Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space
aa r X i v : . [ m a t h . A P ] J a n COMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION WITHMAXWELL REFLECTION BOUNDARY CONDITION IN HALF-SPACE
NING JIANG, YI-LONG LUO ∗ , AND SHAOJUN TANG Abstract.
Starting from the local-in-time classical solution to the compressible Euler sys-tem with impermeable boundary condition in half-space, by employing the coupled weakviscous layers (governed by linearized compressible Prandtl equations with Robin boundarycondition) and linear kinetic boundary layers, and the analytical tools in [22] and some newboundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time ex-istence of Hilbert expansion type classical solutions to the scaled Boltzmann equation withMaxwell reflection boundary condition with accommodation coefficient α ε = O ( √ ε ) whenthe Knudsen number ε small enough. As a consequence, this justifies the corresponding caseof formal analysis in Sone’s books [37, 38]. This also extends the results in [23] from specularto Maxwell reflection boundary condition. Both of this paper and [23] can be viewed asgeneralizations of Caflisch’s classic work [9] to the cases with boundary. Keywords.
Compressible Euler limit; Boltzmann equation; Maxwell reflection boundarycondition; Accommodation coefficients; Hilbert expansion.
AMS subject classifications. Introduction
Boltzmann equation with Maxwell boundary condition.
Hydrodynamic limitsfrom the Boltzmann equation has been one of the central problems in fluid dynamics andkinetic theory since late 1970’s. Among many references, we list some standard books forthorough introduction to the Boltzmann equation and fluid limits, for example, [7, 8, 34].Despite the tremendous progress in this fields (which we will review later), the most importantproblem basically remains open: the limit from the Boltzmann equation to the compressibleEuler system.More precisely, the Boltzmann equation in the Euler scaling is the following form: ( ∂ t F ε + v · ∇ x F ε = ε B ( F ε , F ε ) on R + × R × R ,F ε (0 , x, v ) = F inε ( x, v ) ≥ R × R , (1.1)where the dimensionless number ε > F ε ( t, x, v ) ≥ v ∈ R and position x ∈ R = { x ∈ R ; x > } , a half-space. Moreover, the equation (1.1)is imposed with the Maxwellian reflection boundary condition γ − F ε = (1 − α ε ) Lγ + F ε + α ε Kγ + F ε on R + × Σ − , (1.2)For the derivation from the dimensional Boltzamnn equation to the scaled form (1.1), thereaders can find the details in the aforementioned books, or papers [2, 3].The Boltzmann collision operator is defined as B ( F , F )( v ) = ˆ R ˆ S | v − u | γ F ( u ′ ) F ( v ′ ) b ( θ )d ω d u − ˆ R ˆ S | v − u | γ F ( u ) F ( v ) b ( θ )d ω d u , (1.3) ∗ Corresponding author January 28, 2021. where u ′ = u +[( v − u ) · ω ] ω , v ′ = u +[( v − u ) · ω ] ω , cos θ = ( v − u ) · ω/ | v − u | , 0 < b ( θ ) ≤ C | cos θ | ,and 0 ≤ γ ≤ α ε ∈ [0 ,
1] describes howmuch the molecules accommodate to the state of the wall. The special case α ε = 0 correspondsto specular reflection, while α ε = 1 refers to complete diffusion. Usually this coefficient can betaken the form χε β with β ≥
0. In this paper, we consider the form α ε = √ πε . Analytically,the cases β = , β = 0, the formal analysis will be the same (all the detailsare in Sone’s books [37, 38]), except the leading order involves nonlinear compressible Prandtlequation, whose well-posedness (even local) in Sobolev type space is completely open.We denote n = (0 , , −
1) by the outward normal of R . Let Σ := ∂ R × R be the phasespace boundary of R × R . The phase boundary Σ can be split by outgoing boundary Σ + ,incoming boundary Σ − , and grazing boundary Σ :Σ + = { ( x, v ) : x ∈ ∂ R , v · n = − v > } , Σ − = { ( x, v ) : x ∈ ∂ R , v · n = − v < } , Σ = { ( x, v ) : x ∈ ∂ R , v · n = − v = 0 } . Let γ ± F = Σ ± F . The specular-reflection Lγ + F ε and the diffuse-reflection part Kγ + F ε in(1.2) are Lγ + F ε ( t, x, v ) = F ε ( t, x, R x v ) , R x v = v − v · n ) n = (¯ v, − v ) ,Kγ + F ε ( t, x, v ) = √ πM w ( t, ¯ x, v ) ˆ v · n> γ + F ε ( v · n )d v , respectively, where M w ( t, ¯ x, v ) is the local Maxwellian distribution function corresponding tothe wall (boundary) with the form M w ( t, ¯ x, v ) = ρ w ( t, ¯ x )[2 πT w ( t, ¯ x )] exp n − | v − u w ( t, ¯ x ) | T w ( t, ¯ x ) o . (1.4)The ρ w , u w , T w are, respectively, density, velocity and temperature of the boundary. We alsoassume u w, = 0 , which denotes the boundary wall is fixed. We remark that the wall Maxwellian M w in (1.4)plays a key role in this paper. It matches with the leading local Maxwellian from interior,see (1.71), which gives the compatibility condition between the initial and boundary data.It is one of the main difference or difficulty of the current paper with [23] in which the wallMaxwellian did not appear.1.2. History of compressible Euler limits.
There have been many significant progressin the limits to incompressible fluids, such as Navier-Stokes, Stokes, or even Euler. Weonly list some representative results. One group of results is in the framework of DiPerna-Lions renormalized solutions of the Boltzmann equation [13], i.e., the so-called BGL program,which aimed to justify the limit to Leray solutions of the incompressible Navier-Stokes equa-tions. This was initialized by Bardos-Golse-Levermore [2, 3], and finished by Golse andSaint-Raymond [16, 17]. The corresponding results in bounded domain were carried out in[30, 27]. We also should mention the incompressible Euler limits of Saint-Raymond [32, 33].Another group of results is in the framework of classical solutions, which are based on non-linear energy method, semi-group method or hypercoercivity (the latter two further rely onthe spectral analysis of the linearized Boltzmann operator), see [4, 5, 6, 14, 19, 25, 28].Comparing to the incompressible limits listed above, the limit to the compressible Eulersystem from the Boltzmann equation is much limited. This is mainly due to our still verypoor understanding of the well-posedness of compressible Euler system for which we do not
OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 3 even know how to define weak solutions (at least for multiple spatial dimensions). Theonly available global-in-time solution is the calibrated BV solution of Glimm in 1965 [18].Regarding the higher dimensions, there are local-in-time classical solutions (see standardtextbook [29]). The compressible Euler limit from the Boltzmann equations dates back toJapanese school for analytical data and Caflisch’s Hilbert expansion approach which are basedon the local well-posedness of compressible Euler system, see [9, 31]. Later improvement ofCaflisch result employing the recent progress on the L - L ∞ estimate was also used to helpjustifying the linearized acoustic limit [21, 22]. We should also mention the compressible Eulerlimit in the context of 1-D Riemann problem away from the initial time in [24].In the domain with boundary, the situation is much more complicated. As formally (andnumerically) analyzed by Japanese school (summarized in Sone’s books [37, 38]), to derivethe equations of compressible Euler system from the Boltzmann equations with Maxwellreflection condition (with non-zero accommodation coefficient) using Hilbert type expansion,there needed two coupled boundary layers: viscous and kinetic layers. The formal is alsocalled Prandtl layer , the latter is
Knudsen layer in the physics literatures. The reason thatthese two types of boundary layers are needed can be explained as follows: as well-known,the leading order in interior is compressible Euler system whose natural boundary conditionis impermeable condition u · n = 0. However, the local Maxwellian governed by Euler systemwith this condition does not satisfy the Maxwell reflection boundary condition, except thespecular reflection case (i.e., α ε = 0). So, kinetic layers with thickness ε are needed. In theselayers, each term satisfies the linear kinetic boundary layer equations which requires four solvability conditions (by theorem of Golse-Perthame-Sulem [15]), but the number of boundaryconditions for compressible Euler system is one . This mismatch indicates there should beanother layer with thickness √ ε : Prandtl layer. This name comes from the fact that in thislayer, each term satisfies the famous Prandtl equations of compressible type. Specifically,if α ε = O (1) the leading term satisfies the nonlinear compressible Prandtl equations, thehigher order terms satisfied the linearized compressible Prandtl equations. If α ε = O ( ε β ) with β >
0, the boundary layers are weak, thus all the boundary terms appear in higher order.Thus, nonlinear Prandtl equation does not appear. The current paper aims to give a rigorousjustification for the case α ε = O ( √ ε ).In [23], Guo-Huang-Wang started to justify the compressible Euler limit using Hilbertexpansion approach, formally derived in Sone’s books. They considered the simplest case α ε = 0, i.e., the specular reflection. We emphasize that in this case, the local Maxwelliansgoverned by compressible Euler system with impermeable condition indeed satisfy the specularreflection condition. In this sense, boundary layers seem not needed. However, if the Hilbertexpansion is used, at higher orders, the boundary conditions do not match. As a consequence,both fluid and kinetic layers are needed.In the context of fluid equations, the application of the compressible Prandtl equations isnot new. It has been investigated in the vanishing viscosity limit of compressible Naver-Stokesequations. This phenomena is consistent with paper [23] and the current paper, because aswell-known, the compressible Navier-Stokes system with viscosity and thermal conductivity oforder O ( ε ) can be derived from the Boltzmann equation (more precisely, by Chapman-Enskogexpansion). This means [23] and current paper include this limit, but with different boundaryconditions for compressible Navier-Stokes: paper [23] corresponds to Neumann boundary,the current paper corresponds to mixed Robin boundary. Indeed, at fluid equations level,Xin-Yanagisawa [39] proved the zero viscosity limit of linearized compressible Navier-Stokessystem with Dirichlet condition, in which the linearized compressible Prandtl equations werealso used. This result was extended to the case including temperature equation by Ding andthe first author of the current paper in [12].We would like to mention that the analysis of coupled viscous and kinetic layers was alreadyused in the work of the first author of this paper with Masmoudi [27] in slightly different formwith [23] and this paper. The paper [27] is about the incompressible Navier-Stokes limit, which NING JIANG, YI-LONG LUO, AND SHAOJUN TANG in fact does not need viscous layer. However, the major issue of [27] is to demonstrate the roleplayed by acoustic waves, which is compressible. Thus, the incompressible Navier-Stokes limit(which happens in the time scale O ( ε )) includes the acoustic limit in short time scale O (1). Inthis sense, the two boundary layers in [27] are the same as the current papers and [23] (again,we emphasize that all the layers in [23] and here are linear because they appear in higher orderterms). The main difference is that [27] works in the framework of renormalized solutions,which can not be expanded. So the coupled viscous and kinetic layers are considered in dual form in the sense that these layers appear in test functions. More specifically, the kineticlayers equations are the same as here and [23], but with different boundary conditions: inthis paper, for Maxwell reflection, the wall Maxwellian is local and nontrivial, while in [27],the wall Maxwellian is global, i.e., M ( v ). In [23], it is specular reflection, so there is nowall Maxwellian. For the viscous layer parts, in [27], the stationary version, i.e., degenerateheat operators are used, while in [23] and current paper, the linearized compressible Prandtlequations are used. The more detailed technical difference between [23] and the current paperwill be explained in Subsection 1.5.1.3. Hilbert expansion.
Throughout this paper, we use the notation ¯ U = ( U , U ) for anyvector U = ( U , U , U ) ∈ R . Moreover, for simplicity of presentations, we use the notation V := V | x =0 for any symbol V = V ( x ), which may be a function, vector or operator. Forany derivative operator D x , we denote by D x V = ( D x V ) . For the functional spaces, H s denotes the Sobolev space W s, ( R ) with norm k · k H s , k · k and k · k ∞ are the L -norm and L ∞ -norm in both ( x, v ) ∈ R × R variables, and h· , ·i is the L -inner product.For any function G = G ( t, ¯ x, y, v ), with ( t, ¯ x, y, v ) ∈ R + × R × R + × R , the Taylor expansionat y = 0 is G = G + X ≤ l ≤ N y l l ! G ( l ) + y N +1 ( N +1)! e G ( N +1) , (1.5)where the symbols G ( l ) = ( ∂ ly G )( t, ¯ x, , v ) , e G ( N +1) = ( ∂ N +1 y G )( t, ¯ x, η, v ) for some η ∈ (0 , y ) . In this paper, we take the Hilbert expansion approach to rigorously justify the asymptoticbehaviors of (1.1)-(1.2) as ε →
0. As we already mentioned above, its corresponding formalanalysis has basically been down in Sone’s books [37, 38]. Analytically, the key of this approachis the estimate on the remainder , after suitable truncation.Due to the thickness of viscous boundary layer is √ ε , and the accommodation coefficient α ε = O ( √ ε ), we expand F ε ( t, x, v ) by order √ ε . We remark that if α ε = O ( ε β ) with β = , Interior expansion.
The expansion in interior has the form F ε ( t, x, v ) ∼ X k ≥ √ ε k F k ( t, x, v ) . Plugging into (1.1) and collecting the same orders, √ ε − : 0 = B ( F , F ) , √ ε − : 0 = B ( F , F ) + B ( F , F ) , √ ε : ( ∂ t + v · ∇ x ) F = B ( F , F ) + B ( F , F ) + B ( F , F ) , √ ε : ( ∂ t + v · ∇ x ) F = B ( F , F ) + B ( F , F ) + B ( F , F ) + B ( F , F ) , · · · · · ·√ ε k : ( ∂ t + v · ∇ x ) F k = B ( F , F k +2 ) + B ( F k +2 , F ) + X i + j = k +2 ,i,j ≥ B ( F i , F j ) . (1.6) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 5
Then H -theorem implies that F must be a local Maxwellian: F ( t, x, v ) := M ( t, x, v ) = ρ ( t, x, v )(2 πT ( t, x, v )) exp (cid:16) − | v − u ( t, x, v ) | T ( t, x, v ) (cid:17) . (1.7)Here ( ρ, u , T ) represent the macroscopic density, bulk velocity and temperature, respectively.It is well-known that (see [38], for instance), as ε →
0, the solutions F ε of the Boltzmannequation (1.1) converge to a local Maxwellian M whose parameters ( ρ, u , T ) satisfy the com-pressible Euler system ∂ t ρ + div x ( ρ u ) = 0 ,∂ t ( ρ u ) + div x ( ρ u ⊗ u ) + ∇ p = 0 ,∂ t (cid:2) ρ (cid:0) T + | u | (cid:1)(cid:3) + div x (cid:2) ρ u (cid:0) T + | u | (cid:1)(cid:3) + div x ( p u ) = 0 (1.8)over ( t, x ) ∈ R + × R with the slip boundary condition u · n | x =0 = − u | x =0 = − u = 0 , (1.9)where p = ρT is the pressure. We further impose the initial data (this can be realized bysetting special form of the initial data of the Boltzmann equation (1.1)):( ρ, u , T )(0 , x ) = ( ρ in , u in , T in )( x ) (1.10)with compatibility condition u in · n | x = 0 . By [35] or [11], we have the following proposition.
Proposition 1.1.
Let s ≥ . Assume ( ρ in , u in , T in ) ∈ H s ( R ) satisfies < ρ ≤ ρ in ( x ) ≤ ρ , < T ≤ T in ( x ) ≤ T for some constants ρ , ρ , T and T with ρ √ T < √ . Then there is a τ > such thatthe compressible Euler system (1.8) - (1.9) - (1.10) admits a unique solution ( ρ, u , T ) ∈ C (cid:0) [0 , τ ]; H s ( R ) (cid:1) ∩ C (cid:0) [0 , τ ]; H s − ( R ) (cid:1) such that < ρ ≤ ρ ( t, x ) ≤ ρ , < T ≤ T ( t, x ) ≤ T hold for any ( t, x ) ∈ [0 , τ ] × R . Moreover, the following estimate holds: k ( ρ, u , T ) k C (cid:0) [0 ,τ ]; H s ( R ) (cid:1) ∩ C (cid:0) [0 ,τ ]; H s − ( R ) (cid:1) ≤ C . (1.11) Here the constants τ , C > depend only on the H s -norm of ( ρ in , u in , T in ) . By using the local Maxwellian M ( t, x, v ), the linearized collision operator L is defined as L g = − √ M n B ( M , √ M g ) + B ( √ M g, M ) o . The null space N of L is spanned by (see [9], for instance) √ ρ √ M , v i − u i √ ρT √ M ( i = 1 , , , √ ρ n | v − u | T − o √ M . The weighted L -norm k g k ν = ˆ R ˆ R | g ( x, v ) | ν ( v )d x d v , is defined by the collision frequency ν ( v ) ≡ ν ( M )( v ) ν ( M ) = ˆ R ˆ S b ( θ ) | v − v ′ | γ M ( v ′ )d v ′ d ω . Note that for given 0 ≤ γ ≤ ν ( M ) ∼ ρ h v i γ , (1.12) NING JIANG, YI-LONG LUO, AND SHAOJUN TANG where h v i = p | v | . Let P g be the L v projection with respect to N . Then it is well-known(see for example [9]) that there exists a positive number c > hL g, g i ≥ c k ( I − P ) g k ν . (1.13)For each k ≥ F k √ M can be decomposed as the macroscopic and microscopic parts: F k √ M = P ( F k √ M ) + ( I − P )( F k √ M ) ≡ n ρ k ρ + u k · v − u T + θ k T ( | v − u | T − o √ M + ( I − P )( F k √ M ) . Following [21], for k ≥ I − P ) (cid:0) F k √ M (cid:1) = L − (cid:16) − ( ∂ t + v ·∇ x ) F k − − P i + j = k ,i,j ≥ B ( F i ,F j ) √ M (cid:17) , (1.14)and the fluid variables ( ρ k , u k , θ k ) obeys the following linear hyperbolic system ∂ t ρ k + div x ( ρu k + ρ k u ) = 0 ,ρ (cid:0) ∂ t u k + u k · ∇ x u + u · ∇ x u k (cid:1) − ∇ x ( ρT ) ρ ρ k + ∇ x (cid:16) ρθ k +3 T ρ k (cid:17) = F ⊥ u ( F k ) ,ρ (cid:16) ∂ t θ k + u · ∇ x θ k + (cid:0) θ k div x u + 3 T div x u k (cid:1) + 3 u k · ∇ x T (cid:17) = G ⊥ θ ( F k ) , (1.15)where ( ρ, u , T ) is the smooth solution of compressible Euler equations (1.8), the source terms F ⊥ u ( F k ) and G ⊥ θ ( F k ) are defined as F ⊥ u,i ( F k ) = − X j =1 ∂ x j ˆ R T A ij F k √ M d v ( i = 1 , , , G ⊥ θ ( F k ) = − div x (cid:16) T ˆ R B F k √ M d v + X j =1 T u · ˆ R A F k √ M d v (cid:17) − u · F ⊥ u ( F k ) . (1.16)Here A ∈ R × and B ∈ R are the Burnett functions with entries A ij = n ( v i − u i )( v j − u j ) T − δ ij | v − u | T o √ M (1 ≤ i, j ≤ , B i = v i − u i √ T (cid:16) | v − u | T − (cid:17) √ M (1 ≤ i ≤ . (1.17)Finally, the initial data of (1.15) are imposed on( ρ k , u k , θ k )(0 , x ) = ( ρ ink , u ink , θ ink )( x ) ∈ R × R × R , k = 1 , , , · · · . (1.18)1.3.2. Viscous boundary layer expansion.
In viscous layer, the scaled normal coordinate isneeded: ζ = x √ ε . (1.19)As shown in Sone’s book [38], the viscous boundary layer expansion has the form F bε ( t, ¯ x, ζ ) ∼ X k ≥ √ ε k F bk ( t, ¯ x, ζ, v ) , (1.20)where, throughout our paper, the far field condition is always assumed: F bk ( t, ¯ x, ζ, v ) → , as ζ → + ∞ . (1.21)By similar calculation in [23], plugging F ε + F bε into the Boltzmann equation (1.1) gives √ ε − : 0 = B ( M , F b ) + B ( F b , M ) , OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 7 √ ε : v · ∂ ζ F b = (cid:2) B ( M , F b ) + B ( F b , M ) (cid:3) + [ B ( F , F b ) + B ( F b , F )]+ B ( F b , F b ) + ζ (cid:2) B ( M (1) , F b ) + B ( F b , M (1) ) (cid:3) , √ ε : ∂ t F b + ¯ v · ∇ ¯ x F b + v · ∂ ζ F b = (cid:2) B ( M , F b ) + B ( F b , M ) (cid:3) + ζ (cid:2) B ( M (1) , F b ) + B ( F b , M (1) ) (cid:3) + ζ (cid:2) B ( M (2) , F b ) + B ( F b , M (2) ) (cid:3) + (cid:2) B ( F , F b ) + B ( F b , F ) (cid:3) + (cid:2) B ( F , F b ) + B ( F b , F ) (cid:3) + ζ (cid:2) B ( F (1)1 , F b ) + B ( F b , F (1)1 ) (cid:3) + (cid:2) B ( F b , F b ) + B ( F b , F b ) (cid:3) , · · · · · ·√ ε k : ∂ t F bk + ¯ v · ∇ ¯ x F bk + v · ∂ ζ F bk +1 = X i + j = k +2 ,i,j ≥ B ( F bi , F bj ) + h B ( M , F bk +2 ) + B ( F bk +2 , M ) i + X l + j = k +2 , ≤ l ≤ N ,j ≥ ζ l l ! h B ( M ( l ) , F bj ) + B ( F bj , M ( l ) ) i + X i + j = k +2 ,i,j ≥ h B ( F i , F bj ) + B ( F bj , F i ) i + X i + j + l = k +2 , ≤ l ≤ N ,i,j ≥ ζ l l ! h B ( F ( l ) i , F bj ) + B ( F bj , F ( l ) i ) i for k ≥ , (1.22)where the Taylor expansion at x = 0 is used: M = M + X ≤ l ≤ N ζ l l ! M ( l ) + ζ N +1 ( N +1)! f M ( N +1) ,F i = F i + X ≤ l ≤ N ζ l l ! F ( l ) i + ζ N +1 ( N +1)! e F ( N +1) i . Here the number N ∈ N + will be chosen later.Let f bk = F bk √ M , (1.23)which can be decomposed as f bk = P f bk + ( I − P ) f bk = n ρ bk ρ + u bk · v − u T + θ bk T ( | v − u | T − o √ M + ( I − P ) f bk . Here u bk = ( u bk, , u bk, , u bk, ) ∈ R . Furthermore, let p bk = ρ θ bk +3 T ρ bk . (1.24)Following the Theorem 1.1 of [23], the following relations are obyed u b , ( t, ¯ x, ζ ) ≡ , p b ( t, ¯ x, ζ ) ≡ , ∀ ( t, ¯ x, ζ ) ∈ [0 , τ ] × R × R + , (1.25) NING JIANG, YI-LONG LUO, AND SHAOJUN TANG and ( u bk, , u bk, , θ bk ) ( k ≥
1) satisfy the following linear compressible Prandtl-type equations ρ ( ∂ t + ¯ u · ∇ ¯ x ) u bk,i + ρ ( ∂ x u ζ + u , ) ∂ ζ u bk,i + ρ ¯ u bk · ∇ ¯ x u i + ∂ x p T θ bk = µ ( T ) ∂ ζ u bk,i + f bk − ,i ( i = 1 , ,ρ ∂ t θ bk + ρ ¯ u · ∇ ¯ x θ bk + ρ (cid:0) ∂ x u ζ + u , (cid:1) ∂ ζ θ bk + ρ div x u θ bk = κ ( T ) ∂ ζζ θ bk + g bk − , lim ζ →∞ (¯ u bk , θ bk )( t, ¯ x, ζ ) =0 , (1.26)and ( I − P ) f bk +1 , u bk +1 , , p bk +1 are determined by the equations( I − P ) f bk +1 = ( L ) − n − ( I − P )( v ∂ ζ P f bk ) + ζ √ M (cid:2) B ( M (1) , √ M P f bk )+ B ( √ M P f bk , M (1) ) (cid:3) + √ M (cid:2) B ( F , √ M P f bk ) + B ( √ M P f bk , F ) (cid:3) + √ M (cid:2) B ( √ M f b , √ M P f bk ) + B ( √ M P f bk , √ M f b ) (cid:3)o + J bk − , (1.27)and ∂ ζ u bk +1 , = − ρ ( ∂ t ρ bk + div ¯ x ( ρ ¯ u bk + ρ bk ¯ u )) , lim ζ →∞ u bk +1 , ( t, ¯ x, ζ ) = 0 , (1.28)and ∂ ζ p bk +1 = − ρ ∂ t u bk, − ρ ¯ u ·∇ ¯ x u bk, + ρ ∂ x u u bk, + µ ( T ) ∂ ζζ u bk, − ρ ∂ ζ h(cid:0) ∂ x u ζ + u , (cid:1) u bk, i − ∂ ζ h T A , J bk − i + W bk − , , lim ζ →∞ θ bk +1 ( t, ¯ x, ζ ) = 0 , (1.29)where the source terms f bk − ,i ( i = 1 ,
2) and g bk − are f bk − ,i = − ρ ∂ ζ [( ∂ x u i ζ + u ,i + u b ,i ) u bk, ] − ( ∂ x i − ∂ xi p p ) p bk + W bk − ,i − T ∂ ζ h J bk − , A i i , g bk − = − ρ ∂ ζ h(cid:0) ∂ ζ T ζ + θ + θ b (cid:1) u bk, i + H bk − − ( T ) ∂ ζ h J bk − , B i + (cid:8) ∂ t + 2¯ u · ∇ ¯ x + div x u (cid:9) p bk , (1.30)and W bk − ,i = − X j =1 ∂ x j h T ( I − P ) f bk , A ij i , for i = 1 , , , (1.31) H bk − = − X j =1 ∂ x j h ( T ) B j + X l =1 T u l A jl , ( I − P ) f bk i − u · ¯ W bk − , (1.32)and J bk − =( L ) − n − ( I − P ) (cid:16) √ M (cid:8) ∂ t + ¯ v · ∇ ¯ x (cid:9) F bk − (cid:17) − ( I − P ) (cid:0) v ∂ ζ ( I − P (cid:1) f bk (cid:1) + X l + j = k +1 , ≤ l ≤ N,j ≥ ζ l l ! 1 √ M (cid:2) B ( M ( l ) , √ M f bj ) + B ( √ M f bj , M ( l ) ) (cid:3) In [23], the same system was named as a linear parabolic system.
OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 9 + X l + j = k +1 ,i ≥ ,j ≥ √ M (cid:2) B ( F i , √ M f bj ) + B ( √ M f bj , F i ) (cid:3) + X l + j = k +1 ,i,j ≥ √ M (cid:2) B ( √ M f bi , √ M f bj ) + B ( √ M f bj , √ M f bi ) (cid:3) + X i + j + k = k +1 , ≤ l ≤ N,i,j ≥ √ M ζ l l ! (cid:2) B ( F ( l ) i , √ M f bj ) + B ( √ M f bj , F ( l ) i ) (cid:3) + ζ √ M (cid:2) B ( M (1) , √ M ( I − P ) f bk ) + B ( √ M ( I − P ) f bk , M (1) ) (cid:3) + √ M (cid:2) B ( F , √ M ( I − P ) f bk ) + B ( √ M ( I − P ) f bk , F ) (cid:3) + √ M (cid:2) B ( √ M f b , √ M ( I − P ) f bk ) + B ( √ M ( I − P ) f bk , √ M f b ) (cid:3)o . (1.33)We emphasize that W bk − , H bk − and J bk − depend on f bj (1 ≤ j ≤ k − k = 1, J b = W b = H b = f b = g b = 0. Actually, by (1.50) below, u , = 0 in (1.26). Finally,the initial conditions of (1.26) are imposed on(¯ u bk , θ bk )(0 , ¯ x, ζ ) = (¯ u b,ink , θ b,ink )(¯ x, ζ ) ∈ R × R , k = 1 , , , · · · (1.34)with lim ζ →∞ (¯ u b,ink , θ b,ink )(¯ x, ζ ) = 0.1.3.3. Knudsen boundary layer expansion.
In Knudsen layer, the new scaled normal coordi-nate is introduced: ξ = x ε . (1.35)Then the Knudsen boundary expansion is defined as F bbε ( t, ¯ x, ξ, v ) ∼ X k ≥ √ ε k F bbk ( t, ¯ x, ξ, v ) . (1.36)From plugging F ε + F bε + F bbε in (1.1), √ ε − : v · ∂ ξ F bb = B ( M , F bb ) + B ( F bb , M ) √ ε : v · ∂ ξ F bb − (cid:2) B ( M , F bb ) + B ( F bb , M ) (cid:3) = B ( F + F b, , F bb ) + B ( F bb , F + F b, ) + B ( F bb , F bb ) ...... √ ε k : v · ∂ ξ F bbk +2 − (cid:2) B ( M , F bbk +2 ) + B ( F bbk +2 , M ) (cid:3) = − (cid:8) ∂ t + ¯ v · ∇ ¯ x (cid:9) F bbk + X j +2 l = k +2 , ≤ l ≤ N,j ≥ ξ l l ! (cid:2) B ( M ( l ) , F bbj ) + B ( F bbj , M ( l ) ) (cid:3) + X i + j = k +2 ,i,j ≥ (cid:2) B ( F i + F b, i , F bbj ) + B ( F bbj , F i + F b, i ) + B ( F bbi , F bbj ) (cid:3) + X i +2 l + j = k +2 , ≤ l ≤ N,i,j ≥ ξ l l ! (cid:2) B ( F ( l ) i , F bbj ) + B ( F bbj , F ( l ) i ) (cid:3) + X i + l + j = k +2 , ≤ l ≤ N,i,j ≥ ξ l l ! (cid:2) B ( F b, ( l ) i , F bbj ) + B ( F bbj , F b, ( l ) i ) (cid:3) , (1.37) where the Taylor expansion of F bi at ζ = 0 is utilized: F bi = F b, i + X ≤ l ≤ N ξ l l ! F b, ( l ) i + ξ N +1 ( N +1)! e F b, ( N +1) i . Similar in viscous layer, let f bbk = F bbk √ M , then (1.37) can be rewritten as v ∂ ξ f bbk + L f bbk = S bbk , k ≥ , (1.38)where S bbk = S bbk, + S bbk, with S bbk, = − P n ( ∂ t + ¯ v · ∇ ¯ x ) F bbk − √ M o ∈ N ,S bbk, = X j +2 l = k , ≤ l ≤ N,j ≥ ξ l l ! 1 √ M (cid:2) B ( M ( l ) , √ M f bbj ) + B ( √ M f bbj , M ( l ) ) (cid:3) + X i + j = k ,i,j ≥ √ M (cid:2) B ( F i + F b, i , √ M f bbj ) + B ( √ M f bbj , F i + F b, i ) (cid:3) + X i +2 l + j = k , ≤ l ≤ N,i,j ≥ ξ l l ! 1 √ M (cid:2) B ( F ( l ) i , √ M f bbj ) + B ( √ M f bbj , F ( l ) i ) (cid:3) + X i + l + j = k , ≤ l ≤ N,i,j ≥ ξ l l ! 1 √ M (cid:2) B ( F b, ( l ) i , √ M f bbj ) + B ( √ M f bbj , F b, ( l ) i ) (cid:3) (1.39)+ X i + j = k ,i,j ≥ √ M B ( √ M f bbi , √ M f bbj ) − ( I − P ) n { ∂ t + ¯ v · ∇ ¯ x } F bbk − √ M o ∈ ( N ) ⊥ . Here the notation F bb − = F bb = 0 are used, and S bb = S bb , = S bb , = 0 , S bb , = P S bb = 0 . Lemma 1.1 ([1]) . We assume that S bbk, = (cid:8) a k + b k · ( v − u ) + c k | v − u | (cid:9) √ M (1.40) satisfy lim ξ →∞ e ηξ | ( a k , b k , c k )( t, ¯ x, ξ ) | = 0 for some positive constant η > . Then there exists a function f bbk, = (cid:8) Ψ k v + Φ k, v ( v − u ) + Φ k, v ( v − u ) + Φ k, + Θ k v | v − u | (cid:9) √ M such that v ∂ ξ f bbk, − S bbk, ∈ ( N ) ⊥ , where Ψ k ( t, ¯ x, ξ ) = − ´ + ∞ ξ (cid:0) T a k + 3 c k (cid:1) ( t, ¯ x, s )d s , Φ k,i ( t, ¯ x, ξ ) = − ´ + ∞ ξ T b k,i ( t, ¯ x, s )d s , i = 1 , , Φ k, ( t, ¯ x, ξ ) = − ´ + ∞ ξ b k, ( t, ¯ x, s )d s , Θ k ( t, ¯ x, ξ ) = T ) ´ + ∞ ξ a k ( t, ¯ x, s )d s . (1.41) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 11
Moreover, there holds | v ∂ ξ f bbk, − S bbk, | ≤ C | ( a k , b k , c k )( t, ¯ x, ξ ) |h v i √ M , | f bbk, ( t, ¯ x, ξ, v ) | ≤ C h v i √ M ˆ ∞ ξ | ( a k , b k , c k ) | → as ξ → ∞ . It is easy to know that a = a = 0, b = b = 0, c = c = 0, Ψ = Ψ = 0 , Φ ,i = Φ ,i =0 ( i = 1 , ,
3) and Θ = Θ = 0.Denote by f bbk, = f bbk − f bbk, . We thereby see that v ∂ ξ f bbk, + L f bbk, = S bbk, − ( v ∂ ξ f bbk, − S bbk, ) ∈ ( N ) ⊥ , lim ξ →∞ f bbk, ( t, ¯ x, ξ, v ) = 0 . (1.42)Once we impose the following boundary condition on (1.42): f bbk, ( t, ¯ x, , ¯ v, v ) | v > = f bbk, ( t, ¯ x, , ¯ v, − v ) + f k ( t, ¯ x, ¯ v, − v ) (1.43)for some function f k ( t, ¯ x, ¯ v, v ) only defined for v < v >
0, Golse,Perthame and Sulem [15] proved that the solvability conditions of (1.42)-(1.43) were ˆ R v − ¯ u | v − u | v f k ( t, ¯ x, v ) √ M d v ≡ . (1.44)1.3.4. Expansions of the Maxwell reflection boundary condition (1.2) . In order to give suitableboundary conditions so that the interior expansions, viscous and Knudsen boundary layersare all well-posed, the expansions of F ε + F bε + F bbε will be plugged into the Maxwell reflectionboundary condition (1.2) and then utilize the solvability conditions (1.44) of the Knudsenboundary layer problem (1.42)-(1.43). More precisely, on Σ − , √ ε : L R M = 0 , √ ε : L R ( F + F b + F bb ) = L D M , · · · · · ·√ ε k : L R ( F k + F bk + F bbk ) = L D ( F k − + F bk − + F bbk − ) ( k ≥ , (1.45)where the operators L R and L D are defined as L R F = ( γ − − Lγ + ) F , L D F = √ π ( Kγ + − Lγ + ) F . (1.46)Actually, √ ε -order of (1.45) can imply the slip boundary condition (1.9) of the compress-ible Euler system (1.8). Recalling the definitions f k = F k √ M , f bk = F bk √ M , f bbk = F bbk √ M , f bbk, = f bbk − f bbk, for k ≥
1, where the functions f bbk, ( k ≥
1) are given in Lemma 1.1, we thereby obtain bydirect calculation and (1.45) that the functions f k ( t, ¯ x, ¯ v, v ) ( k ≥
1) in (1.43) are f k ( t, ¯ x, ¯ v, v ) = , if v > , ( f k + f bk + f bbk, )( t, ¯ x, , ¯ v, v ) − ( f k + f bk + f bbk, )( t, ¯ x, , ¯ v, − v )+ √ π (cid:8)(cid:2) h γ + ( f k − + f bk − + f bbk − ) i ∂ R √ M (cid:3) − ( f k − + f bk − + f bbk − ) (cid:9) ( t, ¯ x, , ¯ v, v ) , if v < , (1.47)where the symbol h γ + f i ∂ R means h γ + f i ∂ R = √ π M w ( v ) M ˆ v · n ( x ) > v · n ( x )( γ + f ) √ M d v , and the notations f = √ M , f b = f bb = 0 are used, for simplicity of presentation. Therefore, the following lemma holds:
Lemma 1.2.
Let the local Maxwellian of the boundary M w = M in (1.2) and f k ( t, ¯ x, ¯ v, v ) begiven in (1.47) . Then the solvability conditions (1.44) of the Knudsen boundary layer problem (1.42) - (1.43) imply that for k ≥ , the linear hyperbolic system (1.15) has the following slipboundary condition u k, ( t, ¯ x,
0) = − u bk, ( t, ¯ x, − T (Ψ k + 5 T Θ k )( t, ¯ x, ( ρ √ T +1) ρ √ π ˆ R ˆ −∞ v ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) √ M d¯ v d v = − ˆ + ∞ ρ (cid:2) ∂ t ρ bk − + div ¯ x ( ρ ¯ u bk − + ρ bk − ¯ u ) (cid:3) ( t, ¯ x, ζ )d ζ − T (Ψ k + 5 T Θ k )( t, ¯ x, ( ρ √ T +1) ρ √ π ˆ R ˆ −∞ v ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) √ M d¯ v d v , (1.48) and for k ≥ , the linear Prandtl-type equations (1.26) are of the Robin-type boundary condi-tions (cid:0) ∂ ζ u bk − ,i − ρ √ T (2+ ρ √ T ) µ ( T ) u bk − ,i (cid:1) ( t, ¯ x,
0) = Λ bk − ,i ( t, ¯ x ) , i = 1 , , (cid:0) ∂ ζ θ bk − − ρ √ T κ ( T ) (2 ρ √ T + √ π ρ + ) θ bk − (cid:1) ( t, ¯ x,
0) = Λ bk − ,θ ( t, ¯ x ) , (1.49) where Ψ k and Θ k are given in (1.41) , and Λ bk − ,i ( t, ¯ x ) = ρ √ T µ ( T ) u k − ,i ( t, ¯ x, µ ( T ) n ρ ( T ) Φ k,i + ρ (cid:2) ( u ,i + u b ,i ) u bk − , (cid:3) + T hA i , J bk − + ( I − P ) f k i o ( t, ¯ x, − √ πµ ( T ) ˆ R ˆ −∞ ( v i − u i ) v (cid:2) ( I − P )( f k − + f bk − ) + f bbk − (cid:3) ( t, ¯ x, , ¯ v, v ) √ M d¯ v d v , and Λ bk − ,θ ( t, ¯ x ) = T ) κ ( T ) hB , ( I − P ) f k + J bk − i ( t, ¯ x,
0) + κ ( T ) ρ (cid:2) ( θ + θ b ) u bk − , (cid:3) ( t, ¯ x, ρ T κ ( T ) (cid:8) T ) Θ k − ( + T − ρ ( T ) )(Ψ k − + 5 T Θ k − ) (cid:9) ( t, ¯ x, ρ √ T κ ( T ) (cid:8) ( ρ √ T + √ π ρ + 2) θ k − − √ T − ρ )( T ρ k − + p bk − ) (cid:9) ( t, ¯ x, − √ πκ ( T ) ˆ R ˆ −∞ v ( | v − u | − ρ ( T ) ) × (cid:2) ( I − P )( f k − + f bk − ) + f bbk − (cid:3) ( t, ¯ x, , ¯ v, v ) √ M d¯ v d v + √ πκ ( T ) ( ρ √ T + 1)( + T − ρ ( T ) ) × ˆ R ˆ −∞ v ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) √ M d¯ v d v . In particular, if k = 1 in (1.48) , we have u , = u , ( t, ¯ x,
0) = √ T ( ρ √ T + 1) . (1.50)The proof of the lemma will be given in Section 4. We remark that the Robin-type boundarycondition (1.49) will be Neumann-type boundary values and u , will vanish, if the Maxwellreflection boundary condition (1.2) is replaced by the specular reflection boundary condition,i.e., by letting α ε = 0, see [23]. OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 13
Truncations of the Hilbert expansion.
Our goal is to prove the compressible Euler limitfrom the scaled Boltzmann equation by above Hilbert type expansion. The key point isto prove that the remainders of expansion will go to zero as the Knudsen number ε → ε . Since the more terms areexpanded, the more special the solutions are. We hope that the terms in the expansion areas less as possible.Our truncated Hilbert expansion takes the following form: F ε ( t, x, v ) = M ( t, x, v ) + X k =1 (cid:8) F k ( t, x, v ) + F bk ( t, ¯ x, x √ ε , v ) + F bbk ( t, ¯ x, x ε , v ) (cid:9) + √ ε F R,ε ( t, x, v ) ≥ . (1.51)Here M is governed by the compressible Euler system (1.11)-(1.9)-(1.10).First, the interior expansions F = P ( F √ M ) √ M and F k = (cid:8) P ( F k √ M ) + ( I − P )( F k √ M ) (cid:9) √ M ( k = 2 , I − P )( F k √ M ) are given by (1.14), and the fluid variables ( ρ k , u k , θ k ) (1 ≤ k ≤
3) associated with P ( F k √ M ) are solutions of the linear hyperbolic system (1.15) withinitial data (1.18) and slip boundary condition (1.48). In this sense, F k (1 ≤ k ≤
3) arecompletely specified. For k = 4 ,
5, we take F k such that their kernel (fluid) parts vanish, i.e., F k = [( I − P ) F k √ M ] √ M , which are defined by (1.14). They thereby are not completely knownterms.Second, the viscous boundary layer expansions F b = ( P f b ) √ M and F bk = (cid:8) P f bk + ( I −P ) f bk (cid:9) √ M ( k = 2 , I − P ) f bk ( k = 2 ,
3) are given by (1.27), and P f bk (1 ≤ k ≤
3) correspond to the fluid variables ( ρ bk , u bk , θ bk ). (¯ u bk , θ bk ) are solutions of thelinear Prandtl-type system (1.26) with initial data (1.34) and Robin-type boundary conditions(1.49). The functions ( ρ bk , u bk, ) (1 ≤ k ≤
3) are expressed through equations (1.24), (1.25),(1.28) and (1.29), where the subscript k + 1 is replaced by k . Thus F k (1 ≤ k ≤
3) are allknown terms, so are F bk (1 ≤ k ≤ k = 4 ,
5, we choose F bk such that their fluid variablesvanish, hence, F bk = [( I − P ) f bk ] √ M , which are expressed by (1.27), which mean that theyare not completely specified.Finally, we observe from (1.38)-(1.39) and the boundary values (1.43)-(1.47) that the Knud-sen boundary layer expansions F bbk depends on F bbi ( i ≤ k −
1) and F bj , F j ( j ≤ k ). Here F bi = F bbi = F i − = 0 for subscript i ≤ F = M . Since M , F k and F bk ( k = 1 , , F bbk ( k = 1 , ,
3) are totally solved. Similarly, F bb and F bb are not completely solved. Consequently, the corresponding remainder shall be √ ε F R,ε as in (1.51). We also emphasize that the number N ∈ N + appeared in the Taylor expansionsbefore will be chosen by N = 4 for the Hilbert expansion (1.51).Consequently, from plugging (1.51) into (1.1)-(1.2), we obtain the remainder equation ∂ t F R,ε + v · ∇ x F R,ε − ε [ B ( M , F R,ε ) + B ( F R,ε , M )]= √ ε B ( F R,ε , F
R,ε ) + R ε + R bε + R bbε + X i =1 √ ε i − [ B ( F i + F bi + F bbi , F R,ε ) + B ( F R,ε , F i + F bi + F bbi )] (1.52)with Maxwell reflection type boundary condition γ − F R,ε = (1 − α ε ) Lγ + F R,ε + α ε Kγ + F R,ε + √ ε Γ ε on Σ − , (1.53)where Γ ε = α ε √ ε ( Kγ + − Lγ + )( F + F b + F bb ) , (1.54) and R ε = − ( ∂ t + v · ∇ x )( F + √ εF ) + X i + j ≥ ≤ i,j ≤ √ ε i + j − B ( F i , F j ) , (1.55)and R bε = − ( ∂ t + ¯ v · ∇ ¯ x )( F b + √ εF b ) − v ∂ ζ F b + X j + l ≥ ≤ j ≤ , ≤ l ≤ √ ε j + l − ζ l l ! [ B ( M ( l ) , F bj ) + B ( F bj , M ( l ) )]+ X i + j ≥ ≤ i,j ≤ √ ε i + j − [ B ( F i , F bj ) + B ( F bj , F i ) + B ( F bi , F bj )]+ X i + j + l ≥ ≤ i,j ≤ , ≤ l ≤ √ ε i + j + l − ζ l l ! [ B ( F ( l ) i , F bj ) + B ( F bj , F ( l ) i )]+ ζ
5! 5 X j =1 √ ε j − [ B ( f M (5) + X i =1 √ ε i e F (5) i , F bj ) + B ( F bj , f M (5) + X i =1 √ ε i e F (5) i )] , (1.56)and R bbε = − ( ∂ t + ¯ v · ∇ ¯ x )( F bb + √ εF bb )+ X j +2 l ≥ ≤ j ≤ , ≤ l ≤ √ ε j +2 l − ξ l l ! [ B ( M ( l ) , F bbj ) + B ( F bbj , M ( l ) )]+ X i + j ≥ ≤ i,j ≤ √ ε i + j − [ B ( F i + F b, i , F bbj ) + B ( F bbj , F i + F b, i ) + B ( F bbi , F bbj )]+ X i + j +2 l ≥ ≤ i,j ≤ , ≤ l ≤ √ ε i + j +2 l − ξ l l ! [ B ( e F ( l ) i , F bbj ) + B ( F bbj , e F ( l ) i )]+ X i + j + l ≥ ≤ i,j ≤ , ≤ l ≤ √ ε i + j + l − ξ l l ! [ B ( e F b, ( l ) i , F bbj ) + B ( F bbj , e F b, ( l ) i )]+ ξ
5! 5 X j =1 √ ε j − [ B ( √ ε f M (5) + X i =1 √ ε i +5 e F (5) i + √ ε i e F b, (5) i , F bbj )+ B ( F bbj , √ ε f M (5) + X i =1 √ ε i +5 e F (5) i + √ ε i e F b, (5) i )] . (1.57)Furthermore, the following initial data are imposed on the remainder equation (1.52): F R,ε (0 , x, v ) = F inR,ε ( x, v ) , (1.58)which satisfies the compatibility condition γ − F inR,ε = (1 − α ε ) Lγ + F inR,ε + α ε Kγ + F inR,ε + √ ε Γ ε | t =0 on Σ − .For the remainder F R,ε , let f R,ε = F R,ε √ M , h ℓR,ε = h v i ℓ F R,ε √ M M (1.59)for ℓ ≥ − γ , where the global Maxwellian M M = M M ( v ) is introduced by [9] M M = πT M ) exp n − | v | T M o . (1.60) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 15
Here the constant T M satisfies T M < max t ∈ [0 ,τ ] ,x ∈ R T ( t, x ) < T M . (1.61)Then there exists constants C , C such that for some < z < t, x, v ) ∈ [0 , τ ] × R × R , the following inequality holds: C M M ≤ M ≤ C ( M M ) z . (1.62)1.4. Main results.
