The Boltzmann equation with incoming boundary condition: global solutions and Navier-Stokes limit
aa r X i v : . [ m a t h . A P ] S e p THE BOLTZMANN EQUATION WITH INCOMING BOUNDARYCONDITION: GLOBAL SOLUTIONS AND NAVIER-STOKES LIMIT
NING JIANG AND XU ZHANG
Abstract.
We consider the Boltzmann equations with cutoff collision kernels in bounded domains.For the initial data with finite physical bounds, we prove the existence of global-in-time renormalizedsolutions in the sense of DiPerna-Lions endowed with incoming boundary condition. Moreover, wejustify the limit as the Knudsen number ǫ → O ( ǫ ). Introduction
The Boltzmann (or Maxwell-Boltzmann) equation is an integro-differentiable equation ∂ t F + v ·∇ x F = Q ( F, F ) , (1.1)which models the statistical evolution of a rarefied gas. In equation (1.1), F ( t, x, v ) is a non-negativemeasurable function, which denotes the number density of the gas molecules at time t ≥
0, at theposition x ∈ Ω, with velocity v ∈ R . Here Ω is the space domain which could be the wholespace R , a torus T , or a bounded (or unbounded) domain in R with boundary ∂ Ω = ∅ (in thecurrent paper, Ω will be this case). Furthermore, Q ( F, F ) is the collision operator whose structureis described below.The Boltzmann equation (1.1) is given an initial data which satisfies some natural physicalbounds (bounded mass, momentum, energy and entropy, etc.). More specifically, F | t =0 = F ( x, v ) in Ω × R , (1.2)which satisfies F ≥ Z Z R × Ω F (1 + | x | + | v | + | log F | ) d v d x < ∞ . (1.3)The well-posedness of the Boltzmann equation (1.1) is a fundamental problem in mathematicalphysics. Besides many results on the smooth solutions which required the initial data F is “small”in some functional spaces, the first global in time solution with “large” data, i.e. the initial data F satisfies (1.3): only some finite physical bounds, without any smallness requirements on the size of F , was proved in DiPerna-Lions’ theorem [12] for Ω = R . Since in the natural functional spacesof the number density F ( t, x, v ), say L ∩ L log L , the collision term Q ( F, F ) in (1.1) is not evenlocally integrable, which makes weak solutions to the Boltzmann equation can not be defined in theusual sense. Instead, under the Grad’s angular cutoff assumption and a mild decay condition onthe collision kernel which we will describe in details later, DiPerna and Lions defined the so-calledrenormalized solutions of (1.1) and proved that a sequence of renormalized solutions which satisfyonly the physically natural a priori bounds is weakly compact in L . From this stability theydeduced global existence of renormalized solutions. Later on, in [1] Alexandre and Villani extendedthe results of [12] to the Boltzmann equations with kernels of some long-range interactions.The Boltzmann equation in domains with boundary is more physically relevant, both in appliedfields and theoretical research. For general setting of the initial-boundary value problems, see [9]and [11]. Among many boundary conditions which can be imposed on the Boltzmann equations, we introduce here two simplest prototypes: Maxwell reflection and incoming boundary conditions.The former was proposed by Maxwell in [24], which stated that the gas molecules back to thedomain at the boundary come one part from the specular reflection of the molecules escapingthe domain, the other part from those entering the wall, interacting with the molecules in thewall, and re-evaporating back to the domain with the thermal dynamical equilibrium state of thewall. The latter is more direct: the number density of the gas molecules back to the domain isprescribed. Most of the existing literatures focused on these two boundary conditions, or theirlinear combinations.After DiPerna-Lions’ work, extending their results to domains with boundary is a natural butnontrivial question. One of the main difficulty is that the functional space for renormalized solutionsis very weak (say, L log L ), and the space for the trace of the solutions is even weaker. Thus, besidesthe compactness and stability of the sequence of traces are hard to establish, even the definitionof the trace itself is a nontrivial issue. Most of the previous work could only establish inequalityfor the trace of the boundary values, see [17, 26, 25]. In particular, in [25], an inequality for theboundary condition of the linear combination of Maxwell and incoming boundary conditions wasproved.The first complete answer in this direction is due to Mischler [27], who proved global renormalizedsolutions of many types of the kinetic equations (including Boltzmann equations) with Maxwellcondition, based on some new observations on weak-weak convergence and his previous results onthe traces of kinetic equations [26, 25]. We emphasize that in [27], the equality for the traces wasestablished.In early 90’s, starting from [5, 6] Bardos-Golse-Levermore initialed an program (briefly, BGLprogram) to justify hydrodynamic limits in the framework of renormalized solutions. After manyattempts to overcome the technical difficulties left in [6], see [7, 22, 13, 14, 28], finally, the firstcomplete rigorous Navier-Stokes-Fourier (briefly, NSF) limit was obtained by Golse and Saint-Raymond[15] for a class of bounded collision kernel. Later, they [16] generalized their result to thecutoff hard potential kernel. Levermore and Masmoudi [20] considered the NSF limits for generalpotential kernels including soft potentials.For bounded domains, based on Mischler’s result the Boltzmann equation with Maxwell bound-ary, Masmoudi and Saint-Raymond [23] justified the Stokes limit, and the weak convergence wasextended to NSF limit in [19] and [29]. Furthermore, in [19], by employing the kinetic-fluid bound-ary layer which damped the fast oscillating acoustic waves, the limit was enhanced to strongconvergence.All the above results, both existence of renormalized solutions and hydrodynamic limits inbounded domains, are about the Maxwell reflection boundary. The corresponding results for theincoming boundary are the main concern of the current paper. Overall, regarding to renormalizedsolutions, Mischler’s method in [27] can be employed here. In fact, for the incoming data case wecan obtain better estimates on the traces, and as a consequence, the proof is even shorter. Further-more, we emphasize that for the hydrodynamic limits, the incoming condition has quite differentfeatures with Maxwell condition. We explain the details as follows.We take the acoustic limit of the Boltzmann equation as an example. From Maxwell condition,the acoustic system derived from the Boltzmann equation is endowed with the impermeable bound-ary condition, i.e. u · n = 0 (here u is the velocity and n is the outer normal vector on the boundary),see [18]. However, as formally derived in [2], from the Boltzmann equation with incoming condition,the acoustic system has the jump boundary condition ρ + θ = α (u · n), where the constant α > XISTENCE AND NS LIMITS 3 incompressible Navier-Stokes limits, the boundary condition derived from kinetic equations withincoming data has also many new features comparing with that of Maxwell condition. For moredetails of formal analysis, see Sone group’s works in [30] and [31]. In this sense, the Boltzmannequation with incoming data has the interests of its own to be investigated. This is the mainconcern and the novelty of this paper.This paper includes two main results. The first is global existence of renormalized solutionsfor the Boltzmann equation with incoming data. We prove this basically following the strategy ofMischler [27, 26, 25]. In fact, we proved that the traces of solutions for incoming data have betterestimates, compared to Maxwell condition in the following sense: for the Maxwell condition, usingthe
Darroz`es-Guiraud information, only partial trace estimates are obtained. More specifically, onlythe estimate for the fluctuations to the trace average was established. For incoming data, we obtainthe full estimates for traces. Moreover, we have a new discovery that for local conservation law ofmass, the trace operator is commutative with the integration operator in the direction orthogonal tothe boundary, see Remark 2.8. This new commutative property is very useful in verifying acousticlimit (see [18]), but it was unknown in [27] for Maxwell condition.The second result of this paper is about the incompressible NSF limit from the Boltzmannequation with “well-prepared” incoming boundary, i.e. the boundary data is close to the globalMaxwellian M( v ) = ( √ π ) − exp( − | v | ) in the sense of the incoming boundary relative entropy withorder O ( ǫ ) (while the initial data are close to M( v ) with order O ( ǫ ) respectively). In this case,we justify the convergence to NSF system with homogeneous Dirichlet condition. We emphasizethat for the bounday data not close to M( v ), the uniform entropy bound is not available, whichmakes the incompressible fluid limit very hard to be justified. We leave this to the future work.Nevertheless, to our best acknowledgement, the NSF limit obtained in this paper is the first rigorousjustification of the diffusive limit from the Boltzmann equation with incoming boundary condition.In this rest of this Introduction, we first introduce more detailed information on the Boltzmannequation in particular the collision kernels and the boundary condition so that we can state ourmain results precisely.1.1. Boltzmann collision kernels.
