Global existence of renormalized solutions to Boltzmann equations with incoming boundary condition and non-cutoff kernel
aa r X i v : . [ m a t h . A P ] J a n GLOBAL EXISTENCE OF RENORMALIZED SOLUTIONS TO BOLTZMANNEQUATIONS WITH INCOMING BOUNDARY CONDITION ANDNON-CUTOFF KERNEL
NING JIANG AND XU ZHANG
Abstract.
We prove the existence of global renormalized solutions to the Boltzmann equationin bounded domain with incoming boundary condition, with non-cutoff collision kernels. Thus weextend the results of [1] for whole spaces or periodic domain to bounded domains endorsed withincoming boundary condition.
Contents
1. Introduction 11.1. Collision kernel of the Boltzmann equation 21.2. Boundary conditions 42. Statements of main results 53. Estimates of Approximate system 84. Estimate of Renormalized Formulation for the non-cutoff case 105. Weak compactness and global existence 125.1. Interior domains 125.2. Boundary parts 145.3. Local conservation law 165.3.1. Local mass conservation law 175.3.2. Local momentum conservation law 17References 181.
Introduction
The Boltzmann equation (or Maxwell-Boltzmann ) equation is an integro-differentiable equation ∂ t f + v ·∇ x f = Q ( f, f ) , (1)which models the statistical evolution of a rarefied gas. In equation (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 N , ( N ≥ R N , or a torus T N , or a bounded domain in R N . Furthermore, Q ( f, f ) is the collision operator whose structure isdescribed below.In this work, the Boltzmann equation (1) is given an initial data which satisfies some naturalphysical bounds (bounded mass, momentum, energy and entropy, etc.). More specifically, f | t =0 = f ( x, v ) in Ω × R N , (2)which satisfies f ≥ Z Z Ω × R N f (1 + | x | + | v | + | log f | ) d x d v < ∞ . (3) The well-posedness of the Boltzmann equation (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 (3): only some finite physical bounds, without any smallness requirements on the sizeof f , was proved in the celebrated DiPerna-Lions’ theorem [4] for Ω = R N (with some minormodifications, their proof works also for torus T N ).Since in the natural functional spaces of the number density f ( t, x, v ), say L ∩ L log L , thecollision term Q ( f, f ) in (1) is not even locally integrable, which makes weak solutions to theBoltzmann equation can not be defined in the usual sense. In stead, under the Grad’s angularcutoff assumption and a mild decay condition on the collision kernel which we will describe indetails later, DiPerna and Lions defined the so-called renormalized solutions to the (1) and provethat a sequence of renormalized solutions which satisfy only the physically natural a priori boundsconverge weakly in L . From this stability they deduced global existence of renormalized solutions.In [1], Alexandre and Villani studied the Boltzmann equation without Grad’s angular cutoffassumption. They introduced a new renormalized formulation that allows the cross section to besingular in both the angular and the relative velocity variables, which occur in long-range interac-tions and soft potentials in particular Coulomb interaction. Together with some new estimates, theyprove global existence of renormalized solutions with defect measure. Again, Alexandre-Villani’sresults were proved for the whole space case.Since the obvious importance in applications and theoretical research, extending the global ex-istence results of DiPerna-Lions [4] and Alexandre-Villani [1] for whole space (and torus) to thedomain with boundary is a natural open question since then. A first complete answer in this di-rection is due to Mischler [13], who proved global renormalized solution of the Boltzmann equationwith cutoff collision kernels in a bounded domain endowed with Maxwell reflection boundary con-dition, based on some new observations on weak-weak convergence and his previous results on thetraces of kinetic equations [11, 12]. Maxwell in [10] proposed this boundary condition, which statedthat the gas molecules back to the domain at the boundary come one part from the specular reflec-tion of the molecules escaping the domain, the other part from those entering the wall, interactingwith the molecules in the wall, and re-evaporating back to the domain with the thermal dynamicalequilibrium state of the wall. This boundary condition for the Boltzmann equation can be viewedas an analogue of the Robin condition for the macroscopic equations. In fact, based on Mischler’sglobal renormalized solutions constructed in [13], incompressible Navier-Stokes equations with sev-eral boundary conditions can be justified [9, 14, 7]. In a forthcoming paper, we extend Mischler’sresult to the non-cutoff collision kernel case.Another boundary condition for the kinetic equation is more direct: the number density ofthe gas molecules back to the domain is prescribed. This is the so-called incoming boundarycondition, which has been widely used in applied fields. For more asymptotic analysis, includinghow to derive the boundary conditions for the incompressible Navier-Stokes equations from theBoltzmann equation with incoming data, see Sone’s books [15, 16]. To prove global renormalizedsolutions to the Boltzmann equations with incoming boundary condition, for both the cutoff andnon-cutoff collision kernels, is the main concern of the current paper. We first introduce moredetailed information on the Boltzmann equation in particular the collision kernels and the boundaryconditions so that we can state our main results precisely.1.1. Collision kernel of the Boltzmann equation.