We first introduce some notations for convenience for stating our maintheorem. For multi-indexes α = ( α , α , · · · , α m ) , α ′ = ( α ′ , α ′ , · · · , α ′ m ) ∈ N m , the symbol α ≤ α ′ means α i ≤ α ′ i ( i = 1 , , · · · , m ) and | α | = α + α + · · · + α m .In order to quantitatively describe the linear hyperbolic system (1.15) in the half-space R ,let ∂ αt, ¯ x = ∂ α t ∂ α x ∂ α x , where α = ( α , α , α ) ∈ N , and let k f ( t ) k H k ( R ) = X | α | + i ≤ k k ∂ αt, ¯ x ∂ ix f ( t ) k L ( R ) , k g ( t ) k H k ( R ) = X | α |≤ k k ∂ αt, ¯ x g ( t ) k L ( R ) (1.63)for functions f ( t ) = f ( t, ¯ x, x ) and g ( t ) = g ( t, ¯ x ). We remark that, for a function f = f (¯ x, x ) independent of the variable t , k f k H k ( R ) is equivalent to the standard Sobolev norm k f k H k ( R ) .While characterizing quantitatively the linear Prandtl-type system (1.26) associated withthe macroscopic parts of the viscous boundary layer, some new norms are required to beintroduced. For l ≥
0, the weighted norm is defined as k f k L l = ˆ R ˆ R + (1 + ζ ) l | f (¯ x, ζ ) | d¯ x d ζ . (1.64)We further introduce a weighted Sobolev space H rl ( R ) for any r, l ≥
0. Denote by the 2Dmulti-index β = ( β , β ) ∈ N . For any l, r ≥
0, let l j = l + 2( r − j ) , ≤ j ≤ r . (1.65)We then introduce the norms k f ( t ) k l,r,n = X γ + | β | = r − n k ∂ γt ∂ β ¯ x ∂ nζ f ( t ) k L lr (0 ≤ n ≤ r ) , k f ( t ) k l,r = r X n =0 k f ( t ) k l,r,n = X γ + | β | + n = r k ∂ γt ∂ β ¯ x ∂ nζ f ( t ) k L lr , k f ( t ) k H rl,n ( R ) = r X j =0 k f ( t ) k l,j,n ( n = 0 , , · · · , r ) , k f ( t ) k H rl ( R ) = r X j =0 k f ( t ) k l,j = r X n =0 k f ( t ) k H rl,n ( R ) (1.66)for function f = f ( t, ¯ x, ζ ). For g = g (¯ x, ζ ), let k g k H rl ( R ) = r X j =0 X | β | + n = j k ∂ β ¯ x ∂ nζ g k L lj . (1.67) Similarly, for function h = h ( t, ¯ x ), let k h ( t ) k H r ( R ) = r X j =0 k h ( t ) k ,j = r X j =0 X γ + | β | = j k ∂ γt ∂ β ¯ x h ( t ) k L ( R ) . (1.68)In the above norms, one order time derivative is equivalent to two orders space derivative.We now clarify the initial data of the scaled Boltzmann equation (1.1). Let M in ( x, v ) = ρ in ( x )[2 πT in ( x )] exp n − | v − u in ( x ) | T in ( x ) o . (1.69)For 1 ≤ k ≤ F ink ( x, v ), F b,ink (¯ x, x √ ε , v ) and F bb,ink (¯ x, x ε , v ) can be constructed by the sameways of constructing the expansions F k ( t, x, v ), F bk ( t, ¯ x, x √ ε , v ) and F bbk ( t, ¯ x, x ε , v ) in Subsection1.3.5, respectively. More precisely, for k = 1 , ,
3, it just replaces M , ( ρ k , u k , θ k ) and (¯ u bk , θ bk )by M in , ( ρ ink , u ink , θ ink ) and (¯ u b,ink , θ b,ink ), respectively. Here we further assume the initial data( ρ ink , u ink , θ ink ) and (¯ u b,ink , θ b,ink ) (1 ≤ k ≤
3) compatibly satisfy the conditions (1.48) and (1.49),respectively. We impose the well-prepared initial data on the scaled Boltzmann equation (1.1) F ε (0 , x, v ) = M in ( x, v ) + X k =1 √ ε k (cid:8) F ink ( x, v ) + F b,ink (¯ x, x √ ε , v ) + F bb,ink (¯ x, x ε , v ) (cid:9) + √ ε F inR,ε ( x, v ) ≥ . (1.70)Furthermore, for the local Maxwellian distribution M w ( t, x ) of the boundary given in (1.4),we take the special form M w ( t, ¯ x, v ) = M ( t, ¯ x, v ) . (1.71)We now state our main theorem. Theorem 1.1.
Consider the hard potential interaction (0 ≤ γ ≤ Boltzmann collisionkernel B with an angular cutoff ( see (1.3)) . Let ℓ ≥ − γ , and integers s , s k , s bk , s bbk , l bk (1 ≤ k ≤ be described as in Proposition 2.1. Assume that k ( ρ in , u in , T in ) k H s ( R ) < ∞ and E in := X k =1 n k ( ρ ink , u ink , θ ink ) k H sk ( R ) + k (¯ u b,ink , θ b,ink ) k H sbklbk ( R ) o < ∞ . (1.72) Let ( ρ, u , T ) be the solution to the compressible Euler equations (1.8) over the time interval t ∈ [0 , τ ] constructed in Proposition 1.1, which determines the local Maxwellian M definedin (1.7) . For ≤ k ≤ , let F k ( t, x, v ) , F bk ( t, ¯ x, x √ ε , v ) and F bbk ( t, ¯ x, x ε , v ) be constructed inProposition 2.1. The local Maxwellian M w ( t, ¯ x, v ) of the boundary is assumed as in (1.71) .There is a small constant ε > such that if for ε ∈ (0 , ε ) k F inR,ε √ M in k + √ ε kh v i ℓ F inR,ε √ M M k ∞ < ∞ , then the scaled Boltzmann equation (1.1) with Maxwell reflection boundary condition (1.2) and well-prepared initial data (1.70) admits a unique solution for ε ∈ (0 , ε ) over the timeinterval t ∈ [0 , τ ] with the expanded form (1.51) , i.e., F ε ( t, x, v ) = M ( t, x, v ) + X k =1 (cid:8) F k ( t, x, v ) + F bk ( t, ¯ x, x √ ε , v ) + F bbk ( t, ¯ x, x ε , v ) (cid:9) + √ ε F R,ε ( t, x, v ) ≥ , where the remainder F R,ε ( t, x, v ) satisfies sup t ∈ [0 ,τ ] n k F R,ε ( t ) √ M k + √ ε kh v i ℓ F R,ε ( t ) √ M M k ∞ o ≤ C ( τ, k ( ρ in , u in , T in ) k H s ( R ) , E in ) < ∞ . OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 17
Remark 1.1.
Together with Proposition 2.1, Theorem 1.1 shows that as ε → , sup t ∈ [0 ,τ ] n k ( F ε − M √ M )( t ) k + kh v i ℓ ( F ε − M √ M M )( t ) k ∞ o ≤ C √ ε → . Therefore, while imposed on the well-prepared initial data and accommodation coefficients α ε = O ( √ ε ) as ε → , we have justified the hydrodynamic limit from the Boltzmann equationwith Maxwell reflection boundary condition to the compressible Euler system with slip boundarycondition for the half-space problem. Sketch of ideas and novelties.
Contrast with the work [23], where the specular re-flection boundary condition (in short, SRBC) for the scaled Boltzmann equation (1.1) wasconsidered, we consider the Maxwell reflection boundary condition (briefly, MRBC). We firstpoint out the formal differences between [23] and our work. For the MRBC case, the slipboundary value u , ( t, ¯ x,
0) = √ T ( ρ √ T + 1) > ρ , u , θ ) (see (2.2)) does not vanish, and for the SRBC case, u , ( t, ¯ x, ≡
0. Thetypes of boundary conditions for the linear compressible Prandtl-type equations (1.26) arealso different. For the MRBC case, we obtain Robin-type boundary conditions (see in (1.49)),while the Neumann-type was derived in the SRBC case. We emphasize that the structures ofboundary energy will be more subtle in the Robin-type than that in the Neumann-type whileproving the existence of the system (1.26). Furthermore, the way to deal with the boundaryintegral − ˜ ∂ R × R v | f R,ε ( t, ¯ x, , v ) | d v d¯ x in L estimates on the remainder f R,ε is evenmuch more different. For the SRBC case, the boundary integral is zero. However, for theMRBC case, a key boundary energy dissipative rate ˜ Σ − | v · n ( x ) || Lγ + f R,ε | d σ Σ − is impliedby the boundary integral, provided that the wall Maxwellian M w ( t, ¯ x, v ) in (1.4) is assumedto be M . For details see Lemma 3.1. There are also many other small and not so essentialdifferences, we will not list them all here.In the Hilbert expansion approach, we hope the number of expanded terms as small aspossible. In this paper, we want to employ L - L ∞ framework (see [20, 22], for instance) toprove that the remainder √ ε F R,ε ( t, x, v ) is a higher order infinitesimal as ε →
0. We claimthat the expansion (1.51) is optimal in this framework. The uniform L - L ∞ estimates relyon an interplay between L and L ∞ estimates for the Boltzmann equation. On one hand,following [22], L -estimates will be such that the norm k f R,ε k can be bounded by √ ε k h ℓR,ε k ∞ .On the other hand, the norm √ ε k h ℓR,ε k ∞ will be dominated by √ ε k h ℓR,ε (0) k ∞ + k f R,ε k .For details see Lemma 3.1 and Lemma 3.2 below. Based on the previous interplay, we canobtain the uniform bound of the quantitysup t ∈ [0 ,τ ] n k f R,ε ( t ) k + k√ ε h ℓR,ε ( t ) k ∞ o ≤ C , (1.73)provided that k f R,ε (0) k + k√ ε h ℓR,ε (0) k ∞ ≤ C initially holds uniformly for small ε > √ ε F R,ε is bounded by O ( √ ε ) in L - L ∞ framework.Moreover, the order √ ε occurred in the uniform bound (1.73) is independent of the numberof expanded terms. Indeed, no matter how many terms are expanded, the remainder equationswill contain a common linear structure ∂ t f R,ε + v · ∇ x f R,ε + ε L f R,ε = some other terms . (1.74)In L estimates, the hypocoercivity of the operator L in (1.13) produces a dissipative rate c ε k ( I − P ) f R,ε k ν , which is the key point to deal with the singular terms. However, in L ∞ estimates, after transforming f R,ε to h ℓR,ε (by using (1.59)), the equation (1.74) will be inte-grated along the trajectory [ X cl ( s ; t, x, v ) , V cl ( s ; t, x, v )] (see (3.16) below). We then have thepointwise estimate | h ℓR,ε ( t, x, v ) | ≤ C k h ℓR,ε (0) k ∞ + ε ˆ t exp n − ε ˆ ts ν ( φ )d φ o × ˆ R × R (cid:12)(cid:12) k ℓ (cid:0) V cl ( s ) , v ′ (cid:1) k ℓ ( v ′ , v ′′ ) (cid:12)(cid:12) ˆ s exp n − ε ˆ ss ν ( v ′ )( φ )d φ o × (cid:12)(cid:12) h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) , v ′′ (cid:1)(cid:12)(cid:12) d v ′′ d v ′ d s d s + some other controllable quantities , where k ℓ ( v, v ′ ) satisfies (3.18). As shown in Case 3b of the proof for Lemma 3.2, the secondterm in RHS of the previous inequality will be bounded by CN sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ + C √ ε sup s ∈ [0 ,t ] k f R,ε ( s ) k for N > L estimates, we fail to seek a similar hypoco-ercivity (1.13) in L ∞ estimates, so that the singular term ε L f R,ε is roughly treated as asource term. Therefore, there is a √ ε -order disparity between sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ andsup s ∈ [0 ,t ] k f R,ε ( s ) k . Once we expand less number of the terms than that in (1.51), the corre-sponding remainder √ ε m F R,ε ( m ≤
3) will only be bounded by O ( √ ε m − ), which is failed toachieve our goal. Nevertheless, this disparity maybe not intrinsic, and perhaps be covered insome other frameworks. So, we may expand less terms in the Hilbert expansion when provingthe compressible Euler limit by employing the Hilbert expansion approach.When constructing the viscous boundary layers F bk , one of key points is to focus on theexistence of the smooth solutions to the linear compressible Prandtl-type equations (2.6) (or(1.26)) with Robin-type boundary conditions (2.8) (or (1.49)), which derives from the Maxwellreflection boundary condition (1.2). Note that the linear compressible Prandtl-type system(2.6) is a degenerated parabolic system with nontrivial Robin-type boundary values (2.8).For the linear system, we can introduce some explicit functions to zeroize the inhomogeneousboundary values (see (5.4)), which actually converts the boundary values into the form ofexternal force terms ( ˜ f , ˜ g ), see (5.6) below. We therefore obtain the system (5.5) with zeroboundary values. Due to (5.5) is a degenerated parabolic system, we can construct a linearparabolic approximate system (5.8) while proving the existence of (5.5). When deriving theuniform bounds of the approximate system in the space L ∞ (0 , τ ; H kl ( R )), the key point isto deal with the boundary values of the higher order normal ζ -derivatives on { ζ = 0 } . Theidea is employing the structures of equations to convert the higher normal ζ -derivatives tothe values of tangential ¯ x -derivatives and time derivatives on { ζ = 0 } , see Lemma 5.2 below.Then we can find a key boundary energysup t ∈ [0 ,τ ] k B c ( u, θ )( t ) k H k − ( R ) + ˆ τ k B c ( u, θ )( s ) k H k ( R ) d s , where B c ( u, θ ) = ( u, θ ) | ζ =0 − ( R u b, R θ a ) (see Lemma 2.2).Furthermore, the solution ( u, θ )( t, ¯ x, ζ ) to the linear compressible Prandtl-type equations(2.6) will produce a loss of derivatives with respect to the boundary values and source terms.More precisely, as shown in Lemma 2.2, if ( u, θ )( t, ¯ x, ζ ) is in L ∞ (0 , τ ; H kl ( R )), the sourceterms ( f, g )( t, ¯ x, ζ ) and the boundary values ( b, a )( t, ¯ x ) should be in L ∞ (0 , τ ; H k +1 l ( R )) and L ∞ (0 , τ ; H k +3 ( R )), respectively. The reasons are as follows. For the system (5.5) zeroedboundary values, we shall use the structures of the equations to dominate the boundaryvalues of higher order ζ -derivatives, which involve the new source term ( ˜ f , ˜ g ) restricted on { ζ = 0 } . As shown in Lemma 5.3, in order to control the boundary values in the space L ∞ (0 , τ ; H k ( R )), the ( ˜ f , ˜ g ) | ζ =0 should be also in L ∞ (0 , τ ; H k ( R )). Combining the traceinequalities in Lemma 5.1, the new source terms should be in H k +1 l ( R ), so should be thesource terms ( f, g ). Noticing that ∂ t ( u b , θ a ) ∼ ∂ t ( b, a ) occur in the new source terms ( ˜ f , ˜ g ) anda first order time derivative is equivalent to a second order tangential ¯ x -derivative in the space H k ( R ), we therefore see that the boundary values ( b, a )( t, ¯ x ) should be in L ∞ (0 , τ ; H k +3 ( R )). OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 19
Organization of this paper.
In next section, the uniform bounds to the expansions F k , F bk and F bbk (1 ≤ k ≤
5) appeared in (1.51) is derived. Section 3 devotes to prove Theorem1.1 by using the L - L ∞ arguments. More precisely, the L and L ∞ estimates are given. InSection 4, it is formally derived the boundary conditions for the linear hyperbolic system(2.2) and linear compressible Prandtl-type equations (1.26), namely, prove Lemma 1.2. InSection 5, the proof for the existence of the linear compressible Prandtl-type system (2.6)with Robin-type boundary values (2.8) is given. Hence, we prove Lemma 2.2.2. Uniform bounds to the solutions of expansions
In this section, we aim at deriving the uniform bounds to the expansions F k , F bk and F bbk (1 ≤ k ≤ Some auxiliary systems.
In the expansions (1.51), there are some common typessystems, which are only distinguished by some known coefficients. For 1 ≤ k ≤
3, the fluidparts of F k satisfy a linear hyperbolic system, and that of F bk obey a linear compressiblePrandtl-type systems with Robin-type boundary conditions. Moreover, F bbk (1 ≤ k ≤ ρ, u , T ) over the time interval t ∈ [0 , τ ] is the smooth solution tothe compressible Euler system (1.1) given in Proposition 1.1 and p = ρT . We also employ thenotation E k := sup t ∈ [0 ,τ ] k ( ρ, u , T )( t ) k H k ( R ) (2.1)for any integer k ≥ ρ, ˜ u, ˜ θ )( t, x ) satisfy the following linear hyperbolic system: ∂ t ˜ ρ + div x ( ρ ˜ u + ˜ ρ u ) = 0 ,ρ ( ∂ t ˜ u + ˜ u · ∇ x u + u · ∇ x ˜ u ) − ∇ x pρ ˜ ρ + ∇ (cid:0) ρ ˜ θ +3 T ˜ ρ (cid:1) = f ,ρ (cid:2) ∂ t ˜ θ + (cid:0) ˜ θ div x u + 3 T div x ˜ u (cid:1) + u · ∇ x ˜ θ + 3˜ u · ∇ x T (cid:3) = g , (2.2)with ( t, x ) ∈ (0 , τ ) × R . We impose the boundary condition˜ u ( t, ¯ x,
0) = d ( t, x ) , ∀ ( t, ¯ x ) ∈ (0 , τ ) × R (2.3)and initial condition (˜ ρ, ˜ u, ˜ θ )(0 , x ) = (˜ ρ , ˜ u , ˜ θ )( x ) . (2.4)Recalling the definitions of the space H k ( R ) and H k ( R ) in (1.63), we then have thefollowing lemma for the local existence of smooth solution for the linear hyperbolic system(2.2). Lemma 2.1 ([23]) . Assume that k (˜ ρ , ˜ u , ˜ θ ) k H k ( R ) + sup t ∈ (0 ,τ ) (cid:2) k ( f, g )( t ) k H k +1 ( R ) + k d ( t ) k H k +2 ( R ) (cid:3) < + ∞ with k ≥ , and the compatibility condition is satisfied for the initial data. Then there existsa unique smooth solution to (2.2) - (2.4) with boundary condition (2.3) for t ∈ [0 , τ ] , such that sup t ∈ [0 ,τ ] k (˜ ρ, ˜ u, ˜ θ )( t ) k H k ( R ) ≤ C ( τ, E k +2 ) n k (˜ ρ , ˜ u , ˜ θ )( t ) k H k ( R ) + sup t ∈ [0 ,τ ] (cid:2) k ( f, g )( t ) k H k +1 ( R ) + k d ( t ) k H k +2 ( R ) (cid:3)o . (2.5) As shown in (1.26) and Lemma 1.2, to construct the solution of viscous boundary layer,the following linear compressible Prandtl type system of ( u, θ ) = ( u , u , θ )( t, ¯ x, ζ ) should beconsidered: for i = 1 , ( ρ ∂ t u i + ρ ¯ u · ∇ ¯ x u i + ρ ( ∂ x u ζ + u , ) ∂ ζ u i + ρ u · ∇ ¯ x u i + ∂ xi p T θ = µ ( T ) ∂ ζ u i + f i ,ρ ∂ t θ + ρ ¯ u · ∇ ¯ x θ + ρ ( ∂ x u ζ + u . ) ∂ ζ θ + ρ div x u θ = κ ( T ) ∂ ζ θ + g , (2.6)where ( t, ¯ x, ζ ) ∈ [0 , τ ] × R × R + , u , = √ T (1 + ρ √ T ) given in (1.50), and ( f , f , g )( t, ¯ x, ζ )are the known source functions. By the properties of compressible Euler system, we canassume the viscosity and heat conductivity, respectively, µ ( T ) ≥ µ > , κ ( T ) ≥ κ > µ , κ >
0. The system (2.6) should be imposed on the non-homogeneousRobin boundary conditions, namely, (cid:0) ∂ ζ u i − R u u i (cid:1)(cid:12)(cid:12) ζ =0 = b i ( t, ¯ x ) , (cid:0) ∂ ζ θ − R θ θ (cid:1)(cid:12)(cid:12) ζ =0 = a ( t, ¯ x ) , lim ζ →∞ ( u, θ )( t, ¯ x, ζ ) = 0 , (2.8)where the known functions ( R u , R θ ) = ( R u , R θ )( ρ , u , T ), and R u ≥ R u > , R θ ≥ R θ > R u and R θ . We impose (2.6) with initial data u ( t, ¯ x, ζ ) | t =0 = u (¯ x, ζ ) , θ ( t, ¯ x, ζ ) | t =0 = θ (¯ x, ζ ) , (2.10)which satisfy the corresponding compatibility conditions. We remark that the θ -equation in(2.6)-(2.10) is independent of u -equation. As a result, we can first solve θ , and then solve u ,where θ involved in u -equation can be regarded as the known source function.Recalling the definitions of H kl ( R ) and H k ( R ) in (1.67) and (1.68), we give the followingresults. Lemma 2.2.