In Boltzmann equation (1.1), Q is the Boltzmann collisionoperator, which acts only on the velocity dependence of f quadratically: Q ( F, F ) = Z R × S ( F ′ F ′∗ − F F ∗ ) b ( v − v ∗ , ω )d v ∗ d ω , (1.4)where F ′ = F ( v ′ ), F ′∗ = F ( v ′∗ ), F ∗ = F ( v ∗ ) ( t and x are only parameters), and the formulae ( v ′ = v + v ∗ + | v − v ∗ | ωv ′∗ = v + v ∗ − | v − v ∗ | ω , yields a parametrization of the set of solutions to the conservation laws of elastic collision ( v + v ∗ = v ′ + v ′∗ | v | + | v ∗ | = | v ′ | + | v ′∗ | . Here v and v ∗ denote the velocities of two particle before the elastic collision, and v ′ and v ′∗ denotesthe post-collision velocities. ω is equal to v ′ − v ′∗ | v ′ − v ′∗ | , belonging to S (the unit sphere in R ). Thenonnegative and a.e. finite weight function b ( v − v ∗ , ω ), called cross-section, is assumed to dependonly on the relative velocity | v − v ∗ | and cosine of the derivation angle ( v − v ∗ | v − v ∗ | , ω ). For a giveninteraction model, the cross section can be computed in a semi-explicit way by solving a classicalscattering problem, see for instance, [9]. A typical example is that in dimension 3, for the inverse s -power repulsive forces (where s > κ = v − v ∗ | v − v ∗ | N. JIANG AND X. ZHANG and ω = v ′ − v ′∗ | v ′ − v ′∗ | , b ( v − v ∗ , ω ) = | v − v ∗ | γ b ( κ · ω ) = | v − v ∗ | γ b (cos θ ) , γ = s − s − , (1.5)and sin θb (cos θ ) ≈ Kθ − − s ′ as θ → , where s ′ = s − and K > . (1.6)Notice that, in this particular situation, b ( z, ω ) is not locally integrable, which is not due to thespecific form of inverse power potential. In fact, one can show (see [32] ) that a non-integrablesingularity arises if and only forces of infinite range are present in the gas. Thus, some assumptionsmust be made on the cross section to make the mathematical treatment of the Boltzmann equationconvenient.We first prove the existence of renormalized solutions. For this purpose, the assumption on thecross section is the same as DiPerna and Lions in [12], i.e. Grad’s angular cutoff, namely, that thecross section be integrable, locally in all variables. More precisely, they assumed A ( z ) = Z S b ( z, ω ) d ω ∈ L loc ( R ) , (1.7)together with a condition of mild growth of A :(1 + | v | ) − Z | z − v |≤ R A ( z ) d z → | v | → ∞ , for all R < ∞ . (1.8)In the second part of the paper, we will study the incompressible NSF limit from the Boltzmannequation, for which in addition to the Grad’s cutoff and the assumption (1.8), we need morerestrictions on the cross-section. Since we justify the limit in the framework of [20] for the interiorpart, we make the same assumptions on the kernel as in [20]. For the convenience of the readers,we list as follows: the cross section b satisfies • Assume that ˆ b has even symmetry in ω , ( ≤ b ( z, ω ) = | v − v ∗ | γ ˆ b ( κ · ω ) , a.e. for − < γ ≤ , R S ˆ b ( κ · ω )d ω < + ∞ . (1.9) • Attenuation assumption. Let a ( v ) = R R ¯ b ( v − v ∗ )M( v ∗ )d v ∗ . There exists some constant C a > C a (1 + | v | ) α ≤ a ( v ) , α ∈ R . (1.10) • Loss operator assumption. Assume that there exists s ∈ (1 , + ∞ ] such thatsup v ∈ R (cid:18) Z R | ¯ b ( v − v ∗ ) a ( v ) a ( v ∗ ) | s a ( v ∗ )M( v ∗ )d v ∗ (cid:19) s < + ∞ . (1.11) • Gain operator assumption. The gain operator is given by K + ( g ) = 12 a ( v ) Z S × R ( g ′ + g ∗ ) b ( v − v ∗ , ω )d ω M( v ∗ )d v ∗ . (1.12)The gain operator assumption requires that K + is a compact operator from L ( a Md v ) to L ( a Md v ). XISTENCE AND NS LIMITS 5
Incoming boundary condition.
Let Ω be an open and bounded subset of R and set O =Ω × R and O T = (0 , T ) × O . We assume that the boundary ∂ Ω is sufficiently smooth. Theregularity that we need is that there exists a vector field n ∈ W , ∞ (Ω ; R ) such that n( x ) coincideswith the outward unit normal vector at x ∈ ∂ Ω. We define the outgoing and incoming sets Σ + andΣ − at the boundary ∂ Ω as Σ ± = { ( x, v ) | x ∈ ∂ Ω , v ∈ R , ± v · n( x ) > } , and Σ = Σ ± ∪ Σ . Wealso denote by d σ x the Lebesgue measure on ∂ Ω and d µ = | n( x ) · v | d v .The boundary condition considered in this paper is that the number density on the incomingto the domain is prescribed. More precisely, denoted by γF be the trace of the number density(provided the trace can be defined), and let γ ± F = Σ ± γF . The so-called incoming boundarycondition is that γ − F = Z , (1.13)where Z ≥ Z T Z Σ − Z (1 + | v | + | log Z | ) d v d σ x d t < ∞ for any T > . (1.14)The first question of this paper is to prove the existence of global-in-time renormalized solution ofthe Boltzmann equation with Grad’s angular cutoff assumption on the cross-section and (1.8) withthe incoming data (1.13)-(1.14), and the initial data (1.2)-(1.3).1.3. Incompressible Navier-Stokes limits.
The second part of this paper will focus on theincompressible Navier-Stokes limit. We start from the following scaled Boltzmann equation: ǫ∂ t F ǫ + v ·∇ x F ǫ = ǫ Q ( F ǫ , F ǫ ) ,γ − F ǫ = Z ǫ ,F ǫ (0 , x, v ) = F ǫ ( x, v ) ≥ , (1.15)where ǫ > F ǫ = M(1 + ǫg ǫ ) . (1.16)Under this scaling, it can formally derive the incompressible NSF system with the correspondingboundary conditions, see [5] and [31] for the derivations of the equations and boundary conditionsrespectively. More specifically, it can be shown formally that g ǫ converges to g as the Knudsennumber ǫ →
0, and g must have the form of the infinitesimal Maxwellian: g ( t, x, v ) = v · u( t, x ) + θ ( t, x )( | v | − ) , (1.17)where (u , θ ) obeys the incompressible Navier-Stokes-Fourier (NSF) system ∂ t u + u · ∇ u + ∇ p − ν ∆u = 0 , divu = 0 ,∂ t θ + u · ∇ θ − k ∆ θ = 0 . (1.18)Furthermore, to derive the boundary conditions for (u , θ ), the incoming boundary data must havethe form: Z ǫ = M(1 + ǫz ǫ ), where z ǫ satisfies formally: z ǫ → ρ w + v · u w + θ w ( | v | − ) , (1.19)then ( u, θ ) satisfies the boundary conditions: there exists two positive constants α, β with α + β > α (u · n) − θ = ρ w ,β (u · n) + θ = θ w , u tan = (u w ) tan . (1.20) N. JIANG AND X. ZHANG
The boundary conditions (1.20) are Dirichlet conditions. In (1.20), the constants α , β are de-termined by the solvability of the linear kinetic boundary layer equation investigated by Bardos-Caflisch-Nicolaenko [4]. In particular, if ( ρ w , u w , θ w ) = (0 , , , θ ) satisfies the homoge-neous Dirichlet boundary conditions, i.e. u = 0 and θ = 0 on ∂ Ω . To make the above formal analysis rigorous is the main concern of the second part of the currentwork. The justification of the NSF is the same as [20], so we will omit the details in this paperand focus on the justification of the boundary conditions (1.20). To justify the non-homogeneousDirichlet conditions (1.20), i.e. ( ρ w , u w , θ w ) = (0 , ,
0) is a challenging problem. In this paper,we treat the homogeneous case. From the point view of formal convergence (1.19), this needs thefluctuations of the incoming data ˆ z ǫ is very “small” so that their limit is zero. We will characterizethis smallness of the incoming data by the so-called incoming boundary relative entropy. We callthe boundary data is “well-prepared” if the incoming boundary relative entropy is of order O ( ǫ ),see (2.12). The second main result of this paper is the justifications of NSF with homogeneousDirichlet boundary for well-prepared incoming boundary data.2. Preliminaries and Main results
In this section, we will introduce the definition of traces to solutions of transport equation, thensome useful lemmas from [26, 27] on weak-weak convergence. As mentioned before, the solution F only belongs to L space. So its trace can not be defined in the usual way. In fact, the trace of F can be defined in the weak sense or by employing characteristic line, for instance, see [10, 17, 26, 27]and references therein. If the solutions to transport equation are smooth, then their traces in weaksense are the same to these in usual sense. In this work, we mainly adopt Mischler’s idea to definetraces.First, we introduce some notations. D ((0 , T ) × O ) is made up of smooth function φ with compactsupport satisfying φ (0 , x, v ) = φ ( T, x, v ) = 0 , for all ( x, v ) ∈ O ,φ ( t, x, v ) | ∂ Ω = 0 , for all ( t, v ) ∈ (0 , T ) × R . D ([0 , T ] × ¯Ω × R ) contains smooth functions which have compact support and may not vanish onthe boundary. In the similar way, we can define D ((0 , T ) × ¯Ω).Let P be Leray projection in L (Ω) onto its subspace with divergence-free vector. Denote by L the linear Boltzmann collision operator(linearized around M) given by L g := Z R × S ( g − g ′ + g ∗ − g ′∗ ) b ( v − v ∗ , ω )M( v ∗ )d v ∗ d ω. It is a Fredholm operator. Its kernel space is spanned by linear independent vectors 1, v and | v | − .Specially, the kinetic momentum flux and heat fluxA( v ) = v ⊗ v − | v | , B( v ) = v ( | v | − )lies in the Ker ⊥ ( L ). Thus there exists ˆA and ˆB such that L ˆA = A , L ˆB = B . We write L ( O ; d v d x ) as L ( O ), L ((0 , T ) × Σ ± ; d µ d σ x d s ) as L ((0 , T ) × Σ ± ) for short. Lemma 2.1 (Green Formula [27]) . Let p ∈ [1 , + ∞ ) , g ∈ L ∞ ((0 , T ) , L ploc ( O ) and h ∈ L ((0 , T ) , L ploc ( O ) .Assume that g and h satisfy equation ∂ t g + v · ∇ x g = h, in distributional sense. Then there exists γg well defined on (0 , T ) × Σ which satisfies γg ∈ L loc (cid:0) [0 , T ] × Σ , (n( x ) · v ) d v d σ x d t (cid:1) , XISTENCE AND NS LIMITS 7 and the following Green Formula Z T Z O (cid:0) β ( g )( ∂ t φ + v · ∇ x φ ) + hβ ′ ( g ) φ (cid:1) d v d x d t = (cid:2) Z O φ ( t, · ) β ( g )( τ, · )d v d x (cid:3) | T + Z T Z Z Σ φβ ( γg )d µ d σ x d t, for β ( · ) ∈ W , ∞ loc ( R + ) with sup x ≥ | β ′ ( x ) | < ∞ , and φ ∈ D ([0 , T ] × ¯Ω × R ) . Remark 2.2.