In the Boltzmann equation (1), Q is theBoltzmann collision operator, which acts only on the velocity dependence of f quadratically: Q ( f, f ) = Z R N × S N − ( f ′ f ′∗ − f f ∗ ) b ( v − v ∗ , ω )d v ∗ d ω , (4) NCOMING DATA 3 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 , yield 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. The nonnegative and a.e. finite weight function b ( v − v ∗ , ω ), called cross-section , is assumed to depend only on the relative velocity | v − v ∗ | and cosine of the derivationangle ( v − v ∗ | v − v ∗ | , ω ). For a given interaction model, the cross section can be computed in a semi-explicitway by solving a classical scattering problem, see for instance, [3]. A typical example is that indimension 3, for the inverse s -power repulsive forces (where s > κ = v − v ∗ | v − v ∗ | 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 − , (5)and sin θb (cos θ ) ≈ Kθ − − s ′ as θ → , where s ′ = s − and K > . (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 [17]) 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.There are basically two types of assumptions on the cross section. The main assumption byDiPerna and Lions in [4] on the cross section was Grad’s angular cutoff , namely, that the crosssection be integrable, locally in all variables. More precisely, they assumed A ( z ) = Z S N − b ( z, ω ) d ω ∈ L loc ( R N ) , (7)together with a condition of mild growth of A :(1 + | v | ) − Z | z − v |≤ R A ( z ) d z → | v | → ∞ , for all R < ∞ . (8)However, although the Grad’s angular cutoff assumption (7) has been widely used in this field,it is not satisfactory from the physical point of view. Indeed, as soon as one consider long range interactions, even with a very fast decay at infinity, this assumption is not satisfied. A typicalexample is the that of inverse s -power repulsive forces in dimension 3 mentioned before. Thefunction sin θb (cos θ ) in (6) presents a non-integrable singularity as θ →
0. This regime correspondsto grazing collisions , i.e. collisions in which particles are hardly deviated. Physically speaking,these are the collisions between particles that are microscopically very far apart, with a largeimpact parameter. Another complication arises when dealing with the Coulomb potential: For s = 2 in dimension N = 3 as in (5), one finds a cross-section behaving like | v − v ∗ | − in the relativevelocity variable, hence not locally integrable as a function of the relative velocity (this is calledkinetic singularity). The DiPerna-Lions formulation can not handle this case, which is one of themost important from a physical point of view.In [1], Alexandre and Villani employed several new tools to treat both angular and kineticsingularities and extended the DiPerna-Lions theory to very general, physically realistic long-rangeinteractions, including the Coulomb potential as a limit case. For the readers’ convenience, we listbelow the non-cutoff assumptions made in [1] on the cross-section: N. JIANG AND X. ZHANG (1) Borderline singularity assumption. Assume that the cross section has the following decom-position: b ( z, ω ) = β ( κ · ω ) | z | + B ( z, ω ) , κ = z | z | , (9)for some nonnegative measurable functions β and B , and define µ = Z S β ( κ · ω )(1 − κ · ω ) d ω , (10) M ( | z | ) = Z S B ( z, ω )(1 − κ · ω ) d ω , (11) M ′ ( | z | ) = Z S B ′ ( z, ω )(1 − κ · ω ) d ω , (12)where B ′ ( z, ω ) = sup <λ ≤√ B ( λz, ω ) − B ( z, ω )( λ − | z | . We require that µ < + ∞ and M ( | z | ) , | z | M ′ ( | z | ) ∈ L loc ( R ) . (2) Behavior at infinity assumption. For 0 ≤ α ≤
2, let M α ( | z | ) = Z S b ( z, ω )(1 − κ · ω ) α d ω, κ = z | z | . (13)We require that for some α ∈ [0 , | z | → ∞ , M α ( | z | ) = o ( | z | − α ) , and | z | M ′ ( | z | ) = o ( | z | ) . (14)(3) Angular singularity assumption. B ( z, ω ) ≥ Φ ( | z | ) b ( κ · ω ) , κ = z | z | , (15)where Φ is a continuous function, Φ ( | z | ) > | z | 6 = 0, and Z S b ( κ · ω ) = ∞ . (16)For the inverse s -power repulsive forces in dimension 3, the above three assumptions togetherallow the following range of parameters: γ ≥ − , ≤ s ′ < , s ′ + γ < . Note that when s = 2, γ = −
3, which corresponds to Coulomb interaction. However, thelimiting case s = 2 is not suited for Boltzmann equation as the Boltzmann collision operator shouldbe replaced by the Landau operator in order to handle that situation (see [17]). We will considerthe boundary problem for Landau equation in a separate paper.1.2. Boundary conditions.
As mentioned before, the main concern of the current paper is toextend Alexandre-Villani [1] theories for non-cutoff cross section to the bounded domain withincoming boundary condition.Let Ω be an open and bounded subset of R N and set O = Ω × R and O T = (0 , T ) × Ω × R .We assume that the boundary ∂ Ω is sufficiently smooth. The regularity that we need is that thereexists a vector field n ∈ W , ∞ (Ω ; R N ) such that n( x ) coincides with the outward unit normal vectorat x ∈ ∂ Ω. We define Σ x ± := { v ∈ R N ; ± v · n( x ) > } the sets of outgoing (Σ x + ) and incoming (Σ x − )velocities at point x ∈ ∂ Ω as well as Σ = ∂ Ω × R N andΣ ± = { ( x, v ) ∈ Σ : ± v · n( x ) > } = { ( x, v ) ; x ∈ ∂ Ω , v ∈ Σ x ± } . NCOMING DATA 5
We also denote by d σ x the Lebesgue measure on ∂ Ω.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 = (0 , ∞ ) × Σ ± γf . The so-called incoming boundarycondition is that γ − f = g , (17)where g ≥ Z T Z Σ − g (1 + | v | + | log g | ) d v d σ x d t < ∞ for any T > . (18)In summary, in this paper, we consider the Boltzmann equation (1), with initial condition (2)-(3),and boundary condition (17)-(18). For the non-cutoff kernel, we work in the class of Alexandre-Villani used [1], i.e. the cross-section satisfies the assumptions from (9) to (16). Our main resultsare: under these assumptions on the initial-boundary datum and cross-sections, the Boltzmannequation admits a global in time renormalized solution. Furthermore, this solution admits someconservation laws (or inequalities) of mass, momentum, energy and entropy.2. Statements of main results
