellian M, the weighted inner product in L 2 v space with respect to the Maxwellian M is defined by: for any functions g 1 (v), g 2 (v) such that the above integral is well-defined, and g 1 If M is the local Maxwellian M in (1.10), we shall use the simplified notation ·, · instead of ·, · M if without confusions. With respect to this inner product ·, · , the following five pairwise orthogonal bases span the macroscopic space N (1.12) In terms of above orthogonal bases, the macroscopic projection P 0 from L 2 (R 3 v ) to N and the microscopic projection P 1 from L 2 (R 3 v ) to N ⊥ can be defined as g, χ j χ j , Taking the inner product of the equation (1.13) and the collision invariants ξ i (v) (i = 0, 1, 2, 3, 4) with respect to v over R 3 , one has the following system for the fluid variables (ρ, u, θ): (1.14) where p = 2 3 ρe = Rρθ is the pressure for the mono-atomic gas. However, the above fluid-type system (1.14) is not self-contained and a equation for the microscopic component G is needed, which can be derived by applying the projection operator P 1 into the equation(1.13): where L M is the linearized operator around the local Maxwellian M given by Recall that the linearized collision operator L M is symmetric in L 2 v space and the null space N of L M is exactly spanned by ξ i (v) (i = 0, 1, 2, 3, 4). For the hard sphere model, L M takes form (cf. [10,17]) Here K M (·) = K 2M (·) − K 1M (·) is a symmetric and compact operator in the above weighted L 2 v space. Let k iM (v, v * ) (i = 1, 2) be the kernel of the operator K iM (·) (i = 1, 2). Then the collision frequency ν M (v) and k iM (v, v * ) (i = 1, 2) have the following expressions By (1.17) 1 , we have ν M (v) ∼ (1 + |v|) as |v| → +∞. Furthermore, the celebrated H-theorem implies the strongly dissipative property of the linearized collision operator L M on the non-fluid component, i. e., there exists a positive constant σ 1 > 0 such that for any function g(v) ∈ N ⊥ (cf. [4,10]), it holds that where and in the sequel the subscript M in the collision frequency ν M (v) will often be omitted as the simplified form ν(v) if without confusions. By (1.18), the inverse of the operator L M exists in N ⊥ , which together with (1.15) implies that (1.14), we can obtain the following compressible Navier-Stokes-type equations for the macroscopic fluid quantities (ρ, u, θ): A direct computation gives rise to the diffusion terms with the viscosity coefficient µ(θ) > 0 and the heat conductivity coefficient κ(θ) > 0 being the smooth functions of the temperature θ. where the constant k = 1 2πe and S is the macroscopic entropy. It is straight to calculate that the Euler system (1.4) for (ρ, u 1 , θ) has three distinct eigenvalues with corresponding right eigenvectors such that Thus the two i-Riemann invariants Σ (j) with ρ > 0 and θ > 0 can be defined by (cf. [15]): Without the loss of generality, we consider the stability of 3−rarefaction wave to the Euler system (1.4)-(1.5) in the present paper and the stability of 1−rarefaction wave can be considered similarly. The 3−rarefaction wave to the Euler system (1.4)-(1.5) can be expressed explicitly through the Riemann solution to the inviscid Burgers equation: If w − < w + , then the Riemann problem (1.23) admits a self-similar rarefaction wave fan solution w r (t, Next, we construct a smooth profile to the 3-rarefaction wave defined in (1.25). Motivated by Matsumura-Nishihara [21], the smooth rarefaction wave can be constructed by the Burgers equation where ε > 0 is a small parameter to be determined and k q is a positive constant such that k q ∞ 0 (1 + y 2 ) −q dy = 1 for each q ≥ 2. Note that the solutionw(t, x 1 ) of the problem (1.26) can be given by Correspondingly, the smooth 3-rarefaction wave (ρ,ū,θ)(t, x 1 ) to compressible Euler equations (1.4)− (1.5) can be defined by (1.27). Then the planar 3-rarefaction wave (ρ,ū,θ)(t, x 1 ) satisfies the Euler system x being the standard Sobolev space and L 2 v 1 √ M * being weighted L 2 v space defined in (1.11) for some global Maxwellian M * . Now we can state our main result as follows.