According to the definition, we can find γβ ( F ) = β ( γF ) . While the solution belongs to L ∞ space, its trace also belongs to the corresponding L ∞ space.Besides, the trace also enjoys monotone properties. Lemma 2.3 (Green Formula of L ∞ ) . (I), Let g ∈ L ∞ ((0 , T ) × O ) and h ∈ L ((0 , T ) × O ) . Assume that g and h satisfies equation ∂ t g + v · ∇ x g = h, (2.1) in distributional sense. Then there exists γg ∈ L loc (cid:0) [0 , T ] × Σ , (n( x ) · v ) d v d σ x d t (cid:1) which satisfiesthe following Green Formula Z T Z O β ( g )( ∂ t φ + v · ∇ x φ ) + hβ ′ ( g ) φ d v d x d t = (cid:2) Z O β ( g )( τ, · ) φ d v d x (cid:3) | T + Z T Z Σ β ( γg )d µ d σ x d t, for all β ( · ) ∈ W , ∞ loc ( R + ) with sup x ≥ | β ′ ( x ) | < ∞ and φ ∈ D ([0 , T ] × ¯Ω × R ) .(II), Moreover, assume that there are g and g , h and h such that ∂ t g + v · ∇ x g = h , and ∂ t g + v · ∇ x g = h , hold in distributional sense. If C ≥ g ≥ g ≥ , then C ≥ γ ± g ≥ γ ± g ≥ . Proof.
This lemma can be proved by the argument in [26, 27]. Specially, ( I ) is the same as the onein [26].Choosing the mollifer ρ ǫ such that ρ n ( x, v ) = n ρ ( vn ) n ρ ( xn − n( x ) · n ) , with Supp ρ ⊂ { x ∈ R | | x | < } and R R ρ ( x )d x = 1.Indeed, applying ρ n ( x, v ) to (2.1), then ∂ t ρ n ∗ g + v · ∇ ρ n ∗ g = ρ n ∗ h + r n ( g ) , where r n ( g ) → L ((0 , T ) × Ω) as n goes to infinity.Now for very n ∈ N + , ρ n ∗ g is smooth. The trace of ρ n ∗ g on Σ ± have a clear meaning in theusually sense. Indeed, for all ( x, v ) ∈ Σ ± , γ ( ρ n ∗ g )( t, x, v ) = lim y → xξ → v ( ρ n ∗ g )( t, x, v ) , ( x, ξ ) ∈ O . So if C ≥ g ≥ g ≥
0, then C ≥ γ ( ρ n ∗ g )( t, x, v ) ≥ ( γρ n ∗ g )( t, x, v ) , ( x, v ) ∈ Σ ± . (2.2) N. JIANG AND X. ZHANG
Setting γg i = γ ( ρ n ∗ g i ) , r n,i = r ( g i )( i = 1 , < t ≤ T Z t Z O (cid:18) β ( ρ n ∗ g i )( ∂ t φ + v · ∇ x φ ) + r n,i β ′ ( ρ n ∗ g i ) φ (cid:19) d v d x d τ = (cid:2) Z O β ( ρ n ∗ g i )( τ, · ) φ d v d x (cid:3) | t + Z T Z Σ β (cid:0) γ ( ρ n ∗ g i ) (cid:1) φ d µ d σ x d τ. By the same argument in [26], there exists γg ∈ L loc (cid:0) [0 , T ] × Σ ± , (n( x ) · v ) d v d σ x d t such that γρ n ∗ g → γg, in L loc (cid:0) [0 , T ] × Σ , d µ d σ x d t. This hints that γρ n ∗ g → γg, a.e. on (0 , T ) × Σ . By (2.2) and Lebesgue dominated convergence theorem, We complete the proof. (cid:3)
Recalling that Q ( F, F ) does not belong to L space, β ′ ( F ) Q ( F, F )( β ( · ) ∈ C ( R + ) with sup x ≥ (1+ x ) | β ′ ( x ) | < ∞ ) belongs to L space. So we can only apply Lemma 2.1 to renormalized version ofBoltzmann equation, namely the resulting equation obtained by multiplying (1.1) by β ′ ( F ). So thetrace is defined just for β ( F ) other than f itself. The following r-convergence is very useful whilerecovering γf from γβ ( F ).Let X and Y be a separable and σ − compact topological space. We denote by g n ↑ g that g n converges increasingly to g in some function space to be clear nearby. Definition 2.4 ([27]) . We say that α is a renormalizing function if α ∈ C ( R ) is increasing and ≤ α ( s ) ≤ s for any s ≥ . We say that { α m } is renormalizing sequence if for any m ∈ N + , α m is a renormalizeing function, α m ( s ) ≤ m and α m ↑ s for all s ≥ when m ↑ ∞ . Given anyrenormalizing sequence ( α m ) m ∈ N + , we say that Z n r-converges(r-convergence) to Z if there existsa sequence { ¯ α m } in L ∞ ( Y ) such that α m ( Z n ) ⇀ ¯ α m , in L ∞ ( Y ); and ¯ α m ↑ Z, a.e. in Y. We denote r-convergence by Z n r ⇀ Z . Proposition 2.5. • If Z n r ⇀ Z , then Z is independent of the renormalizing sequence ( α m ) m ∈ N + . This hintsthat one can choose any renormalizing sequence as you like as long as it is a renormalizingsequence. • If h n ⇀ h in L ( X ) , then h n r ⇀ h in X . [27, Lemma 2.7]Now, we introduce the definition of renormalized solutions to Boltzmann equation with incomingboundary condition. Definition 2.6.
Let β ( x ) ∈ C ( R + ) satisfy β ′ ( x ) ≥ , x ≥ and sup x ≥ (1 + x ) β ′ ( x ) < ∞ . Assumethat the cross section satisfies (1.7) and (1.8) . A nonnegative function F ∈ C (cid:0) R + , D ′ ( O ) (cid:1) ∩ L ∞ ( R + ; L ((0 , T ) × O ; d v d x ) (cid:1) is a renormalized solution to initial boundary problem of Boltzmann equation (1.1) with initialdatum (1.3) and boundary condition (1.14) in the distributional sense: if β ′ ( F ) Q ( F, F ) ∈ L ((0 , T ) × O ; d v d x d s ) and there exist traces γ + F ∈ L ((0 , T ) × Σ; d µ d x d s ) such that XISTENCE AND NS LIMITS 9 Z T Z O (cid:0) β ( F )( ∂ t ψ + v · ∇ x ψ ) + Q ( F, F ) β ′ ( F ) ψ (cid:1) d v d x d t = Z O β ( F )( T, · ) ψ ( T )d v d x − Z O β ( F ) ψ (0)d v d x + Z T Z Σ + β ( γ + F ) ψ d µ d σ x d t − Z T Z Σ − β ( Z ) ψ d µ d σ x d t, (2.3) for any test function ψ ∈ D ([0 , T ] × ¯Ω × R ) . Existence of renormalized solutions.
The relative entropy H ( F | M) is defined as H ( F | M) = Z O h ( F/ M)Md v d x, where h ( x ) = x log x − x + 1 , for x ≥ . We also denote by D ( F ) the H-dissipation D ( F ) = Z Ω Z R × R Z S B ( v − v ∗ , ω )( F ′ F ′∗ − F F ∗ ) log F ′ F ′∗ F F ∗ d ω d v d v ∗ d x. Our main result is as follows.
Theorem 2.7.