In this section, we state our main results. The first difficulty we encounter is the definition ofrenormalized solutions. Besides the renormalization process which is the same as the interior partsfor the non-cutoff, in the bounded domain with boundary, the meaning of the “boundary value” isa nontrivial issue since the solutions lie in the functional space L ∩ L log L the element of whichcan not define the trace in an usual way. Moreover, for the non-cutoff kernels, the formulation ofrenormalized solutions needs a defect measure (see [1]) which makes the definition of the trace evenharder.The obtained solution in this work just makes sense in the distribution sense, namely in thedual space of smooth test function. There are mainly two kinds of test function space. One isthe function space D ((0 , T ) × O ) which is made up of smooth function φ with compact supportsatisfying 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 , and there exists R > φ ( t, x ) ⊂ B R ( v ) , for all ( t, x ) ∈ (0 , T ) × Ω , where B R = { v || v | ≤ R } .The other is function space D ([0 , T ] × ¯Ω × R ) which is made up of smooth functions φ satisfyingthat Supp φ ( t, x ) ⊂ B R ( v ) , for all ( t, x ) ∈ [0 , T ] × ¯Ω . In the following, we will specify the definition of trace for the solution to transport equationwhile the solution just belongs to L space. If the solution to transport equation are smooth up toboundary, then the trace defined below concides with the one in usual sense. Lemma 2.1 (Green Formula[13]) . Let p ∈ [1 , + ∞ ) , g ∈ L ∞ ((0 , T ) , L ploc ( O ) and h ∈ L ((0 , T ) , L ploc ( O ) .Assume that g and h satisfies equation ∂ t g + v · ∇ x g = h, in distribution sense. Then there exists γg well defined on (0 , T ) × Σ which satisfies γg ∈ L loc (cid:0) [0 , T ] × Σ , ( n ( x ) · v ) dvdσ x dt (cid:1) , N. JIANG AND X. ZHANG and the following Green Formula Z T Z O (cid:0) β ( g )( ∂ t φ + v · ∇ x φ ) + hβ ′ ( g ) φ (cid:1) d x d v d t = (cid:2) Z O β ( g )( τ, · )d x d v (cid:3) | T + Z T Z Z Σ β ( γg ) dµdσ x dτ, for β ( . ) ∈ W , ∞ loc ( R + ) with sup x ≥ | β ′ ( x ) | < ∞ , and all the test function φ ∈ D ([0 , T ] × ¯Ω × R ) ,the space of functions φ ∈ D ([0 , T ] × ¯Ω × R ) with φ | (0 ,T ) × Σ = 0 . Now we introduce the definition of solutions. The renormalized solutions obtained in [1] satisfythe following inequality ∂ t β ( f ) + v · ∇ x β ( f ) ≥ β ′ ( f ) Q ( f, f )in the sense of distribution for all concave function β with at most logarithm increase rate. Fur-thermore, we don’t know whether β ′ ( f ) Q ( f, f ) belongs to L space. So the definition of solutionare different with these on cut-off kernel in [12, 13]. Definition 2.2.
Assume that the cross section b ( z, ω ) in (4) satisfies the assumptions listed from (9) to (16) and β ∈ C ( R + , R + ) satisfies β (0) = 0 , < β ′ ( f ) < C f , and β ′′ ( f ) < . (19) A nonnegative function f ∈ C (cid:0) R + , D ′ ( O ) (cid:1) ∩ L ∞ ( R + ; L (cid:0) (1 + | v | ) dxdv ) (cid:1) is called a renormalized solution to the Boltzmann equation (1) , with initial condition (2) - (3) , andboundary condition (17) - (18) , if for every renormalization function β satisfying (19) and every time T > , there is a nonnegative finite defect measure on (0 , T ) × O such that the following equation ∂ t β ( f ) + v · ∇ x β ( f ) ≥ β ′ ( f ) Q ( f, f ) , (20) holds in the following sense : there exist a trace defined on (0 , T ) × Σ + denoted by γ + f ∈ L ((0 , T ) × Σ + ) , and for any non-negative test function ψ ∈ D ([0 , T ] × ¯Ω × R ) with ψ | (0 ,T ) × Σ = 0 , Z T Z O (cid:0) β ( f )( ∂ t ψ + v · ∇ x ψ ) + Q ( f, f ) β ′ ( f ) ψ (cid:1) d x d v d t ≤ Z O β ( f )( T, · ) ψ d x d v − Z O β ( f )(0 , · ) ψ d x d v + Z T Z Σ + β ( γ + f ) ψ | n( x ) · · v | d v d σ x d τ − Z T Z Σ − β ( g ) ψ | n( x ) · · v | d v d σ x d τ. (21) Furthermore, f , γ + f and g satisfies the global mass conservation law Z O f ( t )d v d x + Z t Z Σ + γ + f ( s ) | n( x ) · v | d v d σ x d s = Z O f d v d x + Z t Z Σ − g ( s ) | n( x ) · v | d v d σ x d s, t ≤ T. Remark:
From [1], since there exists a defect measure, the (20) which holds in the sense ofdistribution is only an inequality. The only useful information on this defect measure at ourdisposal is that it is a positive measure. The inequality (20) can be multiplied by positive testfunction ψ belonging to D ([0 , T ] × ¯Ω × R ). But there are too many candidates γ + f satisfying (21).It is very natural to assume that at any time t >
0, the sum of the mass in the interior domainand the mass on the out going set should be equal to the sum of initial mass and the mass on the
NCOMING DATA 7 incoming set. Motivated by [1], we introduce the global mass conservation law to define the tracetoo. Besides, if the solution f are smooth and the defect measures in the interior domain vanishes,by the the trace of f on (0 , T ) × Σ + in the usual sense is equal to γ + f . This is why we denote itby γ + f .Before stating our main results, we introduce some notations. Let M be the global Maxwellian,namely, (2 π ) − exp( − | v | ). The relative entropy denoted by H ( f |M ) is defined as H ( f n |M ) = Z O h ( f n M ) M d v d x, where h ( z ) = z log z − z + 1 , z ≥ , f n M = f n M . We also denote by D ( f ) the H-dissipation D ( f ) = Z Ω Z R × R d v d v ∗ d x Z S d ωB ( v − v ∗ , ω )( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ . Theorem 2.3.