Let k ≥ , l ≥ , and the compatibility conditions for the initial data (2.10) besatisfied. Assume E := k u k H kl ( R ) + sup t ∈ [0 ,τ ] (cid:0) k b ( t ) k H k +3 ( R ) + k f ( t ) k H k +1 l ( R ) (cid:1) + k θ k H kl ( R ) + sup t ∈ [0 ,τ ] (cid:0) k a ( t ) k H k +3 ( R ) + k g ( t ) k H k +1 l ( R ) (cid:1) < ∞ . (2.11) Then there exists a unique smooth solution ( u, θ )( t, ¯ x, ζ ) to (2.6) - (2.10) over t ∈ [0 , τ ] satisfy-ing sup t ∈ [0 ,τ ] (cid:16) k ( u, θ )( t ) k H kl ( R ) + k B c ( u, θ )( t ) k H k − ( R ) + ˆ t k ∂ ζ ( u, θ )( s ) k H kl ( R ) + k B c ( u, θ )( s ) k H k ( R ) d s (cid:17) ≤ C ( τ, E k +1 ) E , (2.12) where B c ( u, θ ) = ( u, θ ) | ζ =0 − ( R u b, R θ a ) . The proof of Lemma 2.2 will be given in Section 5.We then introduce a result on the existence of solutions of the Knudsen boundary layerproblem. Consider the following half-space linear equation for f ( t, ¯ x, ξ, v ) over ( t, ¯ x, ξ, v ) ∈ [0 , τ ] × R × R + × R : v ∂ ξ f + L f = S ( t, ¯ x, ξ, v ) ,f ( t, ¯ x, , ¯ v, v ) | v > = f ( t, ¯ x, , ¯ v, − v ) + f k ( t, ¯ x, ¯ v, − v ) , lim ξ →∞ f ( t, ¯ x, ξ, v ) = 0 . (2.13) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 21
For the above equation, Golse, Perthame and Sulem [15] has proved an existence result inthe norm ´ R + × R h v i e ηξ f d v d ξ + ´ R k e ηξ f k L ∞ ξ d v . Due to the higher order derivatives arerequired, Guo, Huang and Wang [23] similarly gave a modified version of existence result,hence the following lemma. Lemma 2.3.
Let < a < , r ≥ and l ≥ . For each ( t, ¯ x ) ∈ [0 , τ ] × R , we assume that S ∈ ( N ) ⊥ , f k ( t, ¯ x, v ) satisfies the solvability condition (1.44) , and E := sup t ∈ [0 ,τ ] X | α |≤ r n kh v i l ( M ) − a ∂ αt, ¯ x f k ( t ) k L ∞ ¯ x,v ∩ L x L ∞ v + kh v i l ( M ) − a e λ ξ ∂ αt, ¯ x S ( t ) k L ∞ ¯ x,ξ,v ∩ L x L ∞ ξ,v o < + ∞ for some positive constant λ > . Then the Knudsen boundary layer equation (2.13) has aunique solution f ( t, ¯ x, ξ, v ) satisfying X | α |≤ r sup t ∈ [0 ,τ ] n kh v i l ( M ) − a ∂ αt, ¯ x f ( t, · , , · ) k L ∞ ¯ x,v ∩ L x L ∞ v + kh v i l ( M ) − a e λξ ∂ αt, ¯ x f ( t ) k L ∞ ¯ x,ξ,v ∩ L x L ∞ ξ,v o ≤ Cλ − λ E (2.14) for all λ ∈ (0 , λ ) , where C > is independence of ( t, ¯ x ) . Moreover, if S is in C (cid:0) [0 , τ ] × R × R + × R (cid:1) and f k ( t, ¯ x, ¯ v, − v ) is in C (cid:0) [0 , τ ] × R × R × R + (cid:1) , then the solution f ( t, ¯ x, ξ, v ) iscontinuous away from the grazing set [0 , τ ] × Σ . We remark that it is hard to obtain the normal derivatives estimates for the boundary valueproblem (2.13). Lemma 2.3 merely give the tangential and time derivatives estimates.2.2.
Control the expansions F k , F bk and F bbk ( ≤ k ≤ ). Based on results in Lemma 2.1,Lemma 2.2 and 2.3, following the analogously arguments in Proposition 5.1 of [23], we obtainthe uniform bounds of the expansions F k , F bk and F bbk (1 ≤ k ≤
5) as follows. For simplicity,we omit the details of proof here.
Proposition 2.1.
Let < z (1 − z ) < a < , where z ∈ ( , is given in (1.62) , and s ∈ N + be in Proposition 1.1. There are s k , s bk , s bbk , l bk ∈ N + , p k , p bk , p bbk ∈ R + (1 ≤ k ≤ satisfying s ≥ s + 10 , s = s b = s bb , s bbk − ≫ s k ≫ s bk ≫ s bbk ≫ ≤ k ≤ ,p k ≫ p bk ≫ p bbk ≫ p k +1 ≫ ≤ k ≤ , and l kj = l bk + 2( s bk − j ) (0 ≤ j ≤ s bk , ≤ k ≤ with l j ≥ , l kj ≥ l k +1 j + 26 (1 ≤ k ≤ , such that if the initial data ( ρ ink , u ink , θ ink ) (1 ≤ k ≤ in (1.18) and (¯ u b,ink , θ b,ink ) (1 ≤ k ≤ in (1.34) satisfy (1.72) , i.e., E in < ∞ , then there are solutions F k = √ M f k , F bk = √ M f bk and F bbk = √ M f bbk (1 ≤ k ≤ constructed in Subsection 1.3.5 over the time interval t ∈ [0 , τ ] subjecting to the uniform bounds sup t ∈ [0 ,τ ] 5 X k =1 n X γ + | β |≤ s k kh v i p k M − a ∂ γt ∂ βx f k ( t ) k L x L ∞ v + s bk X j =0 X γ + | ¯ β | = j kh v i p bk ( M ) − a ∂ γt ∂ ¯ β ¯ x f bk ( t ) k L lkj L ∞ v + X γ + | ¯ β |≤ s bbk k e ξ k − h v i p bbk ( M ) − a ∂ γt ∂ ¯ β ¯ x f bbk ( t ) k L ∞ ¯ x,ξ,v ∩ L x L ∞ ξ,v o ≤ C (cid:16) τ, k ( ρ in , u in , T in ) k H s ( R ) + E in (cid:17) . (2.15)We remark that, for 1 ≤ k ≤
5, the viscous boundary layers F bk decay algebraically associ-ated with ζ >
0, and the Knudsen boundary layers F bbk decay exponentially associated with ξ >
0. These suffice to dominate the quantities R bε and R bbε (in (1.56) and (1.57), respectively)while deriving the uniform L - L ∞ bounds for the remainder F R,ε .3.
Uniform bounds for remainder F R,ε : Proof of Theorem 1.1
In this section, we aim at proving our main theorem by applying the L - L ∞ arguments[20, 21], which is sufficient to estimate k f R,ε ( t ) k and k h ℓR,ε ( t ) k ∞ associated with the remainder F R,ε in (1.52). Here f R,ε and h ℓR,ε are defined in (1.59). The proof relies on an interplaybetween L and L ∞ estimates for the Boltzmann equation. The L norm of f R,ε is controlledby the L ∞ norm of the high-velocity part and vice versa.Note that the remainder equation (1.52) with Maxwell-type reflection boundary conditioncontains the coefficients F k , F bk , F bbk (1 ≤ k ≤
5) and R ε , R bε , R bbε composed of F k , F bk , F bbk (1 ≤ k ≤ L - L ∞ arguments bysome constants independent of ε . For simplicity of presentation, the corresponding quantitieswill be directly bounded by a constant C >
Lemma 3.1 ( L Estimates) . Under the same assumptions in Proposition 2.1 and ℓ ≥ − γ ,let ( ρ, u , T ) be a smooth solution to the Euler equations over t ∈ [0 , τ ] obtained in Proposition1.1. Let c > be mentioned in (1.13) , and the local Maxwellian M w ( t, ¯ x, v ) of the boundarybe assumed as in (1.71) . Then there are constants ε ′ > and C = C (cid:0) M , F k , F bk , F bbk ; 1 ≤ k ≤ (cid:1) > such that for all < ε < ε ′ , dd t k f R,ε k + c ε k ( I − P ) f R,ε k ν + c ¨ Σ − | v · n ( x ) || Lγ + f R,ε | d σ Σ − ≤ C (cid:0) ε k h ℓR,ε k ∞ (cid:1)(cid:0) k f R,ε k + k f R,ε k (cid:1) + C √ ε (3.1) over t ∈ [0 , τ ] , where c = 1 − ρ √ T > and ρ , T > are given in Proposition 1.1. Lemma 3.2 ( L ∞ Estimate) . Under the same assumptions in Lemma 3.1, there are constants ε ′′ > and C = C (cid:0) M , F k , F bk , F bbk ; 1 ≤ k ≤ (cid:1) > such that for all < ε < ε ′′ , sup s ∈ [0 ,τ ] k√ ε h ℓR,ε ( s ) k ∞ ≤ C (cid:16) k√ ε h ℓR,ε (0) k ∞ + sup ≤ s ≤ t k f R,ε ( s ) k + √ ε (cid:17) . (3.2) Proof of Theorem 1.1.
Based on Lemma 3.1 and (3.2), one has dd t k f R,ε k + c ε k ( I − P ) f R,ε k ν + c ¨ Σ − | v · n ( x ) || Lγ + f R,ε | d σ Σ − ≤ C (cid:2) √ ε ( k√ ε h ℓR,ε (0) k ∞ + sup ≤ s ≤ t k f R,ε ( s ) k + √ ε ) (cid:3) ( k f R,ε k + k f R,ε k ) + C √ ε . The Gr¨onwall inequality yields k f R,ε ( t ) k + 1 ≤ ( k f R,ε (0) k + 1) e Ct (cid:8) √ ε + √ ε ( k√ ε k h R,ε (0) k ∞ +sup t ∈ [0 ,τ ] k f R,ε ( t ) k ) (cid:9) . For bounded sup t ∈ [0 ,τ ] k f R,ε ( t ) k and small ε ∈ (0 , min { ε ′ , ε ′′ } ), utilizing the Taylor expansionof the exponential function in the above inequality, one has k f R,ε ( t ) k + 1 ≤ C ( k f R,ε (0) k + 1) (cid:2) √ ε ( k√ ε k h R,ε (0) k ∞ + sup t ∈ [0 ,τ ] k f R,ε ( t ) k ) (cid:3) . OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 23
For t ≤ τ , by using the Young inequality, we conclude the proof of Theorem 1.1 by the boundsup t ∈ [0 ,τ ] k f R,ε ( t ) k ≤ C τ (cid:0) k f R,ε (0) k + k√ ε h ℓR,ε (0) k ∞ (cid:1) . (cid:3) L -estimate: Proof of Lemma 3.1. From (1.52) and (1.59), we see that f R,ε satisfies ∂ t f R,ε + v · ∇ x f R,ε + ε Lf R,ε = − { ∂ t + v ·∇ x }√ M √ M f R,ε + √ ε √ M B ( √ M f R,ε , √ M f R,ε )+ X k =1 √ ε k − √ M n B ( F k + F bk + F bbk , √ M f R,ε ) + B ( √ M f R,ε , F k + F bk + F bbk ) o + √ M ( R ε + R bε + R bbε ) (3.3)with boundary condition f R,ε ( t, ¯ x, , ¯ v, v ) | v > = f R,ε ( t, ¯ x, , ¯ v, − v ) + √ ε ˜Γ ε , (3.4)where˜Γ ε = √ ε √ M Γ ε − α ε √ ε n f R,ε ( t, ¯ x, , ¯ v, − v ) − √ π M w ( v ) √ M ˆ R ˆ + ∞ v f R,ε ( t, ¯ x, , ¯ v, − v ) √ M d¯ v d v o . (3.5)Here Γ ε is given in (1.54).Multiplying the equation (3.3) by f R,ε , integrating the resultant equation over R × R andusing (1.13), one obtains
12 dd t k f R,ε k + c ε k ( I − P ) f R,ε k ν − ¨ ∂ R × R v | f R,ε ( t, ¯ x, , v ) | d v d¯ x = − ¨ R × R ( ∂ t + v ·∇ x ) √ M √ M | f R,ε | d v d x + √ ε ¨ R × R √ M B ( √ M f R,ε , √ M f R,ε ) f R,ε d v d x + X k =1 √ ε k − ¨ R × R √ M (cid:8) B ( F k + F bk + F bbk , √ M f R,ε )+ B ( √ M f R,ε , F k + F bk + F bbk ) (cid:9) f R,ε d v d x + ¨ R × R √ M (cid:0) R ε + R bε + R bbε (cid:1) d v d x . (3.6)Following the similar arguments in Section 2.1 of [22],RHS of (3.6) ≤ C λ ε k h ℓR,ε k ∞ k f R,ε k + C k f R,ε k + C k f R,ε k + Cλ − γ ε k ( I − P ) f R,ε k ν (3.7)for some small λ > − ˜ ∂ R × R v | f R,ε ( t, ¯ x, , v ) | d v d¯ x .Recalling the boundary condition (3.4), one has ¨ ∂ R × R v | f R,ε ( t, ¯ x, , v ) | d v d¯ x = ¨ R × R n ˆ −∞ + ˆ + ∞ o v | f R,ε ( t, ¯ x, , ¯ v, v ) | d v d¯ v d¯ x = ¨ R × R ˆ −∞ v | f R,ε ( t, ¯ x, , ¯ v, v ) | d v d¯ v d¯ x + ¨ R × R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) + √ ε e Γ ε | d v d¯ v d¯ x = √ ε ¨ R × R ˆ + ∞ v e Γ ε f R,ε ( t, ¯ x, , ¯ v, − v )d v d¯ v d¯ x | {z } I + √ ε ¨ R × R ˆ + ∞ v | e Γ ε | d v d¯ v d¯ x | {z } I . (3.8)By using the definition of e Γ ε in (3.5), one has I = − α ε ¨ R × R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v d¯ x + α ε √ π ¨ R × R ˆ + ∞ v f R,ε ( t, ¯ x, , ¯ v, − v ) · M w ( v ) √ M × (cid:16) ˆ R ˆ + ∞ v ′ f R,ε ( t, ¯ x, , ¯ v, − v ′ ) √ M d¯ v d v ′ (cid:17) d v d¯ v d¯ x + 2 √ ε ¨ R × R ˆ + ∞ v f R,ε ( t, ¯ x, , ¯ v, − v )Γ ε d v d¯ v d¯ x (3.9)=: I + I + I . Obviously, the term I is negative, thus it is a good term in the L -estimates as a boundarydecay rate. We then control the quantity I . Recalling the assumption (1.71) of the localMaxwellian M w ( t, ¯ x, v ) of the boundary, we derive from direct calculation that I = √ πα ε ˆ R (cid:16) ˆ R ˆ + ∞ v f R,ε ( t, ¯ x, , ¯ v, − v ) √ M d¯ v d v (cid:17) d¯ x . By H¨older inequality and the fact ´ R ´ + ∞ v M d¯ v d v = ρ √ T √ π , we further infer that | I | ≤ α ε ˆ R ρ √ T (cid:16) ˆ R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v (cid:17) d¯ x ≤k ρ √ T k L ∞ ([0 ,τ ] × R ) · α ε ˆ R ˆ R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v d¯ x ≤ − √ ρ √ T I , (3.10)where the last inequality is derived from the Proposition 1.1. Here the constant 2 √ ρ √ T ∈ (0 ,
1) and the symbol I is given in (3.9).Moreover, it is deduced from the H¨older inequality and Proposition 2.1 that | I | ≤√ ε α ε ˆ R ˆ R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v d¯ x + 14 √ π √ ε ˆ R ˆ R ˆ + ∞ v | Γ ε ( t, ¯ x, v ) | d¯ v d v d¯ x (3.11) ≤ − √ ε I + C √ ε . Together with the bounds (3.9), (3.10) and (3.11), one sees that I ≤ (cid:0) c − ε (cid:1) I + C √ ε . (3.12)where c = 1 − √ ρ √ T > | I | can be bounded by | I | ≤ − C Γ √ εI + C Γ √ ε (3.13) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 25 for some constant C Γ >
0. We consequently derive from substituting (3.12) and (3.13) into(3.8) that − ¨ R × R ˆ + ∞ v | f R,ε ( t, ¯ x, , v ) | d v d¯ x = − ( I + I ) ≥ (cid:0) c − ε − C Γ √ ε (cid:1) α ε ˆ R ˆ R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v d¯ x − C √ ε − C Γ √ ε , We then choose ε ′ = c min (cid:8) , C Γ (cid:9) > . Therefore, for any ε ∈ (0 , ε ′ ), − ¨ ∂ R × R v | f R,ε ( t, ¯ x, , v ) | d v d¯ x ≥ c ¨ R × R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d¯ v d v d¯ x − C √ ε (3.14)for some constant C >
0. Together with (3.6) and (3.7), we have
12 dd t k f R,ε k + c − Cλ − γ ε k ( I − P ) f R,ε k ν + c ¨ R × R ˆ + ∞ v | f R,ε ( t, ¯ x, , ¯ v, − v ) | d v d¯ v d¯ x (3.15) ≤ C λ ε k h ℓR,ε k ∞ k f R,ε k + C k f R,ε k + C k f R,ε k + C √ ε for some small λ > λ > c − Cλ − γ ≥ c >
0, and then finish the proof of Lemma 3.1.3.2. L ∞ estimate for h ℓR,ε : Proof of Lemma 3.2. As in [20] or [23], we introduce thefollowing backward b -characteristics of Boltzmann equation. Given ( t, x, v ), let [ X ( s ) , V ( s )]be determined by ( dd s X ( s ) = V ( s ) , dd s V ( s ) = 0 , [ X ( t ) , V ( t )] = [ x, v ] . The solution is then given by[ X ( s ) , V ( s )] = [ X ( s ; t, x, v ) , V ( s ; t, x, v )] = [ x − ( t − s ) v, v ] . For any fixed ( x, v ) with x ∈ R and v = 0, we define its backward exit time t b ( x, v ) ≥ X ( s ; 0 , x, v ) , V ( s ; 0 , x, v )] remains in x ∈ R . Hence, it holds that t b ( x, v ) = sup (cid:8) τ ≥ x − τ v ∈ R (cid:9) , which means x − t b v ∈ ∂ R , i.e., x − t b v = 0. We also define x b ( x, v ) = x ( t b ) = x − t b v ∈ ∂ R . In the half space, for the case v <
0, the back-time trajectory is a straight line and does nothit the boundary. For the case v >
0, the back-time cycle will hit the boundary for one time.Now let x ∈ R , ( x, v ) ∈ Σ + and ( t , x , v ) = ( t, x, v ). The back-time cycle is defined as X cl ( s ; t, x, v ) = [ t ,t ) ( s ) (cid:8) x − ( t − s ) v (cid:9) + ( −∞ ,t ) ( s ) (cid:8) x − R x b v ( t − s ) (cid:9) ,V cl ( s ; t, x, v ) = [ t ,t ) ( s ) v + ( −∞ ,t ) ( s ) R x b v , where ( t , x b ) = (cid:0) t − t b ( x, v ) , x b ( x, v ) (cid:1) . The explicit formula is t b ( x, v ) = (cid:26) x v , for v > , ∞ , for v < ,V cl ( s ) = ( v , if s ∈ [ t , t ] , (¯ v, − v ) , if s ∈ ( −∞ , t ) , (3.16) X cl ( s ) = ( x − ( t − s ) v , if s ∈ [ t , t ] , (cid:0) ¯ x − ( t − s )¯ v, − x + ( t − s ) v (cid:1) , if s ∈ ( −∞ , t ) . It is easy to see that | V cl ( s ) | ≡ | v | .As in [9], we define L M g = − √ M M (cid:8) B ( M , p M M g ) + B ( p M M g, M ) (cid:9) = (cid:8) ν ( M ) + K (cid:9) g , where Kg = K g − K g with K g = ˆ R × S b ( θ ) | u − v | γ p M M ( u ) M ( v ) √ M M ( v ) g ( u )d ω d u ,K g = ˆ R × S b ( θ ) | u − v | γ M ( u ′ ) √ M M ( v ′ ) √ M M ( v ) g ( v ′ )d ω d u + ˆ R × S b ( θ ) | u − v | γ M ( v ′ ) √ M M ( u ′ ) √ M M ( v ) g ( u ′ )d ω d u . Consider a smooth cutoff function 0 ≤ χ m ≤ m > χ m ( s ) ≡ s ≤ m ,and χ m ( s ) ≡ s ≥ m . Then we introduce (see [22]) K m g = ˆ R × S b ( θ ) | u − v | γ χ m ( | u − v | ) p M M ( u ) M ( v ) M M ( v ) g ( u )d ω d u − ˆ R × S b ( θ ) | u − v | γ χ m ( | u − v | ) M ( u ′ ) √ M M ( v ′ ) √ M M ( v ) g ( v ′ )d ω d u − ˆ R × S b ( θ ) | u − v | γ χ m ( | u − v | ) M ( v ′ ) √ M M ( u ′ ) √ M M ( v ) g ( u ′ )d ω d u , and further define K c g = Kg − K m g . We then have the following lemma.