Under the assumption (1.7) and (1.8) on cross section, if the initial datum F satisfies (1.2) and the boundary condition z satisfies (1.14) , then the initial-boundary problem toBoltzmann equation (1.1) admits a global renormalized solution F ∈ L ∩ L log L .Furthermore, F has the following properties: • Local conservation law of mass: ∂ t Z R F ( t )d v + ∇ · Z R F ( t ) v d v = 0 , in D ′ ((0 , T ) × Ω) . (2.4) • Estimate of traces γ ± F ∈ L (cid:0) (0 , T ) × Σ ± ; (1 + | v | )d µ d σ x d s (cid:1) , for all T > , (2.5) and Z T Z Σ ± γ ± F | log γ ± F | d µ d σ x d s < + ∞ , for all T > . (2.6) • Commutative properties: As for the local conservation law of mass, similar to Lemma2.1, we can use the Green formula to define the trace of R R vF d v on ∂ Ω , Denoting it by γ x ( R R vF d v ) . Moreover n( x ) · γ x ( Z R vF d v ) = n( x ) · Z R vγF d v. This means that the trace operator γ is commutative with integral operator R . This isbecause that the trace of solutions enjoys the full estimate. • Local conservation law of momentum: There is a distribution-value matrix W belonging to D ′ ((0 , T ) × Ω) such that ∂ t Z R vF ( t )d v + ∇ · Z R v ⊗ vF ( t )d v + ∇ · W = 0 , in D ′ ((0 , T ) × Ω) . • Global conservation law of momentum: Z O F ( t ) v d v d x + Z t Z Σ + vγ + F ( s )d µ d σ x d s = Z O F v d v d x + Z t Z Σ − vZ ( s )d µ d σ x d s, t ≤ T. (2.7) • Global energy inequality: Z O F ( t ) | v | d v d x + Z t Z Σ + | v | γ + F ( s )d µ d σ x d s ≤ Z O F | v | d v d x + Z t Z Σ − Z ( s ) | v | d µ d σ x d s, t ≤ T. (2.8) • Global entropy inequality: H ( F | M)( t ) + Z t Z Σ + h ( γ + F/ M)( s )d µ d σ x d s + Z t D ( F )( s )d s ≤ H ( F | M) + Z t Z Σ − h ( Z | M)( s )d µ d σ x d s, t ≤ T. (2.9) Remark 2.8.
First, compared to Maxwell reflection boundary condition, the estimates (2.5) and (2.6) contain full informations of incoming set and out going set other than partial estimates.Furthermore, as a consequence, the trace operator γ is commutative with the integral operator. Remark 2.9.
This result also works for unbounded domain case. While on the unbounded domain,the weight | x | are necessary. Besides, all these result are still correct in R n , n ≥ . Now we state the difficulty and strategy. As mentioned before, the solution obtained in Theorem2.7 only makes sense in the renormalized sense, namely f is a solution to the renormalized versionof (1.1), namely ∂ t β ( F ) + v · ∇ x β ( F ) = β ′ ( F ) Q ( F, F ) . Of course, we can use Lemma 2.1 to get the existence of traces. Noticing that Q ( F, F ) does notbelong to L space, while we only have β ′ ( F ) Q ( F, F ) in L space. So we can only use Lemma2.1 to the renormalized version of (1.1), namely we get the existence of γ ± β ( F ). It is not easy torecover the trace of F satisfying Theorem 2.7 from the trace of β ( F ) while F only belongs L ( O ).Furthermore, since the trace of F on Σ − is fixed, the main goal is to find some γ + F satisfying (2.3).Motivated by [3, 27], we choose a sequence of { β j } with β j ( x ) = jxj + x to fulfill this. By Lemma 2.1,we can get γ ± β j ( F ), the trace of β j ( F ). For any β satisfying the assumption in Def. 2.6, β ( β j ( f )) → β ( F ) , j → ∞ , in L ( O ) , we will show that ( γ + β j ( F )) is a monotone sequence and thus there exists some γ + F such that γ + β j ( F ) ↑ γ + F, on O , and γ + F satisfies (2.3). Moreover we still find that the trace operator are commutative withintegration operator, namely Remark 2.8. It is unknown whether Remark 2.8 is true for theMaxwell reflection boundary conditions, [27]. This is a new discovery. XISTENCE AND NS LIMITS 11
Navier-Stokes limit.
We define G ǫ = F ǫ / M , G ǫ = F ǫ / M . Noticing that if the kernel satisfy assumptions (1.9) to (1.12), then it satisfies (1.7) and (1.8). Theexistence of renormalized solutions (including their traces) to scaled Boltzmann equation (1.15)is ensured by Theorem 2.7. Indeed, there exists a family of renormalized solutions F ǫ ∈ L log L satisfying ǫH ( G ǫ )( t ) + Z t Z Σ + h ( γ + G ǫ )d µ d σ x d s + ǫ Z t D ( F ǫ )d s ≤ ǫH ( G ǫ ) + Z t Z Σ − h ( z ǫ )d µ d σ x d s. (2.10) Theorem 2.10.
Under the assumptions from (1.9) to (1.12) on cross section, for every ǫ > , let F ǫ be DiPerna-Lions renormalized solutions to Boltzmann equation (1.15) . Under the Navier-Stokesscaling (1.16) , let C = sup ǫ> ǫ Z Ω Z R h ( G ǫ )Md v d x < + ∞ , (2.11) and z ǫ be close to M in the relative entropy sense: C = sup ǫ> (cid:0) ǫ Z + ∞ Z Σ − h ( z ǫ )d µ d σ x d s (cid:1) < + ∞ . (2.12) Assume that for some (u , θ ) ∈ L (Ω) , P (cid:0) ǫ R vF ǫ d v (cid:1) and ǫ R ( | v | − f ǫ − M)d v converges to u and θ respectively in the sense of distribution. Then, g ǫ is weakly compact in L loc (d t d x ; L ((1 + | v | )Md v ) . Moreover, for every limiting point g , it is an infinitesimal Maxwellian with form g ( t, x, v ) = u( t, x ) · v + θ ( t, x ) | v | − , (2.13) where (u , θ ) ∈ L ∞ ((0 , ∞ ); L (Ω)) is a Leray solution to NSF system with initial data u(0 , x ) = u , θ (0 , x ) = θ ( x ) , and boundary conditions u | ∂ Ω = 0 , θ | ∂ Ω = 0 . Besides, the viscosity coefficient ν and thermal conductivity k are defined as follows ν = Z R A : ˆAMd v, k = Z R B : ˆBMd v. and for any t > , u and θ satisfy the following energy estimates Z Ω (cid:0) | u | + | θ | (cid:1) ( t, x )d x + Z t Z Ω (cid:0) ν |∇ u | + k |∇ θ | (cid:1) ( s, x )d x d s ≤ C + C . (2.14) Furthermore, for every subsequence g ǫ k of g ǫ converging to g in L loc (d t d x ; L ((1 + | v | )Md v ) spaceas ǫ k → , it also satisfies P h vg ǫ k i → u , h ( | v | − g ǫ i → θ, in C ([0 , ∞ ) , D ′ ( R )) . Remark 2.11. If z ǫ = M , the homogeneous incoming boundary condition, by simple calculation C = 0 . From our result, if the given incoming boundary condition is close to M in the relativeentropy sense, the homogeneous Dirichlet boundary conditions of Navier-Stokes equations can bederived. Furthermore, according to Sone’s book [30, 31] , the jump type boundary conditions of NSFequations can be derived while z ǫ is not close to M in the relative entropy sense, i.e. C = + ∞ . But this case is more challenging. At least, there is no uniform relative entropy bound for fluctuations G ǫ with respect to ǫ . We leave it to the future work. For Navier-Stokes limit, the proof of the interior domain follows the proof of [20]. Here, wefocus on the boundary part, that is to say, how to derive the boundary conditions of NSF fromhomogeneous incoming boundary condition of Boltzmann equation. The idea and strategy will beelaborated during its proof. 3.
Estimates of Approximate system
In this section, we will construct a sequence of approximate solution to Boltzmann equation withmodified collision kernel Q n , namely Q n ( F, F ) = (cid:0)
11 + n R F d v (cid:1) Z R × S B n [ F ( v ′ ) F ( v ∗′ ) − F ( v ) F ( v ∗ )]d v ∗ d ω (3.1)with B n ( v − v ∗ , ω ) = B ( v − v ∗ , ω ) · n ≤| v − v ∗ |≤ n . (3.2)For every n ∈ N + , the initial data approximate system are chosen as these in [12], namely F n = ˜ F n + n exp( − | x | − | v | ) , (3.3)where ˜ F n is obtained by truncating f first and then smoothing it. In details, we will solve thefollowing initial-boundary problem ∂ t F n + v · ∇ x F n = Q n ( F n , F n ) ,F n (0 , x, v ) = F n ( x, v ) ,γ − F n = Z, on Σ − (3.4)where Z satisfies for all t > Z t Z Σ − Z (1 + | v | + | log z | )d µ d σ x d s < C ( t ) < ∞ , (3.5)and F n satisfies Z O F n (1 + | v | + | log F n | )d µ d σ x d s < C < ∞ , for all n ∈ N + . (3.6)For each fixed n ∈ N + , System (3.4) can be solved by the fixed point theorem to followingiteration system ∂ t F n,k +1 + v · ∇ x F n,k +1 = Q n ( F n,k , F n,k ) ,F n,k (0 , x, v ) = F n ( x, v ) ,γ − F n,k = Z, on Σ − . From [12], k Q n ( F, F ) k L ( O ) ≤ C n k F k L ( O ) , then for every k , the existence of the above equation is equal to that of the following equation: ∂ t h + v · ∇ x h = ˜ H,h (0 , x, v ) = h ( x, v ) ,γ − h = z Σ − , (3.7)where for any T > h ∈ L ( O ) , ˜ H ∈ L ((0 , T ) × O ) , z ∈ L ((0 , T ) × Σ + ) . (3.8) XISTENCE AND NS LIMITS 13
Theorem 3.1 (Existence of (3.7)) . Under the assumption (3.8) , there is a unique solution h ∈ L ∞ ([0 , T ] , L ( O )) such that ∂ t h + v · ∇ x h = ˜ H, holds in the sense of distribution. Furthermore, there exists a unique trace γ + h ∈ L ((0 , T ) × Σ + ) to (3.7) such that Z T Z O (cid:0) β ( h )( ∂ t φ + v · ∇ x φ ) + ˜ Hβ ′ ( h ) φ (cid:1) d v d x d t = Z O φ ( T ) β ( h )( T )d v d x − Z O φ (0) β ( h )d v d x + Z T Z Σ + φβ ( γ + h )d µ d σ x d t − Z T Z Σ − β ( z ) φ d µ d σ x d t, for all β ( · ) ∈ W , ∞ loc ( R ) with sup x ∈ R | β ′ ( x ) | < ∞ and all the test function φ ∈ D ([0 , T ] × ¯Ω × R ) .Moreover, the solution and its trace satisfy the following estimate Z O | h | ( t )d v d x + Z t Z Σ + | γ + h | d µ d σ x d s ≤ C (cid:18) Z O T | ˜ H | d v d x d s + Z O | h | d v d x + Z t Z Σ − | z | d µ d σ x d s (cid:19) , t ≤ T. (3.9) Proof.