Under the assumption on the cross section B from (9) to (16) , if the initial datumsatisfies (3) and the incoming boundary condition satisfies (18) , then the initial-boundary problemto Boltzmann equation (1) admits a renormalized solution f . Furthermore, f has the followingproperties: • Regularity of Trace: γ + f ∈ L (cid:0) (0 , T ) × Σ + ; (1 + | v | ) | n( x ) · v | d v d σ x ds (cid:1) , for all T > . and Z T Z Σ + γ + f | log γ + f | d µ d σ x d s < + ∞ , for all T > . • Local conservation law of mass: ∂ t Z R f d v + ∇ · Z R f v d v = 0 , in D ′ ((0 , T ) × Ω) . • 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 d v + ∇ · W = 0 , in , D ′ ((0 , T ) × Ω) . • Global momentum conservation law: Z O f ( t ) v d v d x + Z t Z Σ + vγ + f ( s ) | n( x ) · v | d v d σ x d s = Z O f v d v d x + Z t Z Σ − vg ( s ) | n( x ) · v | d v d σ x d s, t ≤ T. (22) • Global energy inequality: Z O f ( t ) | v | d v d x + Z t Z Σ + | v | γ + f ( s ) | n( x ) · v | d v d σ x d s ≤ Z O f | v | d v d x + Z t Z Σ − g ( s ) | v | | n( x ) · v | d v d σ x d s, t ≤ T. (23) N. JIANG AND X. ZHANG • Global entropy inequality: H ( f |M )( t ) + Z t Z Σ + h ( γ + f M ) | n( x ) · v | d v d σ x d s + Z t D ( f )( s )d x d v d s ≤ H ( f |M ) + Z t Z Σ − h ( g |M ) | n( x ) · v | d v d σ x d s, t ≤ T. (24) Remark 2.4.
As for the local mass conservation law, similar to Lemma 2.1, we can use the Greenformula to define the trace of R R vf d v on ∂ Ω . Denote it by γ x ( R R vf d v ) , n( 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, and n( x ) γ x ( Z R f v d v ) ∈ L (cid:0) (0 , T ) × ∂ Ω; d σ x d s (cid:1) . Remark 2.5.
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 ≥ . Compared to Boltzmann equation with cutoff kernel, from [1], the solution f only satisfies thefollowing inequality ∂ t β ( f ) + v · ∇ x β ( f ) ≥ β ′ ( f ) Q ( f, f )in distributional sense. Besides, β ( f ) Q ( f, f ) doesn’t belongs to L space. It is just a distributionbelonging to D ′ ((0 , T ) × O ). So the trace theory in [13, 11] and references therein on cutoff casecompletely doesn’t work here. It needs some new idea. Since the trace on Σ − is fixed, the maintask is to find some γ + f satisfying (21), conservation law of mass and Theorem 2.3. From ourformer work on Boltzmann equation with cutoff kernel, we can construct a sequence of approximatesolutions whose traces are weakly compact in L space. Noticing that the renormalized function β are convex and the test function are positive, then by the upper semi-continuity of convex function,we complete the proof. 3. Estimates of Approximate system
In this section, we will construct a sequence of approximate solutions to Boltzmann equationwith modified collision kernel Q n , namely Q n ( f, f ) = (cid:0)
11 + n R f n dv (cid:1) Z R × S B n | [ f ( v ′ ) f ( v ∗′ ) − f ( v ) f ( v ∗ )]d v ∗ d ω (25)with B n ( v − v ∗ , ω ) = B ( v − v ∗ , ω ) · I n ≤ θ ≤ π I n ≤| v − v ∗ |≤ n . (26)For every n ∈ N + , the initial data of approximate system are chosen as the one in [4], namely f n = ˜ f n + 1 n exp( − | x | − | v | , (27)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 ) , x ∈ Ω ⊂ R , v ∈ R ,f n (0 , x, v ) = f n ( x, v ) ,γ − f n = g, on Σ − (28) NCOMING DATA 9 where g satisfies Z t Z Σ − g (1 + | v | + | log g | )d µ d σ x d s < C ( t ) < ∞ , for all t > . (29)and f n satisfies Z O f n (1 + | v | + | log f n | )d µ d σ x d s < C < ∞ , for all n. (30)For each fixed n , we can use fixed point theorem to solve system (28). Th detailed proof of theexistence can been found in our former work where we obtain the following theorem about globalexistence: Theorem 3.1 (global-in-time existence) . For any
T > , under the assumption (30) and (29) , forevery n , system (28) 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 ) hold in the sense of distribution. Further, there exists a unique trace γ + f ∈ L ((0 , T ) × Σ + ; d µ d σ x d s ) to (28) such that 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 x d v d t = Z O φ ( T ) β ( f n )( T )d x d v − Z O φ (0) β ( f n )d x d v + Z T Z Σ + φβ ( γ + f n )d µ d σ x d t − Z T Z Σ − β ( g ) φ d µ d σ x d t, for all β ′ ( . ) ∈ L ∞ ( R + ) and all the test function φ ∈ D ([0 , T ] × ¯Ω × R ) , the space of functions φ ∈ D ([0 , T ] × ¯Ω × R ) with φ | (0 ,T ) × Σ = 0 . Furthermore, f n and γ + f n satisfy • global conservation law of mass: Z O f n ( t )d x d v + Z t Z Σ + γ + f n | n( x ) · v | d v d σ x d s = Z O f n d x d v + Z t Z Σ − g | n( x ) · v | d v d σ x d s, t ≤ T. (31) • global conservation law of momentum Z O vf n ( t )d x d v + Z t Z Σ + vγ + f n | n( x ) · v | d v d σ x d s = Z O vf n d x d v + Z t Z Σ − vg | n( x ) · v | d v d σ x d s, t ≤ T. (32) • global conservation las of energy Z O | v | f n ( t )d x d v + Z t Z Σ + | v | γ + f n | n( x ) · v | d v d σ x d s ≤ Z O | v | f n d x d v + Z t Z Σ − | v | g | n( x ) · v | d v d σ x d s, t ≤ T. (33) • global entropy inequality Z O f n log f n ( t )d x d v + Z t Z Σ + γ + f n log γ + f n | n( x ) · v | d v d σ x d s + Z t D ( f n )( s )d x d v d s ≤ Z O f n log f n d x d v + Z t Z Σ − g log g | n( x ) · v | d v d σ x d s, t ≤ T, (34) • global relative entropy inequality H ( f n |M )( t ) + Z t Z Σ + h ( γ + f n M ) | n( x ) · v | d v d σ x d s + Z t D ( f n )( s )d x d v d s ≤ H ( f n |M ) + Z t Z Σ − h ( g |M ) | n( x ) · v | d v d σ x d s, t ≤ T. (35) Remark 3.2.