Lemma 3.3 (Lemma 2.3 of [22]) . | K m g ( v ) | ≤ Cm γ ν ( M ) k g k ∞ (3.17) and K c g ( v ) = ´ R k ( v, v ′ ) g ( v ′ )d v ′ , where the kernel k ( v, v ′ ) satisfies k ( v, v ′ ) ≤ C m exp (cid:8) − c | v − v ′ | (cid:9) | v − v ′ | (1+ | v | + | v ′ | ) − γ (3.18) for some c > .Proof of Lemma 3.2. The proof is similar to [22]. We will only elaborate on the proof ideasand give a sketch of the proof process. The details of calculation will be omitted for simplicityof presentation. We will especially focus on the details of the relationship between k h ℓR,ε k ∞ and k f R,ε k .We first define K ℓ g ≡ h v i ℓ K (cid:16) g h v i ℓ (cid:17) . OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 27
From the remainder equation (1.52) and the definition of h ℓR,ε given in (1.59), we deduce that ∂ t h ℓR,ε + v · ∇ x h ℓR,ε + ν ( M ) ε h ℓR,ε − ε K ℓ h ℓR,ε = √ ε h v i ℓ √ M M B (cid:16) √ M M h v i ℓ h ℓR,ε , √ M M h v i ℓ h ℓR,ε (cid:17) + h v i ℓ √ M M (cid:0) R ε + R bε + R bbε (cid:1) (3.19)+ X i = k √ ε k − h v i ℓ √ M M n B (cid:16) F k + F bk + F bbk , √ M M h v i ℓ h ℓR,ε (cid:17) + B (cid:16) √ M M h v i ℓ h ℓR,ε , F k + F bk + F bbk (cid:17)o . For any ( t, x, v ), integrating (3.19) along the backward trajectory (3.16), one obtains that h ℓR,ε ( t, x, v ) = exp (cid:8) − ε ˆ t ν ( φ )d φ (cid:9) h ℓR,ε (0 , X cl (0) , V cl (0)) | {z } I + ˆ t exp (cid:8) − ε ˆ ts ν ( φ )d φ (cid:9)(cid:0) ε K mℓ h ℓR,ε (cid:1) ( s, X cl ( s ) , V cl ( s ))d s | {z } I + ˆ t exp (cid:8) − ε ˆ ts ν ( φ )d φ (cid:9)(cid:0) ε K cℓ h ℓR,ε (cid:1) ( s, X cl ( s ) , V cl ( s ))d s | {z } I + ˆ t exp (cid:8) − ε ˆ ts ν ( φ )d φ (cid:9)(cid:16) √ ε h v i ℓ √ M M B (cid:0) √ M M h v i ℓ h ℓR,ε , √ M M h v i ℓ h ℓR,ε (cid:1)(cid:17) ( s, X cl ( s ) , V cl ( s ))d s | {z } I + X k =1 √ ε k − ˆ t exp (cid:8) − ε ˆ ts ν ( φ )d φ (cid:9)n h v i ℓ √ M M B (cid:16) F k + F bk + F bbk , √ M M h v i ℓ h ℓR,ε (cid:17) + h v i ℓ √ M M B (cid:16) √ M M h v i ℓ h ℓR,ε , F k + F bk + F bbk (cid:17)o ( s, X cl ( s ) , V cl ( s ))d s := I + ˆ t exp (cid:8) − ε ˆ ts ν ( φ )d φ (cid:9)(cid:16) h v i ℓ √ M M (cid:0) R ε + R bε + R bbε (cid:1)(cid:17)(cid:0) s, X cl ( s ) , V cl ( s ) (cid:1) d s | {z } I , (3.20)where the simplified symbol has been employed: ν ( φ ) = ν ( M ) (cid:0) φ, X cl ( φ ) , V cl ( φ ) (cid:1) . The term I in (3.20) can be bounded by | I | ≤ Cε h v i − γ k h ℓR,ε (0) k ∞ ≤ Cε k h ℓR,ε (0) k ∞ . (3.21)Moreover, for small m >
0, one has | I | ≤ Cm γ sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ , | I | ≤ C √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ , | I | ≤ C √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ , | I | ≤ C √ ε . (3.22)We then focus on the estimate of the term I in (3.20). Let k ℓ ( v, v ′ ) be the correspondingkernel associated with K cℓ . From Lemma 3.3, | k ℓ ( v, v ′ ) | ≤ C h v ′ i ℓ exp (cid:8) − c | v − v ′ | (cid:9) | v − v ′ |h v i ℓ (1+ | v | + | v ′ | ) − γ ≤ C exp {− c | v − v ′ | }| v − v ′ | (1+ | v | + | v ′ | ) − γ . (3.23)We therefore bound I in (3.20) by | I | ≤ ε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R (cid:12)(cid:12) k ℓ ( V cl ( s ) , v ′ ) h ℓR,ε ( s, X cl ( s ) , v ′ ) (cid:12)(cid:12) d v ′ d s . We now use (3.20) again to evaluate h ℓR,ε via replacing ( t, x, v ) with ( s, X cl ( s ) , v ′ ). Togetherwith (3.22), the above can be further bounded by | I | ≤ ε ˆ t ˆ R exp n − ε ˆ ts ν ( φ )d φ − ε ˆ s ν ( v ′ )( φ )d φ o | k ℓ ( V cl ( s ) , v ′ ) |× | h ℓR,ε (cid:0) , X cl (0; s, X cl ( s ) , v ′ ) , v ′ (cid:1) | d v ′ d s := I + ε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R | k ℓ ( V cl ( s ) , v ′ ) | ˆ s exp n − ε ˆ ss ν ( φ )d φ o × (cid:12)(cid:12) K mℓ h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) , V cl ( s ; s, X cl ( s ) , v ′ ) (cid:1)(cid:12)(cid:12) d s d v ′ d s := I + ε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R × R (cid:12)(cid:12) k ℓ (cid:0) V cl ( s ) , v ′ (cid:1) k ℓ ( v ′ , v ′′ ) (cid:12)(cid:12) × ˆ s exp n − ε ˆ ss ν ( v ′ )( φ )d φ o(cid:12)(cid:12) h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) , v ′′ (cid:1)(cid:12)(cid:12) d v ′′ d v ′ d s d s := I + Cε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R | k ℓ ( V cl ( s ) , v ′ ) | d v ′ d s · √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ | {z } I + Cε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R | k ℓ ( V cl ( s ) , v ′ ) | d v ′ d s · √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ | {z } I + Cε ˆ t exp n − ε ˆ ts ν ( φ )d φ o ˆ R | k ℓ ( V cl ( s ) , v ′ ) | d v ′ d s · √ ε | {z } I . (3.24)Except for the term I , the sum of all other terms can be bounded by I + I + I + I + I ≤ C (cid:8) k h ℓR,ε (0) k ∞ + √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ + √ ε sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ + √ ε (cid:9) . (3.25)It remains to control the quantity I in (3.24). As in [22], we will finish our arguments by thefollowing several cases. Let large N > κ ∗ > | v | ≥ N , I ≤ CN sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ . (3.26)Case 2. If either | v | ≤ N , | v ′ | ≥ N or | v ′ | ≤ N , | v ′′ | ≥ N , I ≤ C η e − η N sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ (3.27)for some small η > | v | ≤ N , | v ′ | ≤ N , | v ′′ | ≤ N , s − s ≤ εκ ∗ , I ≤ C N κ ∗ sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ . (3.28)Case 3b. If | v | ≤ N , | v ′ | ≤ N , | v ′′ | ≤ N , s − s ≥ εκ ∗ , together with (1.12), one has I ≤ Cε ˆ t ˆ (cid:8) | v ′ |≤ N , | v ′′ |≤ N (cid:9) ˆ s − εκ ∗ exp n − C h v i γ ( t − s ) ε o exp n − C h v ′ i γ ( s − s ) ε o × (cid:12)(cid:12) k ¯ ℓ ( V cl ( s ) , v ′ ) k ℓ ( v ′ , v ′′ ) h ℓR,ε (cid:0) s ; X cl ( s ; s, X cl ( s ) , v ′ ) , v ′′ (cid:1)(cid:12)(cid:12) d s d v ′ d v ′′ d s . OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 29
We remark that the relation between k h ℓR,ε k ∞ and k f R,ε k comes from this case. By (3.23), k ℓ ( V cl ( s ) , v ′ ) has a possible integrable singularity of | V cl ( s ) − v ′ | . We can choose k N ( V cl ( s ) , v ′ )smooth with compact support such thatsup | p |≤ N ˆ | v ′ |≤ N (cid:12)(cid:12) k N ( p, v ′ ) − k ℓ ( p, v ′ ) (cid:12)(cid:12) d v ′ ≤ N . (3.29)Splitting k ℓ ( V cl ( s ) , v ′ ) k ℓ ( v ′ , v ′′ ) = (cid:2) k ℓ ( V cl ( s ) , v ′ ) − k N ( V cl ( s ) , v ′ ) (cid:3) k ℓ ( v ′ , v ′′ )+ (cid:2) k ℓ ( v ′ , v ′′ ) − k N ( v ′ , v ′′ ) (cid:3) k N ( V cl ( s ) , v ′ ) + k N ( V cl ( s ) , v ′ ) k N ( v ′ , v ′′ ) , we derive from (3.29) that ≤ CN sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ + Cε ˆ t ˆ {| v ′ |≤ N , | v ′′ |≤ N } n ˆ s − κ ∗ ε max { ,s ′ } + ˆ s ′ s ′ > ( s ) o (3.30)exp (cid:8) − C h v i γ ( t − s ) ε (cid:9) exp (cid:8) − C h v ′ i γ ( s − s ) ε (cid:9) × (cid:12)(cid:12) k N ( V cl ( s ) , v ′ ) k N ( v ′ , v ′′ ) h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) , v ′′ (cid:1)(cid:12)(cid:12) d s d v ′ d v ′′ d s , where s ′ = s − t b (cid:0) X cl ( s ; s, X cl ( s ) , v ′ ) , v ′ (cid:1) , and the following bound has been used:sup | v ′ |≤ N ˆ {| v ′′ |≤ N } | k ℓ ( v ′ , v ′′ )d v ′′ + sup | V cl ( s ) | = | v |≤ N ˆ {| v ′ |≤ N } | k N ( V cl ( s ) , v ′ ) | d v ′ ≤ C .
For the integration over v ′ in (3.30), we make a change of variable v ′ → y := X cl (cid:0) s ; s, X cl ( s ) , v ′ (cid:1) .From the explicit formula (3.16), one has ∂y∂v ′ = ( − ( s − s )Diag(1 , , , if max { , s ′ } ≤ s ≤ s − εκ ∗ , − ( s − s )Diag(1 , , − , if 0 ≤ s ≤ s ′ . (3.31)Therefore, it holds that (cid:12)(cid:12)(cid:12) det (cid:16) ∂y∂v ′ (cid:17) ( s ) (cid:12)(cid:12)(cid:12) = ( s − s ) s ≥ ( εκ ∗ ) > s ∈ [0 , s − κ ∗ ε ] , which yields that for any | v ′′ | ≤ N ˆ | v ′ |≤ N (cid:12)(cid:12) h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) , v ′′ (cid:1)(cid:12)(cid:12) d v ′ ≤ C N (cid:16) ˆ | v ′ |≤ N R (cid:0) X cl ( s ; s, X cl ( s ) , v ′ ) (cid:1)(cid:12)(cid:12) h ℓR,ε (cid:0) s , X cl ( s ; s, X cl ( s ) , v ′ ) (cid:1)(cid:12)(cid:12) d v ′ (cid:17) ≤ C N ( κ ∗ ε ) (cid:16) ˆ | y |≤ N ( s − s ) (cid:12)(cid:12) h ℓR,ε (cid:0) s , y, v ′′ ) (cid:12)(cid:12) d y (cid:17) ≤ C N (cid:0) ( s − s ) +1 (cid:1) ( κ ∗ ε ) (cid:16) ˆ R (cid:12)(cid:12) h ℓR,ε (cid:0) s , y, v ′′ ) (cid:12)(cid:12) d y (cid:17) . Together with the definition of f R,ε and h ℓR,ε in (1.59), the last term in (3.30) can be boundedby C N ,κ ∗ ε sup s ∈ [0 ,t ] k f R,ε ( s ) k . One thereby has I ≤ CN sup s ∈ [0 ,t ] k h ℓR,ε ( s ) k ∞ + C N ,κ ∗ ε sup s ∈ [0 ,t ] k f R,ε ( s ) k . (3.32)Collecting the above all estimates, and choosing sufficiently small ε > m > κ ∗ > N >
0, we can finish the proof of Lemma 3.2. (cid:3) Boundary conditions for equations of fluid variables: Proof of Lemma 1.2
In this section, we will formally derive the boundary conditions for the linear hyperbolicsystem (1.15) and linear compressible Prandtl-type equations (1.26), which, respectively, areslip boundary conditions and Robin-type boundary conditions, hence, prove Lemma 1.2.
Proof of Lemma 1.2.
We will prove our conclusion from the solvability condition (1.44), i.e., ˆ R v f k ( t, ¯ x, v ) v − ¯ u ) | v − u | √ M d v = 0 , where f k ( t, ¯ x, v ) is given in (1.47). We split f k ( t, ¯ x, v ) = ˆ g k ( t, ¯ x, v ) + g k ( t, ¯ x, v ) , where, for v < g k ( t, ¯ x, v ) =( f k + f bk + f bbk, )( t, ¯ x, , ¯ v, v ) − ( f k + f bk + f bbk, )( t, ¯ x, , ¯ v, − v ) , g k ( t, ¯ x, v ) = √ π (cid:8) h γ + ( f k − + f bk − + f bbk − ) i ∂ R √ M − ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) (cid:9) , ans they both vanish for v >
0. We remark that g k ( t, ¯ x, v ) ≡ ˆ R v f k ( t, ¯ x, v ) √ M d v = ˆ R v ˆ g k ( t, ¯ x, v ) √ M d v | {z } Q + ˆ R v g k ( t, ¯ x, v ) √ M d v | {z } Q . Following Section 2.4 of [23], Q = ρ (cid:2) u k, + u b, k, + T (Ψ k + 5 T Θ k ) (cid:3) = ρ n u k, + ˆ + ∞ ρ (cid:2) ∂ t ρ bk − + div ¯ x (cid:0) ρ ¯ u bk − + ρ bk − ¯ u (cid:1)(cid:3) d ζ + T (Ψ k + 5 T Θ k ) o . (4.1)Since M w = M and √ π ´ R ´ −∞ v M w ( v )d¯ v d v = − ρ √ T , we derive from the directcalculation that Q = (cid:16) √ π ˆ R ˆ −∞ v M w ( v )d¯ v d v − (cid:17) × √ π ˆ R ˆ −∞ v ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) √ M d¯ v d v = − ( ρ √ T + 1) √ π ˆ R ˆ −∞ v ( f k − + f bk − + f bbk − )( t, ¯ x, , ¯ v, v ) √ M d¯ v d v . (4.2)Consequently, the above three relations imply (1.48), and (1.50) will automatically hold byletting k = 1.Second, for i = 1 ,
2, one has0 = ˆ R ( v i − u i ) v f k ( t, ¯ x, v ) √ M d v = ˆ R ( v i − u i ) v ˆ g k ( t, ¯ x, v ) √ M d v + ˆ R ( v i − u i ) v g k ( t, ¯ x, v ) √ M d v = : Y + Y . (4.3)As shown in [23], together with (1.50), one has Y = (cid:2) − µ ( T ) ∂ ζ u bk − ,i + ρ √ T (1 + ρ √ T ) u bk − ,i (cid:3) ( t, ¯ x,
0) + ρ (cid:2)(cid:0) u ,i + u b, ,i (cid:1) u b, k − , (cid:3) + T hA i , J bk − + ( I − P ) f k i ( t, ¯ x,
0) + ρ ( T ) (cid:0) δ i Φ k, + δ i Φ k, (cid:1) . (4.4) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 31
Since M w = M and √ π ´ R ´ −∞ ( v i − u i ) v M w ( v )d¯ v d v = 0 for i = 1 ,
2, one has Y =2 π ˆ R ˆ −∞ ( v i − u i ) v M w ( v )d¯ v d v ˆ R ˆ −∞ ( − v )( f k − + f bk − + f bbk − ) √ M d¯ v d v − √ π ˆ R ˆ −∞ ( v i − u i ) v ( f k − + f bk − + f bbk − ) √ M d¯ v d v = − √ π ˆ R ˆ −∞ ( v i − u i ) v ( f k − + f bk − + f bbk − ) √ M d¯ v d v . Noticing that for i = 1 , ˆ R ˆ −∞ ( v i − u i ) v P ( f k − + f bk − ) √ M d¯ v d v = − ρ √ T √ π ( u k − ,i + u bk − ,i ) , the Y can be further computed as Y = ρ √ T ( u k − ,i + u b, k − ,i ) − √ π ˆ R ˆ −∞ ( v i − u i ) v [( I − P )( f k − + f bk − ) + f bbk − ] √ M d¯ v d v . (4.5)Then, plugging (4.4) and (4.5) into (4.3) implies the first boundary condition of (1.49).At the end,0 = ˆ R | v − u | v f k ( t, ¯ x, v ) √ M d v = ˆ R | v − u | v ˆ g k ( t, ¯ x, v ) √ M d v | {z } Z + ˆ R | v − u | v g k ( t, ¯ x, v ) √ M d v | {z } Z (4.6)As computed in Section 2.4 of [23], Z = − κ ( T ) ∂ ζ θ bk − ( t, ¯ x,
0) + 2( T ) hB , ( I − P ) f k + J bk − i ( t, ¯ x, ρ [( θ + θ b ) u bk − , ]( t, ¯ x,
0) + 10 ρ ( T ) Θ k ( t, ¯ x, . (4.7)Since M w = M and ˆ R ˆ −∞ v | v − u | M w ( v )d¯ v d v = − ρ ( T ) √ π , one has Z =2 π ˆ R ˆ −∞ v | v − u | M w ( v )d¯ v d v × ˆ R ˆ −∞ ( − v )( f k − + f bk − + f bbk − ) √ M d¯ v d v − √ π ˆ R ˆ −∞ v | v − u | ( f k − + f bk − + f bbk − ) √ M d¯ v d v = − √ π ˆ R ˆ −∞ v ( | v − u | − ρ ( T ) )( f k − + f bk − + f bbk − ) √ M d¯ v d v . A direct calculation implies ˆ R ˆ −∞ v ( | v − u | − ρ ( T ) ) P ( f k − + f bk − ) √ M d¯ v d v = − ρ √ T √ π (cid:0) ρ √ T + √ π ρ + (cid:1) θ bk − + ρ (cid:0) + T − ρ ( T ) (cid:1) ( u k − , + u bk − , ) − ρ √ T √ π (cid:0) ρ √ T + √ π ρ + 2 (cid:1) θ k − + √ T √ π ( ρ √ T − T ρ k − + p bk − ) . One therefore obtains Z = ρ √ T (cid:0) ρ √ T + √ π ρ + (cid:1) θ bk − − √ T ( ρ √ T − T ρ k − + p bk − )+ ρ √ T (cid:0) ρ √ T + √ π ρ + 2 (cid:1) θ k − − √ πρ (cid:0) + T − ρ ( T ) (cid:1) ( u k − , + u bk − , ) − √ π ˆ R ˆ −∞ v ( | v − u | − ρ ( T ) ) × [( I − P )( f k − + f bk − ) + f bbk − ] √ M d¯ v d v . (4.8)Consequently, the second condition of (1.49) is derived from using (1.48) and plugging (4.7)-(4.8) into (4.6), and then the proof of Lemma 1.2 is finished. (cid:3) Existence for the linear compressible Prandtl-type system (2.6) : Proof ofLemma 2.2
In this section, we focus on the existence of the smooth solutions to the linear compressiblePrandtl-type equations (2.6) with Robin-type boundary conditions (2.8), which derives fromthe Maxwell reflection boundary condition (1.2). We remark that if the (1.2) is replaced bythe specular reflection boundary condition, one can obtain Neumann boundary condition, see[23].Observe that the linear compressible Prandtl-type system (2.6) is a degenerated parabolicsystem with nontrivial Robin-type boundary values (2.8). First, we can introduce some ex-plicit functions to zeroize the inhomogeneous boundary values (see (5.4)), which actuallyconverts the boundary values into the form of external force term. We therefore obtain thesystem (5.5) with zero boundary values. We then construct a linear parabolic approximatesystem (5.8) while proving the existence of (5.5). When deriving the uniform bounds of theapproximate system in the space L ∞ (0 , τ ; H kl ( R )), we will employ the structures of equationsto convert the higher normal ζ -derivatives to the values of tangential ¯ x -derivatives and timederivatives on { ζ = 0 } , see Lemma 5.2 below.We first quote the following results to control some integral values on the boundary ∂ R by that in the interior R . More precisely, Lemma 5.1 (Lemma 8.1 of [23]) . Let Ω b := { (¯ x, x ) : ¯ x ∈ R , x ∈ [0 , b ) } with ≤ b ≤ ∞ .We assume f, g ∈ H (Ω b ) . There holds, for any x ∈ [0 , b ) , that (cid:12)(cid:12)(cid:12) ˆ R ( f g )(¯ x, x )d¯ x (cid:12)(cid:12)(cid:12) ≤ k ∂ x ( f, g ) k L (Ω b ) k ( f, g ) k L (Ω b ) + b k f k L (Ω b ) k g k L (Ω b ) . (5.1) For i = 1 , , there holds (cid:12)(cid:12)(cid:12) ˆ R ( ∂ x i f · g )(¯ x, x )d¯ x (cid:12)(cid:12)(cid:12) ≤ k ∂ x ( f, g ) k L (Ω b ) k ∂ x i ( f, g ) k L (Ω b ) + b k ∂ x i f k L (Ω b ) k g k L (Ω b ) . (5.2)We remark that if b = ∞ , Ω ∞ = R and the b -terms in (5.1)-(5.2) will be automaticallyvanished. We will apply Lemma 5.1 by letting b = σ , which means 0 < b ≤ for 0 < σ ≤ OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 33
Proof of Lemma 2.2.