This equation can be solved by characteristic line method. Indeed, for any s ≥ x ∈ Ω with velocity v , we can define its trajectory as S t ( s, x, v ) = ( s + t, x + vt, v ) , S t ∈ O , − t − < t < t , x t = x + vt, where t denotes the maximal forward time of staying in O (with velocity v ) and t − denotes themaximal backward time of staying in O (with velocity − v ). Then integrating over the characteristicline S t , h ( s + t, x + vt, v ) = h ( s, x, v ) + Z ts ˜ H ( s + τ, x + τ v, v )d τ, x ∈ Ω . We just need to pay careful attention to the particles which locate at the boundary or are going tohit the boundary. When the molecule hits the boundary at x with velocity v , the velocity aftercollision is given by the specular reflection formula, R x v = v − n ( x ) v ] n ( x ). Then we can setthe point ( t , x t , R x v ) as new ”initial” point. Similarly, we can define t , the maximal forwardtime of staying in O (with velocity R x t v ) starting from x and the trajectory S t ( s, x, v ) = ( s + t, x + R x t v ( t − t ) , R x t v ) , S t ∈ O , t ≤ t < t . According to the boundary condition, γ − h ( t , x , R x t v ) = z ( t , x , R x t v ) , then for the particle located x ∈ ∂ Ω with velocity R x v and t ≤ t < t , h ( t, x + R x v ( t − t ) , R x v ) = z ( t , x , R x v ) + Z tt ˜ H ( τ, x + R x v ( τ − t ) , R x v )d τ. Inductively, we can define its trajectory S t ( s, x, v ) for any t > S t .We start to deduce some estimates. Let β δ be a sequence of even smooth functions, such that β δ (0) = 0 , β δ ( y ) ≥ , | β ′ δ ( y ) | ≤ , β δ → | y | ( δ → . By Lemma 2.1, multiplying the first equation in 3.7, we can infer that Z O β δ ( h )( t )d v d x + Z t Z Σ + β δ ( γ + h )d µ d σ x d s ≤ Z t Z Σ − β δ ( z )d µ d σ x d s + Z O T | ˜ H | d v d x d s + Z O β δ ( F n )d v d x ≤ Z t Z Σ − | z | d µ d σ x d s + Z O T | ˜ H | d v d x d s + Z O | F n | d v d x. (3.10)We complete the proof by letting δ → (cid:3) For every k ≥ T >
0, if we set F n,k (0) = F n , by Theorem 3.1, the following system ∂ t F n,k +1 + v · ∇ x F n,k +1 = Q n ( F n,k , F n,k ) , v ∈ R ,F n,k (0 , x, v ) = F n ( x, v ) ,γ − F n,k = Z, on Σ − , admits a unique solution F n,k ∈ L ∞ ([0 , T ] , L ( O )).In fact, for any fixed n , as long as the life span is small enough, F n,k is a compact sequence in L ( O ). Theorem 3.2 (Local-in-time existence to (3.4)) . For every n , under the assumptions on initialdata (3.6) and on the incoming boundary condition (3.5) , there exists some T n such system (3.4) admits a unique solution F n ∈ L ∞ ([0 , T n ]; L ( O )) in the sense of distribution. Further, there existsa unique trace γ + F n ∈ L ((0 , T n ) × Σ + ) such that Z T n Z O (cid:0) β ( F n )( ∂ t φ + v · ∇ x φ ) + Q n ( F n , F n ) β ′ ( F n ) φ (cid:1) d v d x d t = Z O φ ( T n ) β ( F n )( T n )d v d x − Z O φ (0) β ( F n )d v d x + Z T n Z Σ + φβ ( γ + F n )d µ d σ x d t − Z T n Z Σ − β ( Z ) φ d µ d σ x d t, for all β ( · ) ∈ W , ∞ loc ( R ) with sup x ∈ R | β ′ ( x ) | < ∞ and all the test function φ ∈ D ([0 , T ] × ¯Ω × R ) .Moreover, there exists a constant C n such that the solution and its trace satisfy the followingestimates Z O | F n | ( t )d v d x + Z t Z Σ + | γ + F n | d µ d σ x d s ≤ C n (cid:18) Z O | F n | d v d x + Z t Z Σ − | Z | d µ d σ x d s (cid:19) , t ≤ T n . (3.11) Proof.
We are going to use iteration methods to prove this theorem. For each n ≥
1, we can obtainthe existence of F n, by applying Theorem 3.1 to the following system: ∂ t F n, + v · ∇ x F n, = Q n ( F , F ) ,F n, (0 , x, v ) = F ( x, v ) ,γ − F n, = Z, on Σ − . (3.12)Recalling kQ n ( F, F ) k L ( O ) ≤ C n k F k L ( O ) , XISTENCE AND NS LIMITS 15 inductively, we can obtain a solution sequence { F n,k } from the following iteration system for each k ≥ ∂ t F n,k +1 + v · ∇ x F n,k +1 = Q n ( F n,k , F n,k ) ,F n,k (0 , x, v ) = F ( x, v ) ,γ − F n,k = Z, on Σ − . (3.13)Moreover, F n,k satisfies sup ≤ s ≤ t Z O | F n,k | ( s )d v d x + Z t Z Σ + | γ + F n,k | d µ d σ x d s ≤ C n (cid:18) Z O T | F n,k − | d v d x d s + Z O | F | d v d x + Z t Z Σ − | Z | d µ d σ x d s (cid:19) . (3.14)Denote: C ,z, = Z O | F | d v d x + Z Z Σ − | Z | d µ d σ x d s. Then, for t ≤
1, sup ≤ s ≤ t Z O | F n,k | ( s )d v d x + Z t Z Σ + | γ + F n,k | d µ d σ x d s ≤ C n (cid:18) Z O t | F n,k − | d v d x d s + C ,z, (cid:19) ≤ C n t Z O t | F n,k − | d v d x d s + (1 + C n t ) C ,z, ≤ ( C n t ) k − t Z O | F | d v d x + (1 + C n t + · · · + ( C n t ) k − ) C ,z, . (3.15)Choosing small enough t n such that C n t n <
1, there exists ˜ C n such that for any k ∈ N + sup ≤ s ≤ t Z O | F n,k | ( s )d v d x + Z t Z Σ + | γ + F n,k | d µ d σ x d s ≤ ˜ C n C ,z, , t ≤ t n . (3.16)In fact, { F n,k } is a convergent sequence in L ∞ ([0 , t n ] , L ( O )). Noticing system (3.13) is a linearequation, we can infer ∂ t ( F n,k +1 − F n,k ) + v · ∇ x ( F n,k +1 − F n,k ) = Q n ( F n,k , F n,k ) − Q n ( F n,k − , F n,k − ) , ( F n,k +1 − F n,k )(0 , x, v ) = 0 ,γ − ( F n,k +1 − F n,k ) = 0 , on Σ − . (3.17)By simple calculation, one getssup ≤ s ≤ t Z O | F n,k +1 − F n,k | ( s )d v d x + Z t Z Σ + | γ + ( F n,k +1 − F n,k ) | d µ d σ x d s ≤ ˜ C n t sup ≤ s ≤ t Z O | F n,k − F n,k − | ( s )d v d x, (3.18)where we have use k Q n ( F, F ) − Q n ( G, G ) k L ( O ) ≤ C n k F − G k L ( O ) . Choosing t n < t n such that t n ˜ C n <
1, then { F n,k } is a convergent sequence. Resetting T n = t n ,we complete the proof. (cid:3) The life span T n in Theorem 3.2 does not have a lower bound with n . This results from the sourceterm. With the help of the symmetry properties of the Boltzmann collision kernel, we conclude Theorem 3.3 (global-in-time existence to (3.4)) . For any
T > , under the assumptions (3.5) and (3.6) , for every n , system (3.4) has a unique solution F n ∈ L ∞ ([0 , T ]; L ( O )) such that ∂ t F n + v · ∇ x F n = Q n ( F n , F n ) holds in the sense of distribution. Further, there exists a unique trace γ + F ∈ L ((0 , T ) × Σ + ) suchthat Z T Z O (cid:0) β ( F n )( ∂ t φ + v · ∇ x φ ) + Q n ( F n , F n ) β ′ ( F n ) φ (cid:1) d v d x d t = Z O φ ( T ) β ( F n )( T )d v d x − Z O φ (0) β ( F n )d v d x + Z T Z Σ + φβ ( γ + F n )d µ d σ x d t − Z T Z Σ − β ( Z ) φ d µ d σ x d t, for all β ( · ) ∈ W , ∞ loc ( R ) with sup x ∈ R | β ′ ( x ) | < ∞ and all the test function φ ∈ D ([0 , T ] × ¯Ω × R ) .Furthermore, F n and γ + F n satisfy • local conservation law of mass: for any φ ∈ D ( ¯Ω) , Z O F n ( t ) φ d v d x + Z t Z Σ + γ + F n φ d µ d σ x d s = Z O F n φ d v d x + Z t Z Σ − Zφ d µ d σ x d s, t ≤ T. (3.19) • global conservation law of mass: Z O F n ( t )d v d x + Z t Z Σ + γ + F n d µ d σ x d s = Z O F n d v d x + Z t Z Σ − Z d µ d σ x d s, t ≤ T. (3.20) • global conservation law of momentum Z O vF n ( t )d v d x + Z t Z Σ + vγ + F n d µ d σ x d s = Z O vF n d v d x + Z t Z Σ − vZ d µ d σ x d s, t ≤ T. (3.21) • global conservation law of energy Z O | v | F n ( t )d v d x + Z t Z Σ + | v | γ + F n d µ d σ x d s ≤ Z O | v | F n d v d x + Z t Z Σ − | v | Z d µ d σ x d s, t ≤ T. (3.22) • global entropy inequality Z O F n log F n ( t )d v d x + Z t Z Σ + γ + F n log γ + F n d µ d σ x d s + Z t D ( F n )( s )d s ≤ Z O F n log F n d v d x + Z t Z Σ − Z log Z d µ d σ x d s, t ≤ T, (3.23) XISTENCE AND NS LIMITS 17 • global relative entropy inequality H ( F n | M)( t ) + Z t Z Σ + h ( γ + F n / M)d µ d σ x d s + Z t D ( F n )( s )d s ≤ H ( F n | M) + Z t Z Σ − h ( Z | M)d µ d σ x d s, t ≤ T. (3.24) Proof.