From (33) and we can infer that sup ≤ s ≤ t Z O f n | log f n | ( s )d x d v + Z t Z Σ + γ + f n | γ + log f n || n( x ) · v | d v d σ x d s ≤ C ( T ) . (36)4. Estimate of Renormalized Formulation for the non-cutoff case
This section is devoted to using the conservation law to bound the source terms β ′ ( f n ) Q n ( f n , f n )where the β ( · ) satisfies the Definition 2.2 while the cross section satisfies all the assumption from(9) to (16). To simplify the notations, we drop the superscript for the time being and just prove β ′ ( f ) Q ( f, f ) can be controlled bysup ≤ s ≤ T Z f ( s )(1 + | v | + | log f | )d x d v ≤ C ( T ) , ∀ t > . (37)Then we will show how to modify three lemmas to the approximate case β ′ ( f n ) Q n ( f n , f n ). As [1],we split β ′ ( f ) Q ( f, f ) into three parts β ′ ( f ) Q ( f, f ) = R + R + R , (38)where R = [ f β ′ ( f ) − β ( f )] Z R × S dv ∗ dωB ( f ′∗ − f ∗ ) , (39) R = Z R × S dv ∗ dωB [ f ′∗ β ( f ′ ) − f ∗ β ( f )] , (40) R = Z R × S dv ∗ dωBf ′∗ (cid:0) β ( f ′ ) − β ( f ) − β ′ ( f )( f ′ − f ) (cid:1) . (41)Moreover, R = [ f β ′ ( f ) − β ( f )] f ∗ v S , (42)where S ( | z | ) = C Z π/ dθ sin θ (cid:2) θ B ( 1cos θ , cos θ ) − B ( | z | , cos θ ) (cid:3) , and Z R R φ ( v ) dv = Z R × R dvdv ∗ f ( v ∗ ) β ( f ) T ( φ ) , (43)where T ( φ ) = Z S B ( v − v ∗ , ω )( φ ′ − φ ) dω = 2 π Z π B ( | v − v ∗ | , cos θ )( φ ′ − φ ) sin θdθ. From [1], we know that S ( | v | ) and T ( φ ) have very good properties even thought there exist angularsingularity and velocity singularity for B . This owes to the Symmetry-Induced Cancellation effects . Proposition 4.1.
Let B satisfies assumption from (9) to (16) , then S is local integrable and S ( | z | ) = o ( | z | ) , | z | → ∞ . NCOMING DATA 11
Proposition 4.2.
Let B satisfies assumptions from (9) to (16) , then for all φ ∈ W , ∞ ( R ) , wehave |T ( φ ) | ≤ C k φ k W , ∞ | v − v ∗ | (1 + | v − v ∗ | ) M ( | v − v ∗ | ) . Alexandre and Villani dealt with the whole space case. In this work, we consider the initial-boundary case. For R and R , ways of using conservation quantities (37), Proposition 4.1 andProposition 4.2 to control R and R in the whole space works for the bounded case. Similarly, wecan infer Lemma 4.3.
Let B satisfy assumptions from (9) to (16) , and let f satisfy (37) . Then R lies inthe function space L ∞ ([0 , T ]; L (Ω × B R ( v ))) for all R > , where B R ( v ) = { v ∈ R | | v | ≤ R } . Lemma 4.4.
Let B satisfy assumptions from (9) to (16) . Then R belongs to the function space L ∞ ([0 , T ]; L (Ω; W − , ( B R ( v )))) for all R > . The term R is different. In [1], they work on the whole space. The advantage is that theydon’t need to consider the boundary effect. In details, after multiplying (38) by some proper testfunction φ ( v ) and integrating over R × R , Z R × R v · ∇ x β ( f ) · φ ( v )d v d x = 0 . (44)But when we consider the initial-boundary problem, Z Ω × R v · ∇ x β ( f ) · ψ ( v )d x d v = 0 . (45)But on the other hand, from (31) and (33), recalling γ − f = g , we can get Z t Z Σ + (1 + | v | ) γ + f | n( x ) · v | d v d σ x d s + Z t Z Σ − (1 + | v | ) γ − f | n( x ) · v | d v d σ x d s ≤ C ( T ) . (46)Then R can be controlled as following: Lemma 4.5.