, then there exist a positive constant ε 0 < 1 and a global Maxwellian M * = M [ρ * ,u * ,θ * ] with ρ * > 0, θ * > 0, such that if ε ≤ ε 0 and the initial values satisfy with some uniform-in-time constant C and the time-asymptotic stability of planar 3-rarefaction wave: Remark 1.1. Theorem 1.1 is the first result on the time-asymptotic stability of basic wave patterns for the three-dimensional Boltzmann equation, even though the corresponding stability result for shock wave or contact discontinuity is still completely open.
The rest part of the paper is arranged as follows. First, we present the local-in-time existence of the solution to 3D Boltzmann equation (1.1)-(1.3), and list some properties for the rarefaction wave and Boltzmann equation's microscopic H-theorem in Section 2. Then, we will prove our main result Theorem 1.1 based on the a priori energy estimates in Section 3. Finally, we give the proof of local-intime existence of solution to Boltzmann equation (1.1)-(1.3) in Appendix.

Preliminaries
In this section, we first present the local-in-time existence of solution to Boltzmann equation (1.1)-(1.3), list some properties for the rarefaction wave and linearized collision operator, and then show the celebrated microscopic H-theorem for Boltzmann equation.
We start from the local-in-time existence of solution to Boltzmann equation (1.1)-(1.3), whose proof will be given in Appendix.
Then we list some properties of the smooth 3−rarefction wave constructed in (1.28) in the following two lemmas.
(2) For any t > 0 and p ∈ [1, ∞], there exists a constant C p,q such that (3) The smooth rarefaction wavew(t, x 1 ) and the original rarefaction wave w r ( x 1 t ) are time-asymptotically equivalent, i.e., lim is the strength of the 3-rarefaction wave (ρ,ū,θ) defined in (1.28), then the following properties hold: (ii) The following estimates hold for all t > 0 and p ∈ [1, ∞]: (iii) Time-asymptotically, the smooth 3-rarefaction wave and the inviscid 3-rarefaction wave are equivalent, i.e., Next, we list some lemmas on the estimates and dissipative properties of the linearized collision operator in the weighted L 2 space, based on the celebrated H-theorem. The first lemma can be found from [18].
where M can be any Maxwellian so that the above integrals are well-defined.
Based on Lemma 2.4, the following three lemmas are taken from [17]. Their proofs are straightforward by using Lemma 2.4 and Cauchy inequality.
Remark 2.1. In Lemmas 2.5-2.7, η 0 may not be sufficiently small positive constant. However, in the proof of Theorem 1.1 in the following sections, the smallness of η 0 is crucially used to close the a priori assumptions (3.3).
3. The Proof of Theorem 1.1 In this section, we prove our main result Theorem 1.1, based on the local-in-time existence of the solution in Lemma 2.1 and the a priori estimates carried out in the following. First set the perturbation around the 3-rarefaction wave by with the correction functionḠ as Based on the local-in-time existence of the solution in Lemma 2.1 and the standard continuum argument, to prove the global existence on the time interval [0, T ] with T > 0 being any positive time and the uniform-in-time estimates (1.31) and then Theorem 1.1, it is sufficient to close the following a-priori assumptions: and verify (1.32), where and in the sequel (∂ α , ∂ β ) = (∂ α t,x , ∂ β t,x ) and χ is a small positive constant depending on the initial data but independent of the time T . Note that the global Maxellian M * in (3.3) is determined in Theorem 1.1. It can be seen easily from (3.3) and Sobolev's inequality that with some positive constant C. Under the a priori assumption (3.3), we can prove that The proof of (3.5) in Theorem 3.1 will be done through the suitable combination of Proposition 3.1 and Proposition 3.2 below by multiplying (3.34) with a large constant C and then adding the resulting equality and (3.8) together.