Noticing that 1 and | v | lay in the kernel of Q n , we infer that Z O F n ( t )d v d x + Z t Z Σ + γ + F n d µ d σ x d s = Z O F n d v d x + Z t Z Σ − Z d µ d σ x d s, t > , (3.25)and Z O | v | F n ( t )d v d x + Z t Z Σ + | v | γ + F n d µ d σ x d s = Z O | v | F n d v d x + Z t Z Σ − | v | Z d µ d σ x d s, t > . (3.26)Then for any t >
0, we findsup ≤ s ≤ t Z O (1 + | v | ) F n ( s )d v d x + Z t Z Σ + (1 + | v | ) γ + F n d µ d σ x d s ≤ Z O (1 + | v | ) F n d v d x + Z t Z Σ − (1 + | v | ) Z d µ d σ x d s. (3.27)Using the same trick as the one in [12, pp.360], one finds that F n ( t, x, v ) > , t ≤ T n , ( x, v ) ∈ O . (3.28)Moreover, γ + F n ( t, x, v ) > , t ≤ T n , ( x, v ) ∈ Σ + . (3.29)For the relative entropy inequality, multiplying the first equation of (3.4) by 1 + log F , log Mrespectively, we have ∂ t ( F n log F n ) + v · ∇ x ( F n log F n ) = Q n ( F n , F n )(1 + log F n ) , (3.30) ∂ t ( F n log M) + v · ∇ x ( F n log M) = Q n ( F n , F n )(log M) . (3.31)In the light of (3.30) and (3.31), we can infer that ∂ t H ( F n | M) + v · ∇ x H ( F n | M) = Q n ( F n , F n ) log f n M . (3.32)For any t >
0, integrating (3.32) over O t , we conclude thatsup ≤ s ≤ t Z O H ( F n | M)( s )d v d x + Z t Z Σ + γ + (cid:0) H ( F n | M) (cid:1) ( s )d µ d σ x d s + Z t D ( F n )( s )d s ≤ Z O H ( F n | M)( s )d v d x + Z t Z Σ − H ( Z | M)d µ d σ x d s. Denoting C H,z,t = Z O H ( F n | M)( t )d v d x + Z t Z Σ − H ( Z | M)d µ d σ x d s, and recalling that relative entropy is always positive, the above inequality indicates that Z O H ( F n | M)( t )d v d x ≤ C H,z,t , (3.33) Z t Z Σ + γ + (cid:18) H ( F n | M) (cid:19) ( s )d µ d σ x d s ≤ C H,z,t . (3.34)Similarly, for any t >
0, integrating (3.30) over O t , we havesup ≤ s ≤ t Z O F n log F n ( s )d v d x + Z t Z Σ + γ + ( F n log F n )( s )d µ d σ x d s + Z t D ( F n )( s )d s ≤ Z O F n log F n d v d x + Z t Z Σ − Z log Z ( s )d µ d σ x d s. (3.35)With the help of (3.27) and (3.35), by simple calculation, we find thatsup ≤ s ≤ t Z O F n | log F n | ( s )d v d x + Z t Z Σ + γ + F n | γ + log F n | d µ d σ x d s ≤ C ( t ) . (3.36)Thus, the life span T n in Theorem 3.2 can be extended to any T > (cid:3) weak compactness and global existence In this section, we prove Theorem 2.7. First, we summarize all these estimates on F n upsup ≤ s ≤ T Z F n ( s )(1 + | v | + | log F n | )d v d x ≤ C ( T ) , (4.1)and Z t Z Σ ± (1 + | v | + | log γ ± F n | ) γ ± F n d µ d σ x d s ≤ C ( T ) . (4.2)From estimates (4.1), for any fixed T >
0, using Dunford-Pettis Lemma, it follows that { F n } and { γ + F n } are weakly compact sequence in L ∞ ((0 , T ); L ( O )) and L (0 , T ) × Σ + ). Further, takingthe whole sequence as example, there exists some f ∈ L ((0 , T ) × O ) such that F n ⇀ F, in L ((0 , T ) × O ) . (4.3)The proof of Theorem 2.7 is split into two parts: the interior domain part and the boundarypart.4.1. Interior domain.Theorem 4.1 (Extended Stability [12, 21]) . Let B in (1.4) satisfy the assumptions (1.7) and (1.8) . Let { F n } be a sequence of solutions obtained in Theorem 3.3 to the approximate Boltzmannequation (3.4) . Then • F n → F in L p ([0 , T ] , L ( O )) , p ≥ ; • for all nonlinearity β ∈ C ( R + , R + ) satisfying ≤ β ′ ( F ) < C (1 + f ) − , then β ′ ( F ) Q ( F, F ) ∈ L ((0 , T ) × O ) , and ∂ t β ( F ) + v · ∇ x β ( F ) = β ′ ( F ) Q ( F, F ) , holds in the sense of distribution. XISTENCE AND NS LIMITS 19 • for all φ ∈ D ([0 , T ] × ¯Ω × R ) , Z t Z O Q n ( F n , F n ) β ′ ( F n ) φ d v d x d s → Z t Z O Q ( F, F ) β ′ ( F ) φ d v d x d s, t ≤ T. Proof.
The first two items directly come from [12, 21]. With the strong convergence of { F n } in L space, the third can be verified by the argument in [12] too. (cid:3) Boundary Parts.
From Theorem 4.1, the solution F n satisfies ∂ t β ( F n ) + v · ∇ x β ( F n ) = β ′ ( F n ) Q ( F n , F n ) , in D ′ ((0 , T ) × O ) . By Lemma 2.1, we can only get the trace of β ( F ) other than that of f . Recall Z O F n ( t )d v d x + Z t Z Σ + γ + F n d µ d σ x d s = Z O F d v d x + Z t Z Σ − Z d µ d σ x d s, t > . (4.4)Since F n → F in L ∞ ([0 , T ] , L (Ω × R )), we get Z O F n ( s )d v d x → Z O F ( s )d v d x, s ≤ T. Then according to (4.1), by Dunford-Pettis Lemma, we can infer: There exists f γ ∈ L ((0 , T ) × Σ + )such that γ + F n ⇀ F γ , in L ((0 , T ) × Σ + ) , Z t Z Σ + γ + F n d µ d σ x d s → Z t Z Σ + F γ d µ d σ x d s. Thus, we infer that Z O F d v d x + Z t Z Σ + F γ d µ d σ x d s = Z O F d v d x + Z t Z Σ − Z d µ d σ x d s, t > . (4.5)In fact, we will show that F γ satisfies (2.3), namely that F γ is the trace of F on Σ + , γ + F := F γ . Theorem 4.2.