Let B satisfy assumption from (9) to (16) , and let f satisfy (37) and (46) . Then R lies in L ∞ ([0 , T ]; L (Ω × B R ( v ))) for all R > .Proof. Firstly, from (38), ∂ t β ( f ) + v · ∇ x β ( f ) ≥ R + R + R , multiplying the above equation with φ ( v ), satisfying φ ( v ) | B R = 1 , φ ( v ) ≥ , φ ( v ) = 0 , | v | ≥ R ,then integrating the resulting equation over O , Z t Z O R ( s ) φ ( v )d v d xds ≤ − Z t Z O φ R d v d xds − Z t Z O φ R d v d xds + Z O β ( f )( t ) dvds − Z O β ( f ) dvds + Z t Z O v · ∇ x ( β ( f ) φ )d v d xds := A ( t ) + A ( t ) + A ( t ) + A ( t ) + A ( t ) . Recalling that 0 < β ( f ) ≤ f, using Lemma 4.3 and Lemma 4.4, then there exists some constant dependent on R and T such that X i =1 | A i ( t ) | ≤ C ( T, R ) , t ≤ T. (47) A is more complicated. Integrating it by parts, recalling0 ≤ φ ( v ) ≤ , ≤ β ( γf ) ≤ γf, γ ± f ≥ , we have Z t v · ∇ x ( β ( f ) φ )d v d xds = − Z t Z Σ − β ( γ − f )( t, x, v ) φ ( v ) | n( x ) · v | d v d σ x d s + Z t Z Σ + β ( γ + f )( t, x, v ) φ ( v ) | n( x ) · v | d v d σ x d s ≤ Z t Z Σ − γ − f ( t, x, v ) | n( x ) · v | d v d σ x d s + Z t Z Σ + γ + f ( t, x, v ) | n( x ) · v | d v d σ x d s ≤ C ( T ) , (48)where we have used (46).Summing (47) and (48) up, recalling that R ≥ , we complete the proof of this lemma. Remark 4.6.
Noticing that the truncated cross section B n also satisfies assumption (7) and (8) and < (cid:0) n R f n dv (cid:1) ≤ , so in the similar way, we can prove that the corresponding R n , R n and R n corresponding to (28) , R n = [ f n β ′ ( f n ) − β ( f n )] Z R × S dv ∗ dωB n ( f n ′ ∗ − f ∗ ) , R n = Z R × S dv ∗ dωB n [ f n ′ ∗ β ( f n ′ ) − f n ∗ β ( f n )] , R n = Z R × S dv ∗ dωB n f n ′ ∗ (cid:0) β ( f n ′ ) − β ( f n ) − β ′ ( f n )( f n ′ − f n ) (cid:1) . also satisfy Lemma 4.3, Lemma 4.4 and Lemma 4.5. (cid:3) Weak compactness and global existence
This whole section is devoted to prove Theorem 2.3. The key tools is L weak compactnesstheorem, Dunford-Pettis Lemma and De la Vall´ee-Poussin uniform integrability criterion. One cancheck [5] for details. We need to consider the interior parts(Theorem 5.1) and boundary parts. Theinterior parts have been done by Alexandre and Villani in [1]. We just quote it.5.1. Interior domains.
From estimates (31), (33) and (36), using Dunford-Pettis Lemma, we canconclude that there exist some f such that f n ⇀ f, L ((0 , T ) × O ) . (49)Moreover, by simple calculation, the approximate cross section chosen B n in (26) also satisfiesthe following assumption. For all n , B n ( z, ω ) ≥ Φ ( | z | ) b ,n ( κ · ω ) , κ = z | z | , (50) NCOMING DATA 13 for some fixed continuous function Φ ( | z | ) such that Φ ( | z | ) > z = 0, and Z S lim inf n →∞ b ,n ( κ · ω ) dω = ∞ . (51) Theorem 5.1 (Extended Stability [1]) . Let B satisfy assumption from (9) to (16) . Let ( f n ) n ∈ N be a sequence of solutions to (28) with initial data (30) and boundary conditions in Theorem 3.1and satisfy the natural ´a priori bounds (37) and (46) . Assume without loss generality that f n ⇀ f in L p ([0 , T ] , L (Ω × R )) , ≤ p < ∞ . Then (1) f n → f in L p ([0 , T ] , L (Ω × R )) ; (2) for all functions β ∈ C ( R + , R + ) satisfying β (0) = 0 , < β ′ ( f ) < C f , β ′′ ( f ) < , there exists a defect measure ν such that the equality ∂ t β ( f ) + v · ∇ x β ( f ) = β ′ ( f ) Q ( f, f ) + ν holds in the sense of distributions. (3) for each φ ∈ D ([0 , T ] × ¯Ω × R ) , Z T Z Ω R n φ d x d v d s → Z T Z Ω R φ d x d v d s, Z T Z Ω R n φ d x d v d s → Z T Z Ω R φ d x d v d s, and moreover, if φ is a non-negative function, Z T Z Ω R φ d x d v d s ≤ lim inf n →∞ Z T Z Ω R n φ d x d v d s. Proof.