We consider the system (2.6)-(2.10), namely, ρ ( ∂ t + ¯ u · ∇ ¯ x ) u i + ρ ( ∂ x u ζ + u , ) ∂ ζ u i + ρ u · ∇ ¯ x u i + ∂ xi p T θ = µ ( T ) ∂ ζ u i + f i ,ρ ( ∂ t + ¯ u · ∇ ¯ x ) θ + ρ ( ∂ x u ζ + u , ) ∂ ζ θ + ρ div x u θ = κ ( T ) ∂ ζ θ + g , ( ∂ ζ θ − R θ θ ) | ζ =0 = a ( t, ¯ x ) , (cid:0) ∂ ζ u i − R u u i (cid:1)(cid:12)(cid:12) ζ =0 = b i ( t, ¯ x ) , i = 1 , , lim ζ →∞ ( u, θ )( t, ¯ x, ζ ) = 0 ,u ( t, ¯ x, ζ ) | t =0 = u (¯ x, ζ ) ∈ R , θ ( t, ¯ x, ζ ) | t =0 = θ (¯ x, ζ ) . (5.3)We introduce the following functions u b ( t, ¯ x, ζ ) = ( u b , u b )( t, ¯ x, ζ ) and θ a ( t, ¯ x, ζ ) as u bi ( t, ¯ x, ζ ) = ( R u + 2 ζ ) b i ( t, ¯ x ) χ ( ζ ) , i = 1 , , θ a ( t, ¯ x, ζ ) = (cid:0) R θ + 2 ζ (cid:1) a ( t, ¯ x ) χ ( ζ ) , where χ ( ζ ) = (cid:26) , ≤ ζ ≤ , , ζ ≥ , is a monotone cut-off function belonging to C ∞ (cid:0) [0 , ∞ ) (cid:1) . It is easy to verify thatlim ζ →∞ ( u b , θ a )( t, ¯ x, ζ ) = 0 , (cid:0) ∂ ζ θ a − R θ θ a (cid:1) | ζ =0 = a ( t, ¯ x ) , ( ∂ ζ u b − R u u b ) | ζ =0 = b ( t, ¯ x ) . (5.4)Let ℧ = u − u b ∈ R and Θ = θ − θ a . Combining with (5.3) implies ∂ t ℧ + ¯ u · ∇ ¯ x ℧ + ( ∂ x u ζ + u , ) ∂ ζ ℧ + ℧ · ∇ ¯ x ¯ u + ∇ ¯ x p T Θ = ˜ µ∂ ζ ℧ + ˜ f ,∂ t Θ + ¯ u · ∇ ¯ x Θ + ( ∂ x u ζ + u , ) ∂ ζ Θ + div x u Θ = ˜ κ∂ ζ Θ + ˜ g , lim ζ →∞ ( ℧ , Θ)( t, ¯ x, ζ ) = 0 , ( ∂ ζ ℧ − R u ℧ ) | ζ =0 = 0 , ( ∂ ζ Θ − R θ Θ) | ζ =0 = 0 , ( ℧ , Θ)( t, ¯ x, ζ ) | t =0 = ( ℧ , Θ)(0) := (cid:0) u (¯ x, ζ ) − u b (0 , ¯ x, ζ ) , θ (¯ x, ζ ) − θ a (0 , ¯ x, ζ ) (cid:1) , (5.5)where˜ µ = ρ µ ( T ) , ˜ f = ρ f − ∂ t u b − ¯ u · ∇ ¯ x u b − ( ∂ x u ζ + u , ) ∂ ζ u b − u b · ∇ ¯ x ¯ u − ∇ ¯ x p ρ T θ a , ˜ κ = ρ κ ( T ) , ˜ g = ρ g − ∂ t θ a − ¯ u · ∇ ¯ x θ a − ( ∂ x u ζ + u , ) ∂ ζ θ a − div x u θ a + ˜ κ∂ ζ θ a . (5.6)From Proposition 1.1 and (2.7), it is easy to see˜ µ ≥ µ ρ := ˜ µ > , ˜ κ ≥ κ ρ := ˜ κ > . (5.7)To prove the existence of smooth solution to (5.5), we first construct the following approx-imate system ∂ t ℧ + ¯ u · ∇ ¯ x ℧ + ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ ℧ + ℧ · ∇ ¯ x ¯ u + ∇ ¯ x p T Θ = ˜ µ∂ ζ ℧ + λ ∆ ¯ x ℧ + ˜ f σ ,∂ t Θ + ¯ u · ∇ ¯ x Θ + ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ Θ + div x u Θ = ˜ κ∂ ζ Θ + λ ∆ ¯ x Θ + ˜ g σ , ( ∂ ζ ℧ − R u ℧ ) | ζ =0 = 0 , ( ∂ ζ Θ − R θ Θ) | ζ =0 = 0 , ( ℧ , Θ)( t, ¯ x, ζ ) | ζ = σ = 0 , ℧ (0 , ¯ x, ζ ) = ( u − u b ) χ σ ( ζ ) , Θ(0 , ¯ x, ζ ) = ( θ − θ a ) χ σ ( ζ ) (5.8)for ( t, ¯ x, ζ ) ∈ [0 , τ ] × R × [0 , σ ] and 0 < σ, λ ≤
1, where χ σ ( ζ ) = χ ( σζ ), ˜ f σ = ˜ f χ σ ( ζ ) and˜ g σ = ˜ gχ σ ( ζ ). One notices that the compatibility condition of initial data at ζ = σ is alsosatisfied due to the property of χ ( ζ ). For the approximate problem (5.8), we can use thestandard linear parabolic theory to obtain the existence of smooth solution in Sobolev spaceprovided that the initial data and ( ρ , u , T ) are suitably smooth. To prove the existenceof the smooth solutions to (5.5), we only need to obtain some uniform estimates of ( ℧ , Θ)associated with σ and λ , then take the limit σ, λ → We will first estimate the uniform bounds on Θ. The derivation of uniform bounds for ℧ is similar to that for Θ. For simplicity, we still employ the notation k f k L l = ˆ R ˆ σ (1 + ζ ) l | f (¯ x, ζ ) | d¯ x d ζ in (1.64) and, correspondingly, employ the notations k f ( t ) k H rl ( R ) and k g k H kl ( R ) in (1.66) and(1.67), respectively. Moreover, the following properties of χ σ ( ζ ) will be frequently used: ζχ σ ( ζ ) | ζ =0 = ζχ σ ( ζ ) | ζ = σ = 0 , (cid:12)(cid:12) ∂ ζ [ ζ (1 + ζ ) l χ σ ( ζ )] (cid:12)(cid:12) ≤ C (1 + ζ ) l for any l ∈ N ∗ , (cid:12)(cid:12) ∂ nζ ( ζχ σ ( ζ )) (cid:12)(cid:12) ≤ C (1 + ζ ) for any n ≥ , [ ∂ nζ , ζχ σ ( ζ )] f | ζ =0 = n∂ n − ζ f | ζ =0 for any n ≥ . (5.9)Furthermore, by letting k ≥ F = F ( ρ , u , T , ∇ x ( ρ , u , T )), and employing the stan-dard Sobolev embedding arguments, one sees that for any 2 γ + | β | + n ≤ k (cid:12)(cid:12) ˆ R ˆ σ (1 + ζ ) l | ∂ γt ∂ β ¯ x ∂ nζ ( F Θ) | d¯ x d ζ (cid:12)(cid:12) ≤ k F k L ∞ t, ¯ x k ∂ γt ∂ β ¯ x ∂ nζ Θ k L l + X = | β ′ | +2 γ ′ ≤| β | +2 γβ ′ ≤ β,γ ′ ≤ γ C β ′ β C γ ′ γ k ∂ γ ′ t ∂ β ′ ¯ x F k L ∞ t L x k (1 + ζ ) l ∂ γ − γ ′ t ∂ β − β ′ ¯ x ∂ nζ Θ k L x L ζ ≤ C ( sup t ∈ [0 ,τ ] k ( ρ, u , T )( t ) k H k +1 ( R ) ) k Θ( t ) k H kl ( R ) ≤ C ( E k +1 ) k Θ( t ) k H kl ( R ) holds for all Θ ∈ H kl ( R ), which will be directly used to deal with the coefficients dependingon ( ρ , u , T ) in what follows. Here the symbol E k +1 is defined in (2.1). Step 1. The zero-derivatives estimates: L l -bounds. For integer l given in (1.65)with s = k , multiplying the Θ-equation of (5.8) with (1 + ζ ) l Θ, and integrating the resultantover [0 , t ] × R × [0 , σ ], one has k Θ( t ) k L l − k Θ(0) k L l = − ˆ t ˆ R ˆ σ ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ Θ(1 + ζ ) l Θd¯ x d ζ d s − ˆ t ˆ R ˆ σ div x u Θ(1 + ζ ) l Θd¯ x d ζ d s + ˆ t ˆ R ˆ σ ˜ κ∂ ζ Θ(1 + ζ ) l Θd¯ x d ζ d s + λ ˆ t ˆ R ˆ σ ∆ ¯ x Θ(1 + ζ ) l Θd¯ x d ζ d s + ˆ t ˆ R ˆ σ ˜ g σ (1 + ζ ) l Θd¯ x d ζ d s =: I + I + I + I + I , where u , = √ T ( ρ √ T + 1) ≥ c > I = − ˆ t ˆ R ( ∂ x u ζχ σ ( ζ ) + u , )(1 + ζ ) l Θ | ζ = 3 σζ =0 d¯ x d s + ˆ t ˆ R ˆ σ { ∂ x u ∂ ζ (cid:2) ζ (1 + ζ ) l χ σ ( ζ ) (cid:3) + l u , (1 + ζ ) l − } Θ d¯ x d ζ d s ≤ − ˆ t ˆ R u , Θ | ζ =0 d¯ x d s + C ( E k +1 ) ˆ t k Θ( s ) k L l d s ≤ − c ˆ t k Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k Θ( s ) k L l d s , (5.10) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 35 where the facts about χ σ ( ζ ) given in (5.9) have been used. Furthermore, | I | ≤ C ( E k +1 ) ˆ t k Θ( s ) k L l d s , | I | ≤ C ˆ t k (Θ , ˜ g σ )( s ) k L l d s . Together with the boundary conditions in (5.8), one can also compute I = ˆ t ˆ R ˜ κ (1 + ζ ) l ∂ ζ ΘΘ | ζ = σ ζ =0 d¯ x d s − ˆ t ˆ R ˆ σ (1 + ζ ) l ( ∂ ζ Θ) d¯ x d ζ d s − ˆ t ˆ R ˆ σ l ˜ κ (1 + ζ ) l − ∂ ζ ΘΘd¯ x d ζ d s ≤ − ˆ t ˜ κ k ∂ ζ Θ( s ) k L l d s + ˆ t ˜ κ k ∂ ζ Θ( s ) k L l d s + C ˆ t k Θ( s ) k L l d s − ˆ t ˆ R ˜ κR θ Θ | ζ =0 d¯ x d s ≤ − ˆ t ˜ κ k ∂ ζ Θ( s ) k L l d s − ˆ t ˜ κ R θ k Θ | ζ =0 ( s ) k L ( R ) d s + C ˆ t k Θ( s ) k L l d s , where the lower bounds (2.9) and (5.7), i.e., 0 < ˜ κ ≤ ˜ κ and 0 < R θ ≤ R θ , are also used. For I , one has I = − λ ˆ t ˆ R ˆ σ (1 + ζ ) l |∇ ¯ x Θ | d¯ x d ζ d s = λ ˆ t k∇ ¯ x Θ( s ) k L l d s . Therefore, one has k Θ( s ) k L l + ˆ t ˜ κ k ∂ ζ Θ( s ) k L l + λ k∇ ˜ x Θ( s ) k L l d s + ˆ t c ′ k Θ | ζ =0 ( s ) k L ( R ) d s ≤ C (cid:0) k Θ(0) k L l + ˆ t k ˜ g σ ( s ) k L l d s (cid:1) + C ( E k +1 ) ˆ t k Θ( s ) k L l d s , (5.11)where c ′ = ˜ κ R θ + c > Step 2. The higher order ( t, ¯ x ) -derivatives estimates: H rl, ( R ) -bounds. For 2 γ + | β | = r with γ ∈ N , β ∈ N and 1 ≤ r ≤ k , applying ∂ γt ∂ β ¯ x to the Θ-equation of (5.8) infersthat ∂ t ∂ γt ∂ β ¯ x Θ + ¯ u · ∇ ¯ x ∂ γt ∂ β ¯ x Θ + ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ ∂ γt ∂ β ¯ x Θ= ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ) + λ ∆ ¯ x ∂ γt ∂ β ¯ x Θ − [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ]Θ − ζχ σ ( ζ )[ ∂ γt ∂ β ¯ x , ∂ x u ∂ ζ ]Θ − [ ∂ γt ∂ β ¯ x , u , ∂ ζ ]Θ − ∂ γt ∂ β ¯ x ( div ¯ x u Θ) + ∂ γt ∂ β ¯ x ˜ g σ , (5.12)where [ X, Y ] = XY − Y X is the commutator operator. Multiplying (5.12) with (1+ ζ ) l r ∂ γt ∂ β ¯ x Θand integrating over [0 , t ] × R × [0 , σ ] reduce to k ∂ γt ∂ β ¯ x Θ( s ) k L lr − k ∂ γt ∂ β ¯ x Θ(0) k L lr = X k =1 II k , (5.13)where II = − ˆ t ˆ R ˆ σ ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ ∂ γt ∂ β ¯ x Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ)(1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = λ ˆ t ˆ R ˆ σ ∆ ¯ x ∂ γt ∂ β ¯ x Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s , II = − ˆ t ˆ R ˆ σ ¯ u · ∇ ¯ x ∂ γt ∂ β ¯ x Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = − ˆ t ˆ R ˆ σ ζχ σ ( ζ )[ ∂ γt ∂ β ¯ x , ∂ x u ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = − ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ( div x u Θ) · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ˜ g σ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s ,II = ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , u , ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x Θd¯ x d ζ d s , We then estimate the terms in the RHS of (5.13) one by one. Integrating by parts withrespect to the variable ζ and employing the similar derivation of I in (5.10) before give us II ≤ − c ˆ t k ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x Θ( s ) k L lr d s . (5.14)For II , a direct calculation infers that II = ˆ t ˆ R ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ)(1 + ζ ) l r ∂ γt ∂ β ¯ x Θ | ζ = σ ζ =0 d¯ x d s − ˆ t ˆ R ˆ σ ˜ κ (1 + ζ ) l r ( ∂ γt ∂ β ¯ x Θ) d¯ x d ζ d s − ˆ t ˆ R ˆ σ (1 + ζ ) l r [ ∂ γt ∂ β ¯ x , ˜ κ∂ ζ ]Θ · ∂ ζ ∂ γt ∂ β ¯ x Θd¯ x d ζ d s =: II + II + II . The quantity II is bounded by − ˆ t ˆ R ˜ κR θ ( ∂ γt ∂ β ¯ x Θ) | ζ =0 d¯ x d s − ˆ t ˆ R [ ∂ γt ∂ β ¯ x , ˜ κR θ ]Θ · ∂ γt ∂ β ¯ x Θ | ζ =0 d¯ x d s ≤ − ˆ t ˜ κ R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k Θ | ζ =0 ( s ) k H r − ( R ) d s , where the boundary values ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ) | ζ =0 = ∂ γt ∂ β ¯ x (˜ κR θ Θ) | ζ =0 and ∂ γt ∂ β ¯ x Θ | ζ = σ = 0 havebeen used. The quantity II is bounded by − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s , and the quantity II is bounded by ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s + C ( E k +1 ) r − X j =1 X γ + | ˜ β | = j ˆ t k ∂ ζ ∂ ˜ γt ∂ ˜ β ¯ x Θ( s ) k L lj +1 d s ≤ ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s , where we require l j ≥ l j +1 . The above estimates also require the lower bounds (2.9) and (5.7).Consequently, II ≤ − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s − ˆ t ˜ κ R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) + k Θ | ζ =0 ( s ) k H r − ( R ) d s . (5.15) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 37
Similarly, one also has II = − λ ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x Θ( s ) k L lr d s , | II | ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) d s , | II | + | II | ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) d s + C ˆ t k ∂ γt ∂ β ¯ x ˜ g σ ( s ) k L lr d s , (5.16)and | II | = (cid:12)(cid:12) ˆ t ˆ R ˆ σ div ¯ x ¯ u (1 + ζ ) l r ( ∂ γt ∂ β ¯ x Θ( s )) d¯ x d ζ d s (cid:12)(cid:12) ≤ C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x Θ( s ) k L lr d s , (5.17)and | II | ≤ C ( E k +1 ) ˆ t ˆ R ˆ σ (1 + ζ ) l r +1 r − X j =0 X γ + | ˜ β | = j | ∂ ζ ∂ ˜ γt ∂ ˜ β ¯ x Θ || ∂ γt ∂ β ¯ x Θ | d¯ x d ζ d s ≤ C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j (cid:16) ˆ t ˆ R ˆ σ (1 + ζ ) l r +2 | ∂ ζ ∂ ˜ γt ∂ ˜ β ¯ x Θ | ¯ x d ζ d s (cid:17) × (cid:16) ˆ t k ∂ γt ∂ β ¯ x Θ( s ) k L lr d s (cid:17) ≤ C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x Θ( s ) k L lr + k ∂ ζ Θ( s ) k H r − l, ( R ) d s . (5.18)Here l r − = l r + 2 is required, which gives the definition of l j in (1.65). Furthermore, thequantity II can easily bounded by II ≤ C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x Θ( s ) k L lr d s . (5.19)It thereby derived from substituting the estimates (5.14), (5.15), (5.16), (5.17), (5.18) and(5.19) into (5.13) that k Θ( t ) k l,r, + ˆ t ˜ κ k ∂ ζ Θ( s ) k l,r, + λ k∇ ¯ x Θ( s ) k l,r, + ( c + ˜ κ R θ ) k Θ | ζ =0 ( s ) k ,r d s ≤ C (cid:16) k Θ(0) k l,r, + ˆ t k ˜ g σ ( s ) k l,r, d s (cid:17) + C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) + k Θ | ζ =0 ( s ) k H r − ( R ) d s (5.20)for 1 ≤ r ≤ k ( k ≥ ≤ l ≤ k ( k ≥
3) to imply k Θ( t ) k H rl, ( R ) + ˆ t k ∂ ζ Θ( s ) k H rl, ( R ) + λ k∇ ¯ x Θ( s ) k H rl, ( R ) + k Θ | ζ =0 ( s ) k H r ( R ) d s ≤ C ( E k +1 ) (cid:16) k Θ(0) k H rl, ( R ) + ˆ t k (Θ , ˜ g σ )( s ) k H rl, ( R ) d s (cid:17) (5.21)for any 0 ≤ l ≤ k ( k ≥ Step 3. The mixed ( t, ¯ x, ζ ) -derivatives estimates: P ≤ n ≤ r H rl,n ( R ) -bounds. Whileestimating the higher order ( t, ¯ x, ζ )-derivatives, we shall subtly deal with boundary values ofhigher order ζ -derivatives of Θ. In order to overcome the issues, we first give the followinglemma, which can be proved directly. Here we omit the details of proof for simplicity ofpresentation. Lemma 5.2.
Assume Θ( t, ¯ x, ζ ) is a smooth solution to the linear parabolic equation on Θ in (5.8) over ( t, ¯ x, ζ ) ∈ [0 , τ ] × R × [0 , σ ] . Then the boundary values of higher oder ζ -derivativesof Θ satisfy that for m ≥ : ( ˜ κ∂ mζ Θ | ζ = σ = 0 , ˜ κ∂ m +1 ζ Θ | ζ = σ = ( ∂ t + ¯ u · ∇ ¯ x − λ ∆ ¯ x + div x u ) ∂ m − ζ Θ | ζ = σ (5.22) and ∂ m +1 ζ Θ | ζ =0 =( L + ˜ L m − ) ∂ m − ζ Θ | ζ =0 + ˆ L ∂ m − ζ ˜ g σ | ζ =0 , (5.23) where ˜ L m − f = ˜ κ − (cid:0) ¯ u · ∇ ¯ x + div x u + ( m − ∂ x u (cid:1) f , L f = ˜ κ − ( ∂ t − λ ∆ ¯ x ) f , ˆ L f = − ˜ κ − f . (5.24) Moreover, if m is odd, there holds ∂ m +1 ζ Θ | ζ =0 = m +12 Y i =1 ( L + ˜ L m +1 − i )Θ | ζ =0 + m − X i =0 i Y j =1 ( L + ˜ L m +1 − j ) ˆ L ∂ m − − iζ ˜ g σ | ζ =0 If m is even, there holds ∂ m +1 ζ Θ | ζ =0 = m Y i =1 ( L + ˜ L m +1 − i )( R θ Θ) | ζ =0 + m − X i =0 i Y j =1 ( L + ˜ L m +1 − j ) ˆ L ∂ m − − iζ ˜ g σ | ζ =0 . One notices that ˜ L m ∼ ˜ L := ˜ κ − (cid:0) ¯ u · ∇ ¯ x + div x u + ∂ x u (cid:1) (5.25)while carrying the operators ˜ L m in the following estimates. For simplicity of presentation,we will employ the relations on the boundary ζ = 0 ∂ m +1 ζ Θ ∼ ( L + ˜ L ) m +12 Θ + m − X i =0 ( L + ˜ L ) i ˆ L ∂ m − − iζ ˜ g σ for odd m , ( L + ˜ L ) m ( R θ Θ) + m − X i =0 ( L + ˜ L ) i ˆ L ∂ m − − iζ ˜ g σ for even m , (5.26)in the rest of the paper. We then return to the proof of Proposition 2.2. Case 1. The first-order ζ -derivatives: H rl, ( R ) -bounds. Applying ∂ γt ∂ βζ ∂ ζ to theΘ-equation of (5.8) with 2 γ + | β | = r −
1, 1 ≤ r ≤ k ( k ≥
3) yields ∂ t ∂ γt ∂ βζ ∂ ζ Θ + ¯ u · ∇ ¯ x ∂ γt ∂ βζ ∂ ζ Θ + ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ γt ∂ βζ ∂ ζ Θ= ∂ γt ∂ βζ (˜ κ∂ ζ Θ) + λ ∆ ¯ x ∂ γt ∂ βζ ∂ ζ Θ − ζχ σ ( ζ )[ ∂ γt ∂ βζ , ∂ x u ∂ ζ ]Θ − ∂ x u ∂ ζ ( ζχ σ ( ζ )) − [ ∂ γt ∂ βζ , ¯ u · ∇ ¯ x ] ∂ ζ Θ − [ ∂ γt ∂ β ¯ x ∂ ζ , u , ∂ ζ ]Θ − ∂ ζ ( ζχ σ ( ζ ))[ ∂ γt ∂ βζ , ∂ x u ∂ ζ ]Θ − ∂ γt ∂ βζ ∂ ζ ( div x u Θ) + ∂ γt ∂ βζ ∂ ζ ˜ g σ . (5.27)By Lemma 5.2, ˜ κ∂ ζ Θ | ζ =0 = L Θ (Θ , ˜ g σ ) | ζ =0 , Θ | ζ = σ = ∂ ζ Θ | ζ = σ = 0 , (5.28)where L Θ (Θ , ˜ g σ ) = (cid:0) ∂ t Θ − λ ∆ ¯ x Θ + ¯ u · ∇ ¯ x Θ + div x u Θ − ˜ g σ (cid:1) . Multiplying (5.27) with (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θ, and integrating over [0 , t ] × R × [0 , σ ], one has k ∂ γt ∂ β ¯ x ∂ ζ Θ( t ) k L lr − k ∂ γt ∂ β ¯ x ∂ ζ Θ(0) k L lr = X k =0 III k , OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 39 where
III = − ˆ t ˆ R ˆ σ ¯ u · ∇ ¯ x ∂ γt ∂ β ¯ x ∂ ζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ γt ∂ β ¯ x ∂ ζ Θ(1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ) · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = ˆ t ˆ R ˆ σ λ ∆ ¯ x ∂ γt ∂ β ¯ x ∂ ζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ ζχ σ ( ζ )[ ∂ γt ∂ β ¯ x , ∂ x u ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ ∂ x u ∂ ζ ( ζχ σ ( ζ )) ∂ γt ∂ β ¯ x ∂ ζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ] ∂ ζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ ∂ ζ ( ζχ σ ( ζ ))[ ∂ γt ∂ β ¯ x , ∂ x u ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ∂ ζ (cid:0) div x u Θ (cid:1) · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ∂ ζ ˜ g σ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s ,III = − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x ∂ ζ , u , ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s . Integration by parts over ¯ x ∈ R or ζ ∈ [0 , σ ] and Sobolev theory yield | III | ≤ C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s , (5.29) III ≤ − c ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s , (5.30)where the properties of χ σ ( ζ ) in (5.9) have been used. For the term III , it is decomposed as III = − ˆ t ˆ R ˆ σ ˜ κ (1 + ζ ) l r ( ∂ γt ∂ β ¯ x ∂ ζ Θ) + ˜ κl r (1 + ζ ) l r − ∂ γt ∂ β ¯ x ∂ ζ Θ ∂ γt ∂ β ¯ x ∂ ζ Θd¯ x d ζ d s − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , ˜ κ∂ ζ ]Θ · [(1 + ζ ) l r + l r (1 + ζ ) l r − ] ∂ γt ∂ β ¯ x ∂ ζ Θ+ ˆ t ˆ R (1 + ζ ) l r ∂ γt ∂ β ¯ x (˜ κ∂ ζ Θ) ∂ γt ∂ β ¯ x ∂ ζ Θ | ζ = σ ζ =0 d¯ x d s . By the lower bounds (5.7), H¨older and Young inequalities, the first term in the right-handside of the decomposition