Once we proved Theorem 3.1, we can finish the proof of Theorem 1.1. The global-in-time existence of solution follows immediately from Lemma 2.1 (Local-in-time existence) and Theorem 3.1 (A priori estimates). Then we only need to justify the time-asymptotic stability of planar rarefaction wave as in (1.32). In fact, from (3.5) it holds that Then by three-dimensional Sobolev's inequality which together with (3.5) and (3.7) yields which verifies (1.32), hence the proof of Theorem 1.1 is completed.
In the following subsections, we will prove the a priori estimates in Theorem 3.1 by the suitable combinations of the lower order estimates in Proposition 3.1 and the higher order estimates in Proposition 3.2.
3.1. Lower order estimates. We start from the lower order estimates.
Proof : First, define the macroscopic entropy by Multiplying the equation (1.1) by ln M and integrating over v, it holds that Direct computations yields Then the conservation law (1.20) can be rewritten as where ∂ j = ∂ x j (j = 1, 2, 3) and I is the 3 × 3 unit matrix. Here and in the sequel div and ∇ denote the divergence and gradient operator with respect to the spatial variable x if without confusions. Define a relative entropy-entropy flux pair (η, q) around the local Maxwellian Here, we can compute that and then where Ψ(s) = s − ln s − 1 is a strictly convex function around s = 1. Then, for any X in the closed and bounded region of = {X : ρ > 0, θ > 0}, there exists a positive constant C such that Direct computations yield that (3.9) There exists a positive constant C > 0 such that Integrating (3.9) with respect to t, x over [0, t] × D yields that

It follows from Cauchy's inequality and (3.3) that
(3.14) Note that by (1.19), it holds that Choose the global Maxellian M * = M [ρ * ,u * ,θ * ] such that Furthermore, one has (3.20) Substituting (3.18)-(3.20) into (3.15) and then into (3.14), one can obtain (3.21) Substituting the estimates for I i (i = 1, 2, 3) in (3.12), (3.13) and (3.21) into (3.10) gives the first-step lower order estimates Since there is no dissipation for the density function, we want to get the estimation of ∇φ next. For this, by the system (1.14) and (1.29), we obtain the following form for the system of the perturbation (φ, ψ, ζ): Then from the structure of perturbation equation (3.23) 2 , it can be seen that the gradient of pressure, i.e., ∇p can deduce the estimation of ∇φ, which is quite different from compressible Navier-Stokes system case in [14]. More precisely, multiplying (3.23) 2 by ∇φ and integrating over [0, t] × D lead to where in the above equality we have used (3.23) 1 and the following fact It follows from (3.24) and Cauchy's inequality that Then we derive the estimation of (φ t , ψ t , ζ t ). Multiplying the equation (3.23) 1 by φ t , (3.23) 2 by ψ t , (3.23) 3 by ζ t respectively, then integrating over [0, t] × D, we have Combining (3.22), (3.25) and (3.26) together gives that Now we do the microscopic estimates. By (1.15) and (3.1), G satisfies the equation Multiplying the equation (3.28) by G M * , and then integrating over [0, t] × D × R 3 yield that By Cauchy's inequality, we have Similar to (3.20) and by Cauchy's inequality and Lemma 2.4, one has (3.31) It follows from the Cauchy's inequality and Lemma 2.3 that (3.32) Substituting (3.30)-(3.32) into (3.29) gives that which along with (3.27) yields (3.8), and the proof of Proposition 3.1 is completed.

3.2.
Higher order estimates. In this subsection, we will consider the higher order energy estimates.

Proposition 3.2. Under the a priori assumption (3.3), it holds that
The proof of Proposition 3.2 is divided into the following five steps.
We want to show by induction that if g 0 S ≤ Ξ 2 √ C 0 with C 0 := 1 min{1,σ} > 1, then g n S ≤ Ξ for all n, provided that Ξ and T are chosen suitably small. In fact, multiplying (4.5) by g n+1 M * and then