For the first two items, they directly come from [1]. Indeed, multiplying the first equationof (28) by φ ∈ D ((0 , T ) × O ), 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 x d v d t = 0 , then the left proof is similar to [1]. Here we focus on the defect measure ν .Recalling Q n ( f n , f n ) β ′ ( f n ) = R n + R n + R n , with the help of Remark 4.6, we can find that ( R n d x d v d s ) is a bounded measure-value sequenceon D ′ ((0 , T ) × O ). Then there exist some measure d m such that Z T Z O R n φ d x d v d s → Z T Z O φ d m , for any φ ∈ D ((0 , T ) × O ) . Then d ν = d m − R d x d v d s. If φ ≥
0, by Fatou Lemma, we can deduce that d ν is a positive measure.As for the third entry, noticing that Lemma 4.3 and Lemma 4.4 only require that the velocityvariable of test function has compact support. Using the strong compactness of f n , the third itemcan be verified too by the argument in [1]. (cid:3) Boundary parts.
The boundary parts are more complicated. First, Lemma 2.1 works onlyfor the equality. But the solution in Theorem 5.1 is just a inequality. So it is rarely possibleto define trace of solutions in Definition 2.2 by Lemma 2.1. In the meantime, we have defined ameaningful trace γ ± f n for the approximate solution f n . The good new is that the trace sequenceis also weakly compact in L space, namely, according to (36), (31) and (33), by Dunford-PettisLemma, we can infer: there exists some f γ ∈ L ((0 , T ) × Σ + , | n( x ) · v | d v d σ x d s ) such that γ + f n ⇀ f γ , L ((0 , T ) × Σ + , | n( x ) · v | d v d σ x d s ) , Z t Z Σ + γ + f n | n( x ) · v | d v d σ x d s → Z t Z Σ + f γ | n( x ) · v | d v d σ x d s. On the other hand, for the approximate solutions ( f n ) and its trace γ + f n , recalling that Z O f n ( t )d x d v + Z t Z Σ + γ + f n | n( x ) · v | d v d σ x d s = Z O f d x d v + Z t Z Σ − g, t > . (52)By theorem 5.1, f n ⇀ f, L ( O ) , thus we get Z O f n ( s )d x d v → Z O f ( s )d x d v, s ≤ T. Then similarly, we can deduce vγ + f n ⇀ vf γ , L ((0 , T ) × Σ + , | n( x ) · v | d v d σ x d s ) , Z t Z Σ + vγ + f n | n( x ) · v | d v d σ x d s → Z t Z Σ + vf γ | n( x ) · v | d v d σ x d s. Recalling that (1 + | v | ) f n → (1 + | v | ) f , in L ( O ), thus, we infer that Z O f d x d v + Z t Z Σ + f γ | n( x ) · v | d v d σ x d s = Z O f d x d v + Z t Z Σ − g | n( x ) · v | d v d σ x d s, t > , (53)and Z O vf d x d v + Z t Z Σ + vf γ | n( x ) · v | d v d σ x d s = Z O vf d x d v + Z t Z Σ − vg | n( x ) · v | d v d σ x d s, t > . (54)The left goal is to show that f γ satisfies (21), namely γ + f := f γ . NCOMING DATA 15
On the other hand, multiplying the first equation in (28) by a positive test function 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 x d v d t = Z T Z Σ + β ( γ + f n )( s ) φ · | n( x ) · v | d v d σ x d s − Z T Z Σ − β ( g )( s ) φ · | n( x ) · v | d v d σ x d s + Z O β ( f n )( T )( τ, · ) φ d x d v − Z O β ( f n )(0)( τ, · ) φ d x d v. (55)As n goes to infinity, there exists a uniform lower bound to the left hand of (55). Indeed, byTheorem 5.1, Z T Z O (cid:0) β ( f )( ∂ t φ + v · ∇ x φ ) + Q ( f, f ) β ′ ( f ) φ (cid:1) d x d v d t ≤ lim inf n →∞ 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 x d v d t. (56)At the same time, we can get a uniform upper bound for the right hand of (55). In details, since( f n ) is a strong convergence sequence in L ∞ ([0 , T ]; L ( O )),lim sup n →∞ (cid:18) Z T Z Σ + β ( γ + f n )( s ) φ · | n( x ) · v | d v d σ x d s − Z T Z Σ − β ( g )( s ) φ · | n( x ) · v | d v d σ x d s + Z O β ( f n )( T )( τ, · ) φ d x d v − Z O β ( f n )(0)( τ, · ) φ d x d v (cid:19) ≤ (cid:18) lim sup n →∞ Z T Z Σ + β ( γ + f n )( s ) φ · | n( x ) · v | d v d σ x d s − Z T Z Σ − β ( g )( s ) φ · | n( x ) · v | d v d σ x d s + lim sup n →∞ Z O β ( f n )( T )( τ, · ) φ d x d v − lim inf n →∞ Z O β ( f n )(0)( τ, · ) φ d x d v (cid:19) ≤ (cid:18) lim sup n →∞ Z T Z Σ + β ( γ + f n )( s ) φ · | n( x ) · v | d v d σ x d s − Z T Z Σ − β ( g )( s ) φ · | n( x ) · v | d v d σ x d s + Z O β ( f )( T )( τ, · ) φ d x d v − Z O β ( f )( τ, · ) φ d x d v (cid:19) . (57)We claim that for concave function β and non-negative test function φ , if γ + f n ⇀ f γ in L ((0 , T ) × Σ + ; | n( x ) · v | d v d σ x d s ), thenlim sup n →∞ Z T Z Σ + β ( γ + f n )( s ) φ · | n( x ) · v | d v d σ x d s ≤ Z T Z Σ + β ( f γ )( s ) φ · | n( x ) · v | d v d σ x d s. (58)Then by (55-58), we finally verify that f γ satisfies (21), namely Z T Z O (cid:0) β ( f )( ∂ t φ + v · ∇ x φ ) + Q ( f, f ) β ′ ( f ) φ (cid:1) d x d v d