III can be bounded by − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s , and the second term can be bounded by C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s . We further denote the third term by
III B . There therefore holds III ≤ III B − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s . (5.31)Due to the boundary values of (5.8) and (5.28), the quantity III B reads III B = − ˆ t ˆ R ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x )Θ · ∂ γt ∂ β ¯ x ( R θ Θ) | ζ =0 d¯ x d s | {z } III B − ˆ t ˆ R ∂ γt ∂ β ¯ x (¯ u · ∇ ¯ x Θ) · ∂ γt ∂ β ¯ x ( R θ Θ) | ζ =0 d¯ x d s | {z } III B − ˆ t ˆ R ∂ γt ∂ β ¯ x (cid:0) div x u Θ (cid:1) · ∂ γt ∂ β ¯ x ( R θ Θ) | ζ =0 d¯ x d s | {z } III B (5.32)+ ˆ t ˆ R ∂ γt ∂ β ¯ x ˜ g σ · ∂ γt ∂ β ¯ x ( R θ Θ) | ζ =0 d¯ x d s | {z } III B . We first deal with the quantity
III B . We deduce from integration by parts over [0 , t ] × R that III B = − ˆ R R θ (cid:0) ∂ γt ∂ β ¯ x Θ (cid:1) | ζ =0 | s = ts =0 d¯ x − ˆ t ˆ R λR θ |∇ ¯ x ∂ γt ∂ β ¯ x Θ | | ζ =0 d¯ x d s + ˆ t ˆ R ( ∂ t + λ ∆ ¯ x ) R θ ( ∂ γt ∂ β ¯ x Θ) | ζ =0 d¯ x d s − ˆ t ˆ R ∂ γt ∂ β ¯ x ∂ t Θ · [ ∂ γt ∂ β ¯ x , R θ ]Θ | ζ =0 d¯ x d s | {z } III B + λ ˆ t ˆ R ∂ γt ∂ β ¯ x ∆ ¯ x Θ · [ ∂ γt ∂ β ¯ x , R θ ]Θ | ζ =0 d¯ x d s | {z } III B ≤ − R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) − ˆ t λR θ k∇ ¯ x ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s (cid:17) + III B + III B , (5.33)where the lower bounds (2.9) and the inequality | − ˆ R R θ ( ∂ γt ∂ β ¯ x Θ) (0) | ζ =0 d¯ x | ≤ C ( E k +1 ) k Θ(0) k H rl ( R ) implied by (5.1) in Lemma 5.1 have been used in the last inequality. For the term III B , if β = 0 , γ = r − III B = − ˆ R ∂ γt Θ · [ ∂ γt , R θ ]Θ | ζ =0 | s = ts =0 d¯ x − ˆ t ˆ R ∂ γt Θ · ∂ t [ ∂ γt , R θ ]Θ | ζ =0 d¯ x d s ≤ R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s (cid:17) , where the lower bounds (2.9) and trace inequality (5.1) in Lemma 5.1 are used. OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 41 If | β | ≥
2, without loss of generality, we assume β ≥ e = 2(1 , β ≥
2. Then, (5.2)in Lemma 5.1 implies
III B = ˆ t ˆ R ∂ x (cid:0) ∂ γt ∂ β − e ¯ x ∂ t Θ (cid:1) · ∂ x (cid:8) [ ∂ γt ∂ β ¯ x , R θ ]Θ (cid:9) | ζ =0 d¯ x d s ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . If | β | = 1 , γ = r −
2, where r is required to be even. Then, III B = − X i =1 δ r ( β = e i ) ˆ t ˆ R ∂ γt ∂ e i ¯ x ∂ t Θ ∂ e i ¯ x R θ ∂ γt Θ | ζ =0 d¯ x d s + ˆ t ˆ R ∂ γt ∂ β ¯ x Θ · ∂ t ∂ β ¯ x [ ∂ γt , R θ ]Θ | ζ =0 d¯ x d s − ˆ R ∂ γt ∂ β ¯ x Θ · ∂ β ¯ x [ ∂ γt , R θ ]Θ | ζ =0 | s = ts =0 d¯ x ≤ X i =1 δ r ( β = e i ) ˆ t ˆ R ∂ γt ∂ t Θ ∂ e i ¯ x ( ∂ e i ¯ x R θ ∂ γt Θ) | ζ =0 d¯ x d s + C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j ˆ t k ∂ ˜ γt ∂ ˜ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) + C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j k ∂ ˜ γt ∂ ˜ β ¯ x Θ | ζ =0 ( t ) k L ( R ) + C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j k ∂ ˜ γt ∂ ˜ β ¯ x Θ | ζ =0 (0) k L ( R ) ≤ R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s (cid:17) for some small ǫ > δ r ( β = e i ) = (cid:26) , r is even and β = e i , , otherwise . (5.34)Consequently, III B ≤ R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s (5.35)+ C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s (cid:17) . For
III B , as similarly in (5.35), one has III B = − λ ˆ t ˆ R ∇ ¯ x ∂ γt ∂ β ¯ x Θ · ∇ ¯ x [ ∂ γt ∂ β ¯ x , R θ ]Θ | ζ =0 d¯ x d s ≤ λR θ ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s (5.36) + C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . Therefore, from plugging (5.35) and (5.36) into (5.33), one has
III B ≤ − R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) − λ ˆ t R θ k∇ ¯ x ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s (cid:17) . (5.37)For the term III B , there holds III B = − ˆ t ˆ R [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ]Θ · ∂ γt ∂ β ¯ x ( R θ Θ) | ζ =0 d¯ x d s + ˆ t ˆ R ∂ ˜ γt ∂ ˜ β ¯ x Θdiv ¯ x (cid:8) ¯ u [ ∂ γt ∂ β ¯ x , R θ ]Θ (cid:9) | ζ =0 d¯ x d s + ˆ t ˆ R div ¯ x ( R θ ¯ u ) (cid:0) ∂ γt ∂ β ¯ x Θ (cid:1) | ζ =0 d¯ x d s ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . (5.38)Moreover, the trace inequality (5.1) in Lemma 5.1 also implies III B ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . (5.39)and III B ≤ C ( E k +1 ) (cid:16) ˆ t k (Θ , ˜ g σ )( s ) k H rl, ( R ) + k (Θ , ˜ g σ )( s ) k H rl, ( R ) d s (cid:17) . (5.40)It is thereby derived from substituting the bounds (5.37)-(5.40) into (5.32), and combiningwith (5.31) that III ≤ − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s − R θ k ∂ γt ∂ β ¯ x Θ | ζ =0 ( t ) k L ( R ) − λR θ ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x Θ | ζ =0 ( s ) k L ( R ) d s + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k (Θ , ˜ g σ )( s ) k H rl, ( R ) + k (Θ , ˜ g σ )( s ) k H rl, ( R ) d s (cid:17) . (5.41)Direct calculation gives us III = − λ ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s . (5.42)For III and III , one has III ≤ C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j ˆ t ˆ R ˆ σ (1 + ζ ) l r +1 | ∂ ˜ γt ∂ ˜ β ¯ x ∂ ζ Θ || ∂ γt ∂ β ¯ x ∂ ζ Θ | d¯ x d ζ d s ≤ C ( E k +1 ) r − X j =0 X γ + | ˜ β | = j (cid:16) ˆ t ˆ R ˆ σ (1 + ζ ) l r +2 | ∂ ζ ∂ ˜ γt ∂ ˜ β ¯ x ∂ ζ Θ | d¯ x d ζ d s (cid:17) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 43 × (cid:16) ˆ t ˆ R ˆ σ (1 + ζ ) l r | ∂ γt ∂ β ¯ x ∂ ζ Θ | d¯ x d ζ d s (cid:17) ≤ C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s , (5.43)and III ≤ C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ ζ Θ( s ) k L lr d s . (5.44)Similarly, one also has III + III + III ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) d s , (5.45)and III + III ≤ C ˆ t k ∂ γt ∂ β ¯ x ∂ ζ (˜ g σ , Θ)( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) . (5.46)In summary, it is derived from (5.29)-(5.30), (5.41)-(5.46) and summing up for 2 γ + | β | = r − k Θ( s ) k l,r, + ˆ t ˜ κ k ∂ ζ Θ( s ) k l,r, + λ k∇ ¯ x Θ( s ) k l,r, d s + R θ k Θ | ζ =0 ( t ) k ,r − + λR θ ˆ t k∇ ¯ x Θ | ζ =0 ( s ) k ,r − d s ≤ C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l, ( R ) d s + C ( E k +1 ) k Θ | ζ =0 ( t ) k H r − ( R ) + ǫ X γ + | β | = r − X i =1 δ r ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s (5.47)+ C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k (Θ , ˜ g σ )( s ) k H rl, ( R ) + k (Θ , ˜ g σ )( s ) k H rl, ( R ) d s (cid:17) for all 1 ≤ r ≤ k ( k ≥
3) and some undetermined small constant ǫ >
0. Together with (5.11)and (5.47), the induction for 1 ≤ r ≤ k implies that k Θ( s ) k H rl, ( R ) + ˆ t k ∂ ζ Θ( s ) k H rl, ( R ) + λ k∇ ¯ x Θ( s ) k H rl, ( R ) d s + k Θ | ζ =0 ( t ) k H r − ( R ) + λ ˆ t k∇ ¯ x Θ | ζ =0 ( s ) k H r − ( R ) d s ≤ C ( E k +1 ) ǫ r X j =0 X γ + | β | = j − X i =1 δ j ( β = e i ) ˆ t k ∂ γ +1 t Θ | ζ =0 ( s ) k L ( R ) d s | {z } =: A rB, (Θ)( t ) + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t k (Θ , ˜ g σ )( s ) k H rl, ( R ) + k (Θ , ˜ g σ )( s ) k H rl, ( R ) d s (cid:17) (5.48)for some small ǫ > Case 2. n -order ζ -derivatives: H rl,n ( R ) -bounds ( ≤ n ≤ r ). For 2 ≤ n ≤ r with2 γ + | β | = r − n , applying ∂ γt ∂ β ¯ x ∂ nζ to the Θ-equation of (5.8) infers that ∂ t ∂ γt ∂ β ¯ x ∂ nζ Θ + ¯ u · ∇ ¯ x ∂ γt ∂ β ¯ x ∂ nζ Θ + [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ] ∂ nζ Θ + ∂ x u [ ∂ nζ , ζχ σ ( ζ )] ∂ ζ ∂ γt ∂ β ¯ x Θ+ ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ + ∂ nζ (cid:8) ζχ σ ( ζ )[ ∂ γt ∂ β ¯ x , ∂ x u ] ∂ ζ Θ (cid:9) = ∂ γt ∂ β ¯ x (˜ κ∂ n +2 ζ Θ) + λ ∆ ¯ x ∂ γt ∂ β ¯ x ∂ nζ Θ + ∂ γt ∂ β ¯ x ∂ nζ ˜ g σ − ∂ γt ∂ β ¯ x ∂ nζ ( div x u Θ) − [ ∂ γt ∂ β ¯ x ∂ nζ , u , ∂ ζ ]Θ . (5.49)From taking inner product with (1 + γ ) l r ∂ γγ ∂ β ¯ x ∂ nζ Θ over [0 , t ] × R × [0 , σ ] in (5.49), one obtains k ∂ γt ∂ β ¯ x ∂ nζ Θ( t ) k L lr − k ∂ γt ∂ β ¯ x ∂ nζ Θ(0) k L lr = X k =1 IV k , where IV = − ˆ t ˆ R ˆ σ ¯ u · ∇ ¯ x ∂ γt ∂ β ¯ x ∂ nζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , ¯ u · ∇ ¯ x ] ∂ nζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ ( ∂ x u ζχ σ ( ζ ) + u , ) ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ ∂ x u [ ∂ nζ , ζχ σ ( ζ )] ∂ ζ ∂ γt ∂ β ¯ x Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ∂ nζ ( div x u Θ) · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ ∂ nζ (cid:8) ζχ σ ( ζ )[ ∂ γt ∂ β ¯ x , ∂ x u ] ∂ ζ Θ (cid:9) · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x (˜ κ∂ n +2 ζ Θ) · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = ˆ t ˆ R ˆ σ λ ∆ ¯ x ∂ γt ∂ β ¯ x ∂ nζ Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x ∂ nζ ˜ g σ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s ,IV = − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x ∂ nζ , u , ∂ ζ ]Θ · (1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s . It is easy to estimate that | IV | + | IV | ≤ C ( E k +1 ) ˆ t k Θ( s ) k l,r,n d s . Together with the properties of χ σ ( ζ ) in (5.9), IV ≤ − c ˆ t k ∂ γt ∂ β ¯ x ∂ nζ Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k ∂ γt ∂ β ¯ x ∂ nζ Θ( s ) k L lr d s , OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 45 and | IV | ≤ C ( E k +1 ) n − X ˜ n =0 ˆ t (1 + ζ ) l r +1 | ∂ γt ∂ β ¯ x ∂ ˜ n +1 ζ Θ || ∂ γt ∂ β ¯ x ∂ nζ Θ | d s ≤ C ( E k +1 ) n X ˜ n =1 ˆ t k Θ( s ) k H rl, ˜ n ( R ) d s . Moreover, we similarly have | IV | ≤ C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l,n ( R ) + C ( E k +1 ) n X ˜ n =1 ˆ t k Θ( s ) k H rl, ˜ n ( R ) d s , and | IV | ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl,n ( R ) d s , IV = − λ ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ∂ nζ Θ( s ) k L lr d s , | IV | + | IV | ≤ C ˆ t k ∂ γt ∂ β ¯ x ∂ nζ (˜ g σ , Θ)( s ) k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l,n ( R ) d s . It remains to control the term IV , which can be decomposed as follows: IV = ˆ t ˆ R ∂ γt ∂ β ¯ x (˜ κ∂ n +1 ζ Θ)(1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θ | ζ = σ ζ =0 d¯ x d s | {z } IV B − ˆ t ˆ R ˆ σ ˜ κ (1 + ζ ) l r ( ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ) d¯ x d ζ d s − ˆ t ˆ R ˆ σ [ ∂ γt ∂ β ¯ x , ˜ κ∂ n +1 ζ ]Θ · (1 + ζ ) l r ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s − ˆ t ˆ R ˆ σ ∂ γt ∂ β ¯ x (˜ κ∂ n +1 ζ Θ) · l r (1 + ζ ) l r − ∂ γt ∂ β ¯ x ∂ nζ Θd¯ x d ζ d s . (5.50)Due to the lower bound (5.7), the second term in the right-hand side of (5.50) can be boundedby − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ k L lr d s , and the last two terms in that can be bounded by ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ k L lr d s + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l,n ( R ) + k Θ( s ) k H rl,n ( R ) d s . Together with the bound of IV B in Lemma 5.3 below, one has IV ≤ − ˆ t ˜ κ k ∂ ζ ∂ γt ∂ β ¯ x ∂ nζ Θ k L lr d s − E γ,βB,n (Θ)( t ) + C ( τ, E k +1 ) sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H r +1 l ( R ) + ǫ A B,n (Θ)( t ) + C ( E k +1 ) ˆ t k ∂ ζ Θ( s ) k H r − l,n ( R ) + k Θ( s ) k H rl,n ( R ) d s + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) for some undetermined small constant ǫ > Collecting the above bounds of IV i (1 ≤ i ≤
9) and summing up for 2 γ + | β | = r − n reduceto k Θ( t ) k l,r,n + ˆ t ˜ κ k ∂ ζ Θ( s ) k l,r,n + λ k∇ ¯ x Θ( s ) k l,r,n d s + X γ + | β | + n = r E γ,βB,n (Θ)( t ) ≤ C ( E k +1 ) (cid:16) X γ + | β | + n = r ǫ A B,n (Θ)( t ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t k ∂ ζ Θ( s ) k H r − l,n ( R ) (cid:17) + C ( τ, E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t X ≤ ˜ n ≤ n k Θ( s ) k H rl, ˜ n ( R ) d s + sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H r +1 l ( R ) (cid:17) for 2 ≤ r ≤ k ( k ≥
3) and 2 ≤ n ≤ r . Together with (5.11), (5.21) and (5.48), one derivesfrom the induction for 1 ≤ r ≤ k ( k ≥
3) that k Θ( t ) k H rl,n ( R ) + r X j =0 X γ + | β | = j − n E γ,βB,n (Θ)( t )+ ˆ t k ∂ ζ Θ( s ) k H rl,n ( R ) + λ k∇ ¯ x Θ( s ) k H rl,n ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) ǫ A rB,n (Θ)( t ) + ǫ A rB, (Θ)( t ) + k Θ(0) k H rl ( R ) (cid:17) + C ( τ, E k +1 ) (cid:16) X ≤ ˜ n ≤ n ˆ t k Θ( s ) k H rl, ˜ n ( R ) + sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H r +1 l ( R ) (cid:17) (5.51)for 2 ≤ r ≤ k ( k ≥ ≤ n ≤ r and some small ǫ > A rB,n (Θ)( t ) = r X j =0 X γ + | β | = j − n A γ,βB,n (Θ)( t )= r X j =0 X γ + | β | = j − n X i =1 δ j ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s . (5.52)Finally, collecting the estimates (5.11), (5.21), (5.48) and (5.51) implies k Θ( t ) k H rl ( R ) + ˆ t k ∂ ζ Θ( s ) k H rl ( R ) + λ k∇ ¯ x Θ( s ) k H rl ( R ) d s + e E B,r (Θ)( t ) + λ ˆ t k∇ ¯ x Θ | ζ =0 ( s ) k H r − ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) ǫ r X n =1 A rB,n (Θ)( t ) + k Θ(0) k H rl ( R ) (cid:17) + C ( τ, E k +1 ) (cid:16) ˆ t k Θ( s ) k H rl ( R ) + sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H r +1 l ( R ) (cid:17) (5.53)for any 1 ≤ r ≤ k ( k ≥ e E B,r (Θ)( t ) = r X j =0 r X n =2 X γ + | β | = j − n E γ,βB,n (Θ)( t ) + X n =0 , δ n k Θ | ζ =0 ( t ) k H r − ( R ) + X n =0 , ˆ t (1 − δ n ) k Θ | ζ =0 ( s ) k H r ( R ) + δ n λ k∇ ¯ x Θ | ζ =0 ( s ) k H r − ( R ) d s . (5.54)Here δ n is defined in Lemma 5.3, i.e., δ n = 0 for even n and δ n = 1 for odd n . In particular, weset r = k ≥ P kn =1 A kB,n (Θ)( t ) in the right-hand OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 47 side of (5.53), namely, k X n =1 A kB,n (Θ)( t ) = k X n =1 k X j =0 X γ + | β | = j − n X i =1 δ j ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s . It is easily derived from the trace inequality (5.1) in Lemma 5.1 that k X n =1 k − X j =0 X γ + | β | = j − n X i =1 δ j ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s ≤ C ˆ t k Θ( s ) k H kl ( R ) d s . (5.55)We now consider the case j = k . Recall the definition δ k ( β = e i ) in (5.34), i.e., δ k ( β = e i ) = 1for even k and β = e i , otherwise δ k ( β = e i ) = 0. If k is odd, k X n =1 X γ + | β | = k − n X i =1 δ k ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s = 0 . If k is even, k X n =1 X γ + | β | = k − n X i =1 δ k ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s ≤ C ′ k ˆ t k Θ | ζ =0 ( s ) k H r ( R ) d s ≤ C k e E B,r (Θ)( t )for some constant C k >
0. There therefore holds C ( τ, E k +1 ) ǫ k X n =1 A kB,n (Θ)( t ) ≤ ǫ C ( τ, E k +1 ) C k e E B,r (Θ)( t ) + C ( τ, E k +1 ) ˆ t k Θ( s ) k H kl ( R ) d s . (5.56)We now choose an ǫ ∈ (0 ,
1) such that0 < ǫ C ( τ, E k +1 ) C k ≤ . By plugging (5.56) into the inequality (5.53) with r = k , one has k Θ( t ) k H kl ( R ) + ˆ t k ∂ ζ Θ( s ) k H kl ( R ) + λ k∇ ¯ x Θ( s ) k H kl ( R ) d s + e E B,k (Θ)( t ) + λ ˆ t k∇ ¯ x Θ | ζ =0 ( s ) k H k − ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) k Θ(0) k H kl ( R ) + ˆ t k Θ( s ) k H kl ( R ) d s + sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H k +1 l ( R ) (cid:17) . (5.57)By the analogous arguments of the bound (5.57) for Θ, one can also obtain the uniformbounds for ℧ k ℧ ( t ) k H kl ( R ) + ˆ t k ∂ ζ ℧ ( s ) k H kl ( R ) + λ k∇ ¯ x ℧ ( s ) k H kl ( R ) d s + e E B,k ( ℧ )( t ) + λ ˆ t k∇ ¯ x ℧ | ζ =0 ( s ) k H k − ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) k ℧ (0) k H kl ( R ) + ˆ t k ( ℧ , Θ)( s ) k H kl ( R ) d s + sup t ∈ [0 ,τ ] k ˜ f σ ( t ) k H k +1 l ( R ) (cid:17) . (5.58) Here the quantity ´ t k ( ℧ , Θ)( s ) k H kl ( R ) d s is resulted from the term ∇ ¯ x p T Θ in the ℧ -equationof (5.5). From (5.57) and (5.58), k ( ℧ , Θ)( t ) k H kl ( R ) + ˆ t k ∂ ζ ( ℧ , Θ)( s ) k H kl ( R ) + λ k∇ ¯ x ( ℧ , Θ)( s ) k H kl ( R ) d s + e E B,k ( ℧ , Θ)( t ) + λ ˆ t k∇ ¯ x ( ℧ , Θ) | ζ =0 ( s ) k H k − ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) k ( ℧ , Θ)(0) k H kl ( R ) + ˆ t k ( ℧ , Θ)( s ) k H kl ( R ) d s + sup t ∈ [0 ,τ ] k ( ˜ f σ , ˜ g σ )( t ) k H k +1 l ( R ) (cid:17) , which implies by the Gr¨onwall inequality that k ( ℧ , Θ)( t ) k H kl ( R ) + ˆ t k ∂ ζ ( ℧ , Θ)( s ) k H kl ( R ) + λ k∇ ¯ x ( ℧ , Θ)( s ) k H kl ( R ) d s + e E B,k ( ℧ , Θ)( t ) + λ ˆ t k∇ ¯ x ( ℧ , Θ) | ζ =0 ( s ) k H k − ( R ) d s ≤ C ( τ, E k +1 ) (cid:16) k ( ℧ , Θ)(0) k H kl ( R ) + sup t ∈ [0 ,τ ] k ( ˜ f σ , ˜ g σ )( t ) k H k +1 l ( R ) (cid:17) . (5.59)Based on the uniform estimates (5.59), we can first take the limit σ → λ → (cid:3) Now we give the following lemma to control the boundary term IV B . Lemma 5.3.