Assume that β ∈ C [0 , + ∞ ) be a non-linear function with β ′ ( x ) ≥ and sup x ≥ (1 + x ) β ′ ( x ) < ∞ . Then for any φ ∈ D ([0 , T ] × ¯Ω × R ) and any T > , Z T Z O (cid:0) β ( F )( ∂ t φ + v · ∇ x φ ) + Q ( f, f ) β ′ ( F ) φ (cid:1) d v d x d t = Z O β ( F )( T ) φ d v d x − Z O β ( F ) φ (0)d v d x + Z T Z Σ + β ( F γ ) φ d µ d σ x d t − Z T Z Σ − β ( Z ) φ d µ d σ x d t. Proof.
According to Theorem 3.3, for all
T >
0, as long as 0 ≤ β ′ ( x ) ≤ C (1 + x ) − , then for φ ∈ D ([0 , T ] × ¯Ω × R ), Z T Z O (cid:0) β ( F n )( ∂ t φ + v · ∇ x φ ) + Q n ( F n , F n ) β ′ ( F n ) φ (cid:1) d v d x d t = Z O φ ( T ) β ( F n )( T )d v d x − Z O φ (0) β ( F n )d v d x + Z T Z Σ + φβ ( γ + F n )d µ d σ x d t − Z T Z Σ − β ( Z ) φ d µ d σ x d t, Choose a sequence of concave function β j ( x ) = x xj , obviously, β j ( f ) ≤ j, β n (0) = 0 , < β ′ j = j ( j + f ) . Besides, 0 ≤ β j ( γ + F n ) ≤ γ + F n , since { γ + F n } is a weakly compact sequence, for any fixed j ∈ N + , there exists F γ,j ∈ L ((0 , T ) × Σ + )such that β j ( γ + F n ) ⇀ F γ,j , in L ((0 , T ) × Σ + ) . Let n go to infinity, with the help of Theorem 4.1, Z T Z O (cid:0) β j ( F )( ∂ t φ + v · ∇ x φ ) + Q ( F, F ) β ′ j ( F ) φ (cid:1) d v d x d t = Z O φ ( T ) β j ( f )( T )d v d x − Z O φ (0) β j ( F )d v d x + Z T Z Σ + φf γ,j d µ d σ x d t − Z T Z Σ − β j ( Z ) φ d µ d σ x d t, Noticing that β j ( x ) ≤ j , then by Lemma 2.3, γ + β j ( f ) = f γ,j and F γ,j +1 ≥ F γ,j , a.e. on (0 , T ) × Σ + . Furthermore, for all β ∈ C ( R + ) with 0 ≤ β ′ ( x ) and sup x ≥ (1 + x ) β ′ ( x ) < ∞ , Z T Z O (cid:0) β ( β j ( F ))( ∂ t φ + v · ∇ x φ ) + Q ( F, F ) β ′ j ( F ) β ′ ( β j ( f )) φ (cid:1) d v d x d t = Z O φ ( T ) β ( β j ( F ))( T )d v d x − Z O φ (0) β ( β j ( F ))d v d x + Z T Z Σ + φβ ( F γ,j )d µ d σ x d t − Z T Z Σ − β ( β j ( Z )) φ d µ d σ x d t, Recalling for any j ∈ N + , β ( β j ( F )) ≤ β ( F ), by Lebesgue dominated convergence theorem, β ( β j ( F )) → β ( F ) , in L ((0 , T ) × O ; d v d x d t ) , and β ( β j ( F )) → β ( F ) , in L ∞ ((0 , T ); L ( O ; d x d v )) . For the collision term, recalling that there exist a constant such that | β ′ ( β j ( F )) | ≤ C (1 + β j ( F )) − , then |Q ( F, F ) β ′ j ( F ) β ′ ( β j ( F )) | = |Q ( F, F ) (cid:0) Fn (cid:1) − β ′ ( β j ( F )) |≤ C | Q ( F, F )1 + F F (1 + Fn ) ×
11 + F Fn ≤ C | Q ( F, F )1 + F F (1 + Fn ) + f (1 + Fn ) | . XISTENCE AND NS LIMITS 21 ≤ C | Q ( F, F )1 + F | . It follows that Q ( F, F ) β ′ j ( F ) β ′ ( β j ( F )) → Q ( F, F ) β ′ ( F ) , a.e. on O T , as j → ∞ . Thus, by Lebesgue dominated convergence theorem again Z T Z O Q ( F, F ) β ′ j ( F ) β ′ ( β j ( F )) φ d v d x d t → Z T Z O Q ( F, F ) β ′ ( F ) φ d v d x d t. According to the definition (2.4) and recalling β j ( F ) ≤ j, β n (0) = 0 , < β ′ j = j ( j + F ) , based on the above analysis, we can infer that { β j } is a renormalizing sequence and { F γ,j } isa increasing sequence on (0 , T ) × Σ + , together with γ + F n ⇀ f γ ( L ((0 , T ) × Σ + )), by virtual ofProposition 2.5, we conclude F γ,j ↑ F γ , on (0 , T ) × Σ + . Let j → + ∞ , finally, we obtain Z T Z O (cid:0) β ( F )( ∂ t φ + v · ∇ x φ ) + Q ( F, F ) β ′ ( F ) φ (cid:1) d v d x d t = Z O β ( F )( T ) φ d v d x − Z O β ( F ) φ (0)d v d x + Z T Z Σ + β ( F γ ) φ d µ d σ x d t − Z T Z Σ − β ( z ) φ d µ d σ x d t. This means γ + F = F γ , γ − F = Z. (4.6) (cid:3) Then (4.5) becomes Z O F d v d x + Z t Z Σ + γ + F d µ d σ x d s = Z O F d v d x + Z t Z Σ − Z d µ d σ x d s, t > . (4.7)As for the energy inequality, recalling that γ + F n ⇀ γ + F, in L ((0 , T ) × Σ + ) , then for any fixed m ∈ N + , denoting by m the characteristic function of ball in R with radius m , { v : | v | ≤ m } , we can infer that | v | m γ + F n ⇀ | v | m γ + F, in L ((0 , T ) × Σ + ) , and | v | m F n ⇀ | v | m F, in L ∞ ((0 , T ); L ( O )) . By the lower semi-continuity of norm,sup ≤ s ≤ t Z O m | v | F ( s )d v d x + Z t Z Σ + m | v | γ + F d µ d σ x d s ≤ Z O | v | F d v d x + Z t Z Σ − | v | Z d µ d σ x d s, t > . Taking m to infinity, by Fatou lemma, we deducesup ≤ s ≤ t Z O | v | f ( s )d v d x + Z t Z Σ + | v | γ + F d µ d σ x d s ≤ Z O | v | F d v d x + Z t Z Σ − | v | Z d µ d σ x d s, t > . (4.8)For the relative entropy inequality, noticing that h ( z ) is a positive convex function, by the lowersemi-continuity of convex functions with respect to weak convergence, we deduce that H ( F | M) + Z t Z Σ + h ( γ + F/ M)d µ d σ x d s + Z t D ( F )( s )d s ≤ H ( F | M) + Z t Z Σ − h ( Z | M)d µ d σ x d s, t ≥ . (4.9)4.3. Local conservation laws.
In this subsection, we focus on the local conservation laws.4.3.1.
Local conservation law of mass.
Choosing some function φ ∈ D ([0 , T ] × ¯Ω), multiplying thefirst equation (3.4) by φ , integrating by parts, then we have Z O F n ( T, x, v ) φ ( T, x, v )d v d x − Z O F n (0 , x, v ) φ (0 , x )d v d x − Z T Z O F n ( s, x, v ) ∂ t φ ( s, x )d v d x d s = Z T Z O F n ( s, x, v ) v · ∇ φ ( s, x )d v d x d s − Z T Z Σ γF n ( s, x, v ) φ ( s, x )n( x ) · v d v d σ x d s. As (1 + | v | ) F n → (1 + | v | ) F, in L (cid:0) (0 , T ) × Ω (cid:1) , and γ ± F n ⇀ γ ± F, in L (cid:0) (0 , T ) × Σ ± (cid:1) , taking n → ∞ , we have Z O F ( T, x, v ) φ ( T, x )d v d x − Z O f (0 , x, v ) φ (0 , x )d v d x − Z T Z O f ( s, x, v ) ∂ t φ ( s, x )d v d x d s = Z T Z O F ( s, x, v ) v · ∇ φ ( s, x )d v d x d s − Z T Z Σ ± γF ( s, x, v ) φ ( s, x )n( x ) · v d v d σ x d s. Noticing that φ is independent of v , then it can be rewritten as Z Ω φ ( T, x ) · (cid:0) Z R F d v (cid:1) ( T, x )d x − Z Ω φ (0 , x ) · (cid:0) Z R F d v (cid:1) (0 , x )d x − Z T Z ∂ Ω ∂ t φ ( s, x ) · (cid:0) Z R F d v (cid:1) ( s, x )d x d s = Z T Z Ω ∇ φ ( s, x ) · (cid:0) Z R F v d v (cid:1) ( s, x )d x d s − Z T Z ∂ Ω n( x ) · (cid:0) Z R γF v d v (cid:1) ( s, x ) φ ( s, x )d σ x d s. (4.10) XISTENCE AND NS LIMITS 23 If φ ( s, x ) | ∂ Ω = 0 for any 0 ≤ s ≤ T and φ (0 , x ) = φ ( T, x ) = 0, we can conclude: In thedistribution sense ∂ t Z R F n d v + ∇ · Z R vF n d v = 0 . As for the local conservation law of mass, similar to Lemma 2.1, we can use the Green formula todefine the trace of R R vF d v on ∂ Ω, Denoting it by γ x ( R R vf d v ).From (4.10), we can infer thatn( x ) · γ x ( Z R vF d v ) = Z Σ x + γ + F | n( x ) · v | d v − Z Σ x − γ − F | n( x ) · v | d v, = n( x ) · Z R vγf d v. This means that the trace operator γ is commutative with integral operator R . This is becausethat the traces of solutions have the full estimate. Besides, the estimates of γF gives,n( x ) γ x ( Z R F ( s ) v d v ) ∈ L (cid:0) (0 , T ) × ∂ Ω; d σ x d s (cid:1) . Local conservation law of momentum.