t ≤ Z T Z Σ + β ( γ + f )( s ) φ · | n( x ) · v | d v d σ x d s − Z T Z Σ − β ( g )( s ) φ · | n( x ) · v | d v d σ x d s + Z O β ( f )( T )( τ, · ) φ d x d v − Z O β ( f )( τ, · ) φ d x d v. Proof of (58) . Recalling that γ + f n ⇀ f γ , for each positive φ with compact support, if we set φ · | n( x ) · v | d v d σ x d s as a new measure denoted by d φ , we can infer that up to a subsequence γ + f n ⇀ f γ , in L ((0 , T ) × Σ + ; d φ ) . Secondly, noticing that β is a concave function, by the lower upper semi-continuity of concavefunction with respect to weak convergence,lim sup n Z T Z Σ + β ( γ + f n )d φ ≤ Z T Z Σ + β ( f γ )d φ . (cid:3) So we can choose γ + f := f γ . Then (53) becomes Z O f d x d v + Z t Z Σ + γ + f | n · v | d v d σ x d s = Z O f d x d v + Z t Z Σ − g | n · v | d v d σ x d s, t > . (59)As for the energy inequality, recalling that γ + f n ⇀ γ + f, in , L ((0 , T ) × Σ + ; | n · v | d v d σ x d s ) , then for any fixed m ∈ N + , on the characteristic function of ball in R with radius m , { v || v | ≤ m } denoted by m , we can infer that | v | m γ + f n ⇀ | v | m γ + f, in , L ((0 , T ) × Σ + ; | n · v | d v d σ x d s ) , and | v | m f n ⇀ | v | m f, in , L ∞ ((0 , T ); L ( O ; d v d x )) . By the lower semi-continuity of norm,sup ≤ s ≤ t Z O m | v | f ( s )d x d v + Z t Z Σ + m | v | γ + f | n( x ) · v | d v d σ x d s ≤ Z O | v | f d x d v + Z t Z Σ − | v | g, t > . Taking m to infinity, by Fatou lemma, we deducesup ≤ s ≤ t Z O | v | f ( s )d x d v + Z t Z Σ + | v | γ + f | n( x ) · v | d v d σ x d s ≤ Z O | v | f d x d v + Z t Z Σ − | v | g | n · v | d v d σ x d s, t > . (60)For the relative entropy inequality, noticing that h ( z ) is a positive convex function, by the lowersemi-continuity of convex functions, we deduce that H ( f |M ) + Z t Z Σ + h ( γ + f M ) | n( x ) · v | d v d σ x d s + Z t D ( f )( s )d x d v d s ≤ H ( f |M ) + Z t Z Σ − h ( g |M ) | n( x ) · v | d v d σ x d s, t ≥ . (61)5.3. Local conservation law.
In this subsection, we focus on the local conservation law: localconservation law and global conservation law.
NCOMING DATA 17
Local mass conservation law.
Choosing function φ ∈ D ([0 , T ] × ¯Ω), multiplying the firstequation 28 by φ , integrating by parts, then we have Z O f n (0) φ (0) dvdx − Z O f n ( T ) φ ( T ) dvdx − Z T Z O f n ( s ) ∂ t φ ( s ) dvdxds = Z T Z O f n ( s ) v · ∇ φ ( s ) dvdxds − Z T Z Σ γf n ( s ) φ ( s ) v · n dvdσ x ds As (1 + | v | ) f n → (1 + | v | ) f, in L (cid:0) (0 , T ) × Ω; dvdxds (cid:1) , and γ ± f n → γ ± f, in L (cid:0) (0 , T ) × Σ ± ; dµdxds (cid:1) , taking n → ∞ , we have Z O f ( T ) φ ( T ) dvdx − Z O f ( T ) φ ( T ) dvdx − Z T Z O f ( s ) ∂ t φ ( s ) dvdxds = Z T Z O f ( s ) v · ∇ φ ( s ) dvdxds − Z T Z Σ γf ( s ) φ ( s ) v · n dvdσ x ds. Noticing that φ is independent of v , then it can be rewritten as Z Ω φ · (cid:0) Z R f dv (cid:1) ( T ) dx − Z Ω φ · (cid:0) Z R f dv (cid:1) (0) dx − Z T Z ∂ Ω ∂ t φ ( s ) · (cid:0) Z R f dv (cid:1) ( s ) dxds = Z T Z Ω ∇ φ ( s ) · (cid:0) Z R f vdv (cid:1) ( s ) dxds − Z T Z ∂ Ω n · (cid:0) Z R γf vdv (cid:1) ( s ) φ dσ x ds. If φ ( s ) | ∂ Ω = 0 for any 0 ≤ s ≤ T , we can conclude that the following holds in the distributionsense ∂ t Z R f n ( t )d v + ∇ · Z R vf n d v = 0 . For the general case, by the Green formula, the trace of R R vf n ( t ) dv n · γ (cid:0) Z R vf n ( t ) dv (cid:1) = n · (cid:0) Z R γf v d v (cid:1) ( t )= (cid:0) Z R γf v · nd v (cid:1) ( t )= (cid:0) Z R γ + f d µ (cid:1) ( t ) − (cid:0) Z R g d µ (cid:1) ( t )5.3.2. Local momentum conservation law.
Different with the local conservation law, we need to addsome defect measure to deduce the local momentum conservation law. Similarly, multiplying thefist equation of (28) by vφ with φ ∈ D ((0 , T ) × Ω), we have 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 = 0Recalling that vf n ⇀ vf, in , L ∞ ((0 , T ); L ( O )); f n → f, a.e, 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. The only things at our disposal are (37) and f n → f, in , L ((0 , T ) × O ) . With these estimates, we can only prove that there exist distribution-value matrix M with M 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. + < M , ∇ φ > . All together, we conclude the local conservation law of momentum.
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School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China
E-mail address : [email protected] (Xu Zhang) School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China
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