Let e = (1 , , e = (0 , ∈ N . For ≤ n ≤ r and γ + | β | + n = r , let IV B be given in (5.50) , i.e., IV B = ˆ t ˆ R ∂ γt ∂ β ¯ x (˜ κ∂ n +1 ζ Θ)(1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θ | ζ = σ ζ =0 d¯ x d s . There hold IV B ≤ − E γ,βB,n (Θ)( t ) + ǫ A γ,βB,n (Θ)( t ) + C ( τ, E k +1 ) sup t ∈ [0 ,τ ] k ˜ g σ ( t ) k H r +1 l ( R ) + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) (5.60) for some small ǫ > to be determined, where the symbols A γ,βB,n (Θ)( t ) and E γ,βB,n (Θ)( t ) are A γ,βB,n (Θ)( t ) = X i =1 δ r ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s , (5.61) and E γ,βB,n (Θ)( t ) = δ n c n k ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( t ) k L ( R ) + λδ n c n ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( s ) k L ( R ) d s + (1 − δ n ) c ˆ t ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n Θ | ζ =0 ( s ) k L ( R ) d s (5.62) for some constant c , c n > . Here δ n = 1 for odd n and δ n = 0 for even n . The symbol δ r ( β = e i ) is defined in (5.34) . Namely, its value is 1 if r is even and β = e i . Otherwise, itsvalue will vanish. OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 49
Remark 5.1.
When r is even and | β | = 1 , the relation γ + | β | + n = r means that n must beodd. In other words, the quantity A B,n (Θ)( t ) will be zero for the case of even n . Furthermore,in order to obtain the bound of IV B in (5.60) , the source term ˜ g σ shall assume ( r + 1) -orderderivatives (i.e., in H r +1 l ( R ) ). The details can be found in (5.76) below. Here a first ordertime derivative is equivalently regarded as a second order ¯ x -derivative.Proof. From Lemma 5.2, we see that for 2 ≤ n ≤ r , ∂ γt ∂ β ¯ x (˜ κ∂ n +1 ζ Θ)(1 + ζ ) l r ∂ γt ∂ β ¯ x ∂ nζ Θ | ζ = σ = 0 . One therefore has IV B = − ˆ t ˆ R ∂ γt ∂ β ¯ x (˜ κ∂ n +1 ζ Θ) ∂ γt ∂ β ¯ x ∂ nζ Θ | ζ =0 d¯ x d s . (5.63) Case 1. n ∈ [2 , r ] is even. From Lemma 5.2 and remarks (5.26), we have 2 γ + | β | + n = r and IV B = − ˆ t ˆ R ˜ κR θ { ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ } | ζ =0 d¯ x d s − ˆ t ˆ R ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ · ( L ⋆ Θ + L ⋆ ˜ g σ + ˜ κR θ L ⋆ ˜ g σ ) | ζ =0 d¯ x d s − ˆ t ˆ R ( L ⋆ Θ + L ⋆ ˜ g σ ) · L ⋆ ˜ g σ | ζ =0 d¯ x d s =: IV e B + IV e B + IV e B , (5.64)where L ⋆ Θ = ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) n , R θ ]Θ + [ ∂ γt ∂ β ¯ x , ˜ κR θ ]( L + ˜ L ) n Θ , L ⋆ ˜ g σ = n − X i =0 ∂ γt ∂ β ¯ x { ˜ κ ( L + ˜ L ) i ˆ L ∂ n − − iζ ˜ g σ } , L ⋆ ˜ g σ = n − X i =0 ∂ γt ∂ β ¯ x ( L + ˜ L ) i ˆ L ∂ n − − iζ ˜ g σ . (5.65)From the lower bounds (2.9) and (5.7), we know ˜ κR θ ≥ ˜ κ R θ := c ′ >
0. Then the quantity IV e B can be bounded by IV e B ≤ − c ′ ˆ t k ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ | ζ =0 ( s ) k L ( R ) d s . (5.66)One further observes that L ⋆ is an at most ( r − L ⋆ is ( r − L ⋆ is ( r − IV e B ≤ c ′ ˆ t k ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) + X ≤ ˜ n ≤ n k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s (5.67)and IV e B ≤ C ( E k +1 ) ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) + X ≤ ˜ n ≤ n k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s , (5.68)where the trace inequalities (5.1)-(5.2) in Lemma 5.1 have been utilized. Consequently, thebounds (5.66), (5.67) and (5.68) tell us IV B ≤ − c ′ ˆ t k ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) + X ≤ ˜ n ≤ n k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s , when n ∈ [2 , r ] is even. Recalling the definitions of L and ˜ L in Lemma 5.2, one easilycomputes ∂ γt ∂ β ¯ x ( L + ˜ L ) n = ˜ κ − n ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n + L ⋆ , where L ⋆ =[ ∂ γt ∂ β ¯ x , ˜ κ − n ]( ∂ t − λ ∆ ¯ x ) n + n X p =1 C p n ∂ γt ∂ β ¯ x L n − p ˜ L p + ∂ γt ∂ β ¯ x (cid:0) L n − ˜ κ − n ( ∂ t − λ ∆ ¯ x ) n (cid:1) is an ( r − c ′′ k ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n Θ | ζ =0 ( s ) k L ( R ) ≤ k ˜ κ − n ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n Θ | ζ =0 ( s ) k L ( R ) ≤ k ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ | ζ =0 ( s ) k L ( R ) + 2 k L ⋆ Θ | ζ =0 ( s ) k L ( R ) ≤ k ∂ γt ∂ β ¯ x ( L + ˜ L ) n Θ | ζ =0 ( s ) k L ( R ) + C ( E k +1 ) X p =0 , k Θ( s ) k H rl,p ( R ) for some constant c ′′ >
0. We thereby know IV B ≤ − c ˆ t ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) + X ≤ ˜ n ≤ n k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s , (5.69)where c = c ′ c ′′ > Case 2. n ∈ [2 , r ] is odd. It is further derived from Lemma 5.2 and remarks (5.26) that IV B = − ˆ t ˆ R ∂ γt ∂ β ¯ x (cid:8) ˜ κ ( L + ˜ L ) n +12 Θ (cid:9) · ∂ γt ∂ β ¯ x ( L + ˜ L ) n − ( R θ Θ) | ζ =0 d¯ x d s − n − X j =0 ˆ t ˆ R ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) n +12 Θ] · ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 d¯ x d s − n − X i =0 ˆ t ˆ R ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) i ˆ L ∂ n − − iζ ˜ g σ ] × ∂ γt ∂ β ¯ x ( L + ˜ L ) n − ( R θ Θ) | ζ =0 d¯ x d s − n − X i =0 n − X j =0 ˆ t ˆ R ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) i ˆ L ∂ n − − iζ ˜ g σ ] (5.70) × ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 d¯ x d s =: IV B + IV B + IV B + IV B , where the differential operators L , ˆ L and ˜ L are defined in (5.24) and (5.25). While 2 γ + | β | + n = r , one observes that the differential operators ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) i ˆ L ∂ n − − iζ ] (0 ≤ i ≤ n − )and ∂ γt ∂ βζ ( L + ˜ L ) n − are all ( r − ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ (0 ≤ j ≤ n − ) are OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 51 ( r − | IV B + IV B | ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) + n X ˜ n =0 k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s . (5.71)Next we control the quantity IV B . One first computes that ∂ γt ∂ β ¯ x [˜ κ ( L + ˜ L ) n +12 ] =˜ κ − n − ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n +12 + L ⋆ =˜ κ − n − ∂ γ + n +12 t ∂ β ¯ x + ˜ κ − n − ∆ ¯ x L ⋆ + L ⋆ , (5.72)where L ⋆ = n +12 X j =1 C j n +12 ( − λ ) j ∂ γ + n +12 − jt ∂ β ¯ x ∆ j − x , L ⋆ =[ ∂ γt ∂ β ¯ x , ˜ κ − n − ]( ∂ t − λ ∆ ¯ x ) n +12 + n +12 X j =1 C j n +12 ∂ γt ∂ β ¯ x (˜ κ L n +12 − j ˜ L j )+ ∂ γt ∂ β ¯ x (cid:8) ˜ κ L n +12 − ˜ κ − n − ( ∂ t − λ ∆ ¯ x ) n +12 (cid:9) . (5.73)We observe that L ⋆ is ( r − L ⋆ is r -order differentialoperator, where the highest order derivatives must contain ∇ ¯ x . Consequently, the traceinequalities (5.1)-(5.2) in Lemma 5.1 imply that − n − X j =0 ˆ t ˆ R (cid:16) ˜ κ − n − ∆ ¯ x L ⋆ + L ⋆ (cid:17) Θ · ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 d¯ x d s = n − X j =0 ˆ t ˆ R ∇ ¯ x L ⋆ Θ · ∇ ¯ x n ˜ κ − n − ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ o | ζ =0 d¯ x d s − n − X j =0 ˆ t ˆ R L ⋆ Θ · ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 d¯ x d s ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) + n X ˜ n =0 k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s . (5.74)For the quantity IV B , it remains to estimate the term IV B ( β ) =: − n − X j =0 ˆ t ˆ R ˜ κ − n − ∂ γ + n +12 t ∂ β ¯ x Θ · ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 d¯ x d s . If | β | ≥
1, similar arguments in (5.74) reduce to IV B ( β ) = n − X j =0 ˆ t ˆ R ∂ γ + n − t ∂ β ¯ x Θ · ∂ t (cid:8) ˜ κ − n − ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ (cid:9) | ζ =0 d¯ x d s − n − X j =0 ˆ R ∂ γ + n − t ∂ β ¯ x Θ · ˜ κ − n − ∂ γt ∂ β ¯ x ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 | s = ts =0 d¯ x ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) + n X ˜ n =0 k ˜ g σ ( s ) k H rl, ˜ n ( R ) d s + C ( E k +1 ) (cid:0) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + sup t ∈ [0 ,τ ] n X ˜ n =0 k ˜ g σ ( t ) k H rl, ˜ n ( R ) (cid:1) . If β = 0, we have 2 γ = r − n . Here r is also odd. Then ∂ t (cid:8) ˜ κ − n − ∂ γt ( L + ˜ L ) j ˆ L ∂ n − − jζ (cid:9) (0 ≤ j ≤ n − ) are r -order differential operators. It is therefore derived from the traceinequality (5.1) that k ∂ t (cid:8) ˜ κ − n − ∂ γt ( L + ˜ L ) j ˆ L ∂ n − − jζ (cid:9) ˜ g σ | ζ =0 ( t ) k L ( R ) ≤ C ( E k +1 ) n X ˜ n =0 k ˜ g σ ( t ) k H r +1 l, ˜ n ( R ) . (5.75)In other words, the source term ˜ g σ ( t, ¯ x, ζ ) shall be in H r +1 l ( R ). Consequently, IV B ( β ) = n − X j =0 ˆ t ˆ R ∂ γ + n − t Θ · ∂ t (cid:8) ˜ κ − n − ∂ γt ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ (cid:9) | ζ =0 d¯ x d s − n − X j =0 ˆ R ∂ γ + n − t Θ · ˜ κ − n − ∂ γt ( L + ˜ L ) j ˆ L ∂ n − − jζ ˜ g σ | ζ =0 | s = ts =0 d¯ x ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) + n X ˜ n =0 k ˜ g σ ( s ) k H r +1 l, ˜ n ( R ) d s + C ( E k +1 ) (cid:0) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + sup t ∈ [0 ,τ ] n X ˜ n =0 k ˜ g σ ( t ) k H rl, ˜ n ( R ) (cid:1) . In summary, one sees that for all β ∈ N with 2 γ + | β | + n = rIV B ( β ) ≤ C ( E k +1 ) (cid:16) X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s + k Θ(0) k H rl ( R ) + (1 + t ) sup t ∈ [0 ,τ ] n X ˜ n =0 k ˜ g σ ( t ) k H r +1 l, ˜ n ( R ) (cid:17) . (5.76)Then, (5.74)-(5.76) reduce to IV B ≤ C ( E k +1 ) (cid:16) X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s + k Θ(0) k H rl ( R ) + (1 + t ) sup t ∈ [0 ,τ ] n X ˜ n =0 k ˜ g σ ( t ) k H r +1 l, ˜ n ( R ) (cid:17) . (5.77)We next control the term IV B in (5.70). By direct calculation, ∂ γt ∂ β ¯ x ( L + ˜ L ) n − ( R θ · )= R θ ˜ κ − n − ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − + [ ∂ β ¯ x , R θ ˜ κ − n − ] ∂ γt ( ∂ t − λ ∆ ¯ x ) n − + L ⋆ , (5.78)where L ⋆ = n − X q =1 C q n − ∂ β ¯ x (cid:8) R θ ∂ γt L n − − q ˜ L q (cid:9) + ∂ β ¯ x [ ∂ γt ( L + ˜ L ) n − , R θ ]+ ∂ β ¯ x n R θ ∂ γt (cid:0) L n − − ˜ κ − n − ( ∂ t − λ ∆ ¯ x ) n − (cid:1) + R θ [ ∂ γt , ˜ κ − n − ]( ∂ t − λ ∆ ¯ x ) n − o (5.79)is an ( r − L in (5.25), thehighest order (( r − L ⋆ must contain ∇ ¯ x . Together with (5.72) and OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 53 (5.78), we can decompose the term IV B as IV B = − ˆ t ˆ R L ⋆ Θ · ∂ γt ∂ β ¯ x ( L + ˜ L ) n − ( R θ Θ) | ζ =0 d¯ x d s − ˆ t ˆ R ˜ κ − n − ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n +12 Θ · L ⋆ Θ | ζ =0 d¯ x d s − ˆ t ˆ R ˜ κ − n − ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n +12 Θ × [ ∂ β ¯ x , R θ ˜ κ − n − ] ∂ γt ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 d¯ x d s − ˆ t ˆ R R θ ˜ κ − n +1 ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n +12 Θ · ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 d¯ x d s =: IV B + IV B + IV B + IV B . (5.80)It is thereby implied by the trace inequalities (5.1)-(5.2) in Lemma 5.1 that IV B ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s , (5.81)and IV B = − ˆ R ˜ κ − n − ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ · L ⋆ Θ | ζ =0 | s = ts =0 d¯ x + ˆ t ˆ R ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ · ( ∂ t + λ ∆ ¯ x ) (cid:8) ˜ κ − n − L ⋆ Θ (cid:9) | ζ =0 d¯ x d s ≤ C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) . (5.82)If β = 0, we have IV B = 0. If | β | ≥
2, we assume β ≥ e = 2(1 , IV B = − ˆ t ˆ R ∂ γt ∂ β − e ¯ x ( ∂ t − λ ∆ ¯ x ) n +12 Θ × ∂ e ¯ x (cid:8) ˜ κ − n − [ ∂ β ¯ x , R θ ˜ κ − n − ] ∂ γt ( ∂ t − λ ∆ ¯ x ) n − Θ (cid:9) | ζ =0 d¯ x d s ≤ C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . If | β | = 1, the facts that 2 γ + | β | + n = r and n ∈ [2 , r ] is odd imply that r is even. We thenhave 2 γ + n = r − IV B = X i =1 δ r ( β = e i ) ˆ t ˆ R ∂ γ + n +12 t Θ × ∂ e i ¯ x (cid:8) ˜ κ − n − ∂ e i ¯ x ( R θ ˜ κ − n − ) ∂ γt ( ∂ t − λ ∆ ¯ x ) n − Θ (cid:9) | ζ =0 d¯ x d s + X i =1 δ r ( β = e i ) ˆ t ˆ R n +12 X q =1 C q n +12 ( − λ ) q ∇ ¯ x ∂ γ + n +12 − qt ∆ q − x Θ · ∇ ¯ x ∂ e i ¯ x (cid:8) ˜ κ − n − ∂ e i ¯ x ( R θ ˜ κ − n − ) ∂ γt ( ∂ t − λ ∆ ¯ x ) n − Θ (cid:9) | ζ =0 d¯ x d s ≤ ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s for some small ǫ > δ r ( β = e i ) is given in (5.34). As a result, IV B ≤ ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) ˆ t k Θ( s ) k H rl, ( R ) + k Θ( s ) k H rl, ( R ) d s . (5.83)The lower bounds (2.9) and (5.7) yield that there is a c n > R θ ˜ κ − n +1 ≥ c n > . Together with Lemma 5.1, one obtains IV B = − ˆ R R θ ˜ κ − n +1 { ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ } | ζ =0 d¯ x | s = ts =0 − ˆ t ˆ R λR θ ˜ κ − n +1 (cid:12)(cid:12) ∇ ¯ x ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ (cid:12)(cid:12) | ζ =0 d¯ x d s + ˆ t ˆ R ( ∂ t + λ ∆ ¯ x )( R θ ˜ κ − n +1 ) { ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ } | ζ =0 d¯ x d s ≤ − c n k ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( t ) k L ( R ) − λc n ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) . (5.84)From (5.81), (5.82), (5.83) and (5.84), one thereby knows IV B ≤ − c n k ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( t ) k L ( R ) − λc n ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( s ) k L ( R ) d s + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s (5.85)+ C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) . Consequently, if n ∈ [2 , r ] is odd, IV B ≤ − c n k ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( t ) k L ( R ) − λc n ˆ t k∇ ¯ x ∂ γt ∂ β ¯ x ( ∂ t − λ ∆ ¯ x ) n − Θ | ζ =0 ( s ) k L ( R ) d s + ǫ X i =1 δ r ( β = e i ) ˆ t k ∂ γ + n +12 t Θ | ζ =0 ( s ) k L ( R ) d s + C ( E k +1 )(1 + t ) sup t ∈ [0 ,τ ] n X ˜ n =0 k ˜ g σ ( t ) k H r +1 l, ˜ n ( R ) (5.86)+ C ( E k +1 ) (cid:16) k Θ(0) k H rl ( R ) + X p =0 , k Θ( t ) k H rl,p ( R ) + ˆ t X p =0 , k Θ( s ) k H rl,p ( R ) d s (cid:17) OMPRESSIBLE EULER LIMIT FROM BOLTZMANN EQUATION 55 for some small ǫ > (cid:3) Acknowledgement
This work is supported by the grants from the National Natural Foundation of China undercontract Nos. 11971360 and 11731008. This work is also supported by the Strategic PriorityResearch Program of Chinese Academy of Sciences, Grant No. XDA25010404.
References [1] C. Bardos, R. E. Caflisch and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equationof a hard sphere gas.
Comm. Pure Appl. Math. , (1986), no. 3, 323-352.[2] C. Bardos, F. Golse, and C. D. Levermore, Fluid Dynamic Limits of Kinetic Equations I: Formal Deriva-tions. J. Stat. Phys. , (1991), 323-344.[3] C. Bardos, F. Golse, and C. D. Levermore, Fluid Dynamic Limits of Kinetic Equations II: ConvergenceProof for the Boltzmann Equation. Commun. Pure and Appl. Math. , (1993), 667-753.[4] C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math.Models Methods Appl. Sci. , (1991), no. 2, 235-257.[5] M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: aquantitative error estimate. J. Differential Equations , (2015), no. 11, 6072-6141.[6] M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes inSobolev spaces with polynomial weight. Anal. Appl. (Singap.) , (2019), no. 1, 85-116.[7] C. Cercignani, The Boltzmann equation and its applications.
Applied Mathematical Sciences, . Springer-Verlag, New York, 1988.[8] C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases.
Applied MathematicalSciences, . Springer-Verlag, New York, 1994.[9] R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation.
Comm. Pure Appl. Math. , (1980), no. 5, 651-666.[10] S. X. Chen, On the initial-boundary value problems for quasilinear symmetric hyperbolic systems andapplications. Chinese Ann. Math. , (1980), no. 3–4, 511-521.[11] S. X. Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with charac-teristic boundary. Front. Math. China , (2007), no. 1, 87-102.[12] Y. T. Ding and N. Jiang, Zero viscosity and thermal diffusivity limit of the linearized compressible Navier-Stokes-Fourier equations in the half plane. arXiv:1402.1390 [math.AP] Accepted by
Asymptotic Analysis. [13] R. DiPerna and P.-L. Lions, On the Cauchy problem for the Boltzmann equation: global existence andweak stability.
Ann. of Math. (1989), 321-366.[14] I. Gallagher and I. Tristani, On the convergence of smooth solutions from Boltzmann to Navier-Stokes.
Ann. H. Lebesgue (2020), 561-614.[15] F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Ration. Mech. Anal. , (1988), no. 1, 81-96.[16] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collisionkernels. Invent. Math. (2004), no. 1, 81–161.[17] F. Golse and L. Saint-Raymond, The Incompressible Navier-Stokes Limit of the Boltzmann Equation forHard Cutoff Potentials.
J. Math. Pures Appl. (9) (2009), no. 5, 508–552.[18] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. (1965), 697-715.[19] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. (2006), no. 5, 626-687.[20] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. , (2010), no. 2, 713-809.[21] Y. Guo, J. Jang and N. Jiang, Local Hilbert expansion for the Boltzmann equation. Kinet. Relat. Models , (2009), no. 1, 205-214.[22] Y. Guo, J. Jang and N. Jiang, Acoustic limit for the Boltzmann equation in optimal scaling. Comm. PureAppl. Math. , (2010), no. 3, 337-361.[23] Y. Guo, F. Huang and Y. Wang, Hilbert expansion of the Boltzmann equation with specular boundarycondition in half-space. arXiv:2008.07705 [math.AP] [24] F. Huang, Y. Wang, Y. Wang and T. Yang, The limit of the Boltzmann equation to the Euler equationsfor Riemann problems. SIAM J. Math. Anal. (2013), no. 3, 1741-1811.[25] J. Jang and C. Kim, Incompressible Euler limit from Boltzmann equation with Diffuse Boundary Conditionfor Analytic data. arXiv:2005.12192 [math.AP] [26] N. Jiang, C. D. Levermore and N. Masmoudi, Remarks on the acoustic limit for the Boltzmann equation. Comm. Partial Differential Equations (2010), no. 9, 1590-1609.[27] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltz-mann equation in bounded domain I. Comm. Pure Appl. Math. (2017), no. 1, 90-171.[28] N. Jiang, C-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation:classical solutions. Indiana Univ. Math. J. (2018), no. 5, 1817-1855.[29] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables . SpringerAppl. Math. Sci., , 1984.[30] N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in abounded domain. Comm. Pure Appl. Math. (2003), no. 9, 1263-1293.[31] T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressibleEuler equation. Comm. Math. Phys. (1978), no. 2, 119-148.[32] L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal. (2003), no. 1, 47-80.[33] L. Saint-Raymond, Hydrodynamic limits: some improvements of the relative entropy method.
Ann. Inst.H. Poincar´e Anal. Non Lin´eaire (2009), no. 3, 705-744.[34] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation.
Lecture Notes in Mathematics, .Springer-Verlag, Berlin, 2009.[35] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and theincompressible limit.
Comm. Math. Phys. , (1986), 49-75.[36] P. Secchi, Linear symmetric hyperbolic systems with characteristic boundary. Math. Methods Appl. Sci. (1995), no. 11, 855-870.[37] Y. Sone, Kinetic theory and fluid dynamics.
Modeling and Simulation in Science, Engineering and Tech-nology. Birkh¨auser Boston, Inc., Boston, MA, 2002.[38] Y. Sone,
Molecular gas dynamics. Theory, techniques, and applications.
Modeling and Simulation in Sci-ence, Engineering and Technology. Birkh¨auser Boston, Inc., Boston, MA, 2007.[39] Z. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes Equations for a compressibleviscous fluid in the half-plane.
Comm. Pure Appl. Math. (1999), no. 4, 479-541(Ning Jiang) School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China
Email address : [email protected] (Yi-Long Luo) School of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China
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