Different with the local conservation law of mass,there exists defect measure in the conservation law of momentum. For any fixed
T >
0, multiplyingthe first equation of (3.4) by vφ with φ ∈ D ((0 , T ) × Ω), we have after integrating by part Z T Z Ω ∂ t φ ( s ) Z R vF n ( s )d v d x d s + Z T Z Ω ∇ φ ( s ) Z R F n ( s ) v ⊗ v d v d x d s = 0 . (4.11)Recalling that vF n ⇀ vF, in L ∞ ((0 , T ); L ( O )); F n → F, a.e on O T , by Vitalli convergence theorem, we can deduce vF n → vF, in L ∞ ((0 , T ); L ( O )) . Thus, Z T Z Ω ∂ t φ ( s ) Z R vF n ( s )d v d x d s → Z T Z Ω ∂ t φ ( s ) Z R vF ( s )d v d x d s. For the second term in (4.11), the only thing at our disposal is vF n → vF, in L ((0 , T ) × O ) . With these estimates, we can only prove that there exists distribution-value matrix W with W i,j ( i, j = 1 , , ∈ D ′ ((0 , T ) × Ω) such that while n → ∞ Z T Z O F n v ⊗ v · ∇ φ d v d x d s → Z T Z O ( F v ⊗ v ) · ∇ φ d v d x d s + ( W , ∇ φ ) , where ( · , · ) denotes the action between distribution and test function.Thus, we conclude the local conservation law of momentum.5. Proof of Theorem 2.10
In this section, we prove Theorem 2.10. The proof is mainly made up of two parts. The first partconsists in proving the fluctuations { g ǫ } tend to g with form (2.13) in the interior domain. Besides,the coefficient u and θ are weak solutions to NSF equations. The proof of this part is the same asthat in [20]. The second part is devoted to show u and θ satisfy Dirichlet boundary conditions. From now on, We denote d ς = M | n( x ) · v | d v . Here, we sketch the proof of the first part. From(1.16) and (1.15), the equation of g ǫ is ǫ∂ t g ǫ + v ·∇ x g ǫ + ǫ L g ǫ = B ( g ǫ , g ǫ ) ,γ − g ǫ = z ǫ ,g ǫ (0 , x, v ) = ǫ F ǫ − MM , where the scaled collision operator B is defined as B ( G, G ) = Q (M G, M G ) . Because of the order ǫ before linear Boltzmann operator, g ǫ tends to an infinitesimal Maxwellian g in the sense of distribution, the kernel space of linear Boltzmann collisional operator L , g ( t, x, v ) = ρ ( t, x ) + u( t, x ) · v + θ ( t,x )2 ( | v | − . (5.1)According to the uniform relative entropy inequality(2.10), by the argument as [29, Lemma3.1.2], it follows that g ǫ ∈ L ∞ ((0 , + ∞ ) , L ( O , (1 + | v | )Md v d x ) , and g ǫ belongs to L space up to a L perturbation of order ǫ . Moreover, { g ǫ } is weakly compactin L ∞ ((0 , + ∞ ) , L ( O , (1 + | v | )Md v d x ). Thus, there exist ρ, u, θ ∈ L ∞ ((0 , ∞ ); L (Ω)) such that g ( t, x, v ) = ρ ( t, x ) + u( t, x ) · v + θ ( t,x )2 ( | v | − , (5.2)and g ǫ ⇀ g, in L ∞ ((0 , + ∞ ) , L ( O , (1 + | v | )Md v d x ) . Now, we turn to derive equations of ρ , u and θ . Let β ( z ) = z − z − , z > . Denoting ˜ g ǫ = ǫ β ( G ǫ ) and N ǫ = 1 + ǫ g ǫ , then by simple computation˜ g ǫ = g ǫ N ǫ , ǫg ǫ > − . By the relative entropy estimates, for any
T > g ǫ ∈ L ∞ ((0 , T ); L (Md v d x )) . (5.3)By simple computation, ˜ g ǫ ⇀ g, in L ∞ ((0 , T ); L (Md v d x )) . Denoting h f i := Z R f Md v, the equations of ρ , u and θ can be derived from equations of h ˜ g ǫ i , h v ˜ g ǫ i and h ( | v | − g ǫ i . Indeed, bythe existence Theorem 2.7, for every ǫ >
0, the initial boundary problem (1.15) admits renormalizedsolutions g ǫ such that ∂ t ˜ g ǫ + ǫ v · ∇ x ˜ g ǫ = ǫ ( N ǫ − N ǫ ) B ( G ǫ , G ǫ ) (5.4)holds in the sense of distribution.Multiplying the above equation by 1 , v and | v | respectively, we get the approximate conservationlaws of moments: ∂ t h ˜ g ǫ i + ǫ div x h v ˜ g ǫ i = ˜D(1) ,∂ t h v ˜ g ǫ i + ǫ div x h A( v )˜ g ǫ i + ǫ ∇ x h| v | ˜ g ǫ i = ˜D( v ) ,∂ t h ( | v | − g ǫ i + ǫ div x h B( v )˜ g ǫ i + ǫ ∇ x h v ˜ g ǫ i = ˜D(( | v | − , XISTENCE AND NS LIMITS 25 with ˜D( ξ ) = ǫ h ξ ( N ǫ − N ǫ ) B ( G ǫ , G ǫ ) i . Moreover, from [20, Sec. 6], for every subsequence { g ǫ k } converging to g in L ∞ ((0 , + ∞ ) , L ( O , (1 + | v | )Md v d x ) as ǫ k →
0, the subsequence also enjoys P h vg ǫ k i → u , h ( | v | − g ǫ i → θ, in C ([0 , ∞ ) , D ′ ( R )) , and in the distributional sense, P ( ǫ div x h A( v )˜ g ǫ i ) → u · ∇ u − ν ∆u , ǫ div x h B( v )˜ g ǫ i → u · ∇ θ − κ ∆ θ. Moreover, ρ, u , θ in (5.2) satisfy divu = 0 , ρ + θ = 0 . With the notation q ǫ for ǫ ( N ǫ − N ǫ ) B ( G ǫ , G ǫ ), q ǫ ⇀ q = v · ∇ x g, in L (((0 , T ) × O ); Md v d x d t ) . (5.5)The energy estimate of u and θ (2.14) can be obtained by combining the relative entropy in-equality and (5.5), see [20, Sec. 6]. Verify Dirichlet boundary condition.
In the interior domain, the fluctuations { g ǫ } tend to an infinitesimal Maxwellian as ǫ tends tozero. At the mean time, NSF equations can be derived. In what follows, we try to derive theboundary conditions of u and θ . First, we prepare some estimates on the traces of ˜ g ǫ . Lemma 5.1.
Under the same assumptions of Theorem 2.10, there exists a constant C such that k γ (˜ g ǫ )( t ) k L (d ς d σ x ) ≤ C · C ǫ. (5.6) Remark 5.2.
From [23, pp.1273] , for the solutions with Maxwell reflection boundary, they couldonly infer some estimates like ( γ + g ǫ − h γ + g ǫ i| ∂ Ω ) from the relative entropy estimates. But forincoming boundary condition, we can directly deduce some full estimate of γg ǫ from the relativeentropy estimate (2.10) .Proof. ˜ g ǫ = (cid:0) g ǫ ǫg ǫ ) (cid:1) = g ǫ ǫg ǫ · ǫg ǫ (1 + ( ǫg ǫ ) ) ≤ C · g ǫ ǫg ǫ . where C = sup z ≥− z (1+ z ) .On the other hand, from [3, Lemma 8.1], g ǫ ǫg ǫ ≤ h ( G ǫ ) ǫ . Recalling that 1 ǫ Z t Z Σ ± h ( γG ǫ )d ς d σ x d s ≤ C ǫ, we complete the proof. (cid:3) With the estimates (2.14) and (5.6) at our disposal, we can get the boundary conditions of u and θ . Lemma 5.3.
Under the same assumptions of Theorem 2.10, the traces of the limiting fluctuation g | ∂ Ω ∈ L loc (d σ x d t ; L (d ς )) satisfy the identity g | ∂ Ω = u | ∂ Ω · v + θ | ∂ Ω | v | − . Furthermore, u | ∂ Ω = θ | ∂ Ω = 0 . Proof.
For any T ≥
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