Discrete velocity Boltzmann eqations in the plane:stationary solutions for a generic class
aa r X i v : . [ m a t h - ph ] J a n Discrete velocity Boltzmann equations in the plane: stationarysolutions for a generic class
Abstract
The paper proves existence of renormalized stationary solutions for a dense class of discrete velocityBoltzmann equations in the plane with given ingoing boundary values. The proof is based onthe construction of a sequence of approximations with L compactness for the integrated collisionfrequency and gain term. Compactness is obtained using the Kolmogorov-Riesz theorem, whichreplaces the L compactness of velocity averages in the continuous velocity case, not available whenthe velocities are discrete.. The Boltzmann equation is the fundamental mathematical model in the kinetic theory of gases.Replacing its continuum of velocities with a discrete set of velocities is a simplification preservingthe essential features of free flow and quadratic collision term. Besides this fundamental aspectthey can approximate the Boltzmann equation with any given accuracy [9], and are thereby usefulfor approximations and numerics. In the quantum realm they can also be more directly connectedto microscopic quasi/particle models.A discrete velocity model of a kinetic gas in the plane, is a system of partial differential equationshaving the form, ∂f i ∂t ( t, z ) + v i · ∇ z f i ( t, z ) = Q i ( f )( t, z ) , t > , z ∈ Ω , ≤ i ≤ p, where f i ( t, z ), 1 ≤ i ≤ p , are phase space densities at time t , position z and velocity v i . The spatialdomain is Ω ⊂ R . The given discrete velocities are v i ∈ R , 1 ≤ i ≤ p . For f = ( f i ) ≤ i ≤ p , thecollision operator Q = ( Q i ) ≤ i ≤ p with gain part Q + , loss part Q − , and collision frequency ν , is Key words; stationary Boltzmann equation, discrete coplanar velocities, normal model, entropy. Q i ( f ) = p X j,l,m =1 Γ lmij ( f l f m − f i f j )= Q + i ( f ) − Q − i ( f ) ,Q + i ( f ) = p X j,l,m =1 Γ lmij f l f m , Q − i ( f ) = f i ν i ( f ) , ν i ( f ) = p X j,l,m =1 Γ lmij f j , i = 1 , ..., p. The collision coefficients satisfyΓ lmij = Γ lmji = Γ ijlm ≥ . (1.1)If a collision coefficient Γ lmij is non-zero, then the conservation laws for momentum and energy, v i + v j = v l + v m , | v i | + | v j | = | v l | + | v m | , (1.2)are satisfied. The discrete velocity model (DVM) is called normal (see [10]) if any solution of theequationsΨ( v i ) + Ψ( v j ) = Ψ( v l ) + Ψ( v m ) , where the indices ( i, j ; l, m ) take all possible values satisfying Γ lmij >
0, is given byΨ( v ) = a + b · v + c | v | , for some constants a, c ∈ R and b ∈ R . The above description of a discrete velocity model alsoholds for spatial and velocity dimension bigger than two. For any discrete velocity model, conditions(1.2) imply that the velocities can be split into κ groups of pairs ( v i , v j ) belonging to the samecircle of center v κ , κ ∈ { , · · · , κ } . A single velocity v i , 1 ≤ i ≤ p can belong to several circles.This paper studies stationary solutions to normal coplanar discrete velocity models, in a strictlyconvex bounded open subset Ω ⊂ R , with C boundary ∂ Ω and given boundary inflow. We considerthe generic situation whereno pair of velocities v i , v j , ≤ i = j ≤ p, are parallel , (1.3)every velocity v i belongs to a single circle of center v κ for some κ ∈ { , · · · , κ } . (1.4) Remark 1.1
This set of discrete velocity models is dense in the set of discrete velocity models in ( R ) p . Indeed,if a discrete velocity model with velocities ( v i ) ≤ i ≤ p is such that some v i belongs to two circles inthe pairs ( v i , v j ) , ( v i , v j ′ ) , then after a rotation of the two pairs around their respective circle centeran angle ǫ > small enough, the new discrete velocity model satisfies (1.3) - (1.4) . Denote by n ( Z ) the inward normal to Z ∈ ∂ Ω. Denote the v i -ingoing (resp. v i -outgoing) part ofthe boundary by ∂ Ω + i = { Z ∈ ∂ Ω; v i · n ( Z ) > } , (resp. ∂ Ω − i = { Z ∈ ∂ Ω; v i · n ( Z ) < } ) . s + i ( z ) = inf { s > z − sv i ∈ ∂ Ω + i } , s − i ( z ) = inf { s > z + sv i ∈ ∂ Ω − i } , z ∈ Ω . Write z + i ( z ) = z − s + i ( z ) v i (resp. z − i ( z ) = z + s − i ( z ) v i ) (1.5)for the ingoing (resp. outgoing) point on ∂ Ω of the characteristics through z in direction v i . Thestationary boundary value problem v i · ∇ f i ( z ) = Q i ( f )( z ) , z ∈ Ω , (1.6) f i ( z ) = f bi ( z ) , z ∈ ∂ Ω + i , ≤ i ≤ p, (1.7)is considered in L in one of the following equivalent forms ([12]);the exponential multiplier form, f i ( z ) = f bi ( z + i ( z )) e − R s + i ( z )0 ν i ( f )( z + i ( z )+ sv i ) ds + Z s + i ( z )0 Q + i ( f )( z + i ( z ) + sv i ) e − R s + i ( z ) s ν i ( f )( z + i ( z )+ rv i ) dr ds, a.a. z ∈ Ω , ≤ i ≤ p, (1.8)the mild form, f i ( z ) = f bi ( z + i ( z )) + Z s + i ( z )0 Q i ( f )( z + i ( z ) + sv i ) ds, a.a. z ∈ Ω , ≤ i ≤ p, (1.9)the renormalized form, v i · ∇ ln(1 + f i )( z ) = Q i ( f )1 + f i ( z ) , z ∈ Ω , f i ( z ) = f bi ( z ) for z ∈ ∂ Ω + i , ≤ i ≤ p, (1.10)in the sense of distributions.Denote by L (Ω) the set of non-negative integrable functions on Ω. For a distribution function f = ( f i ) ≤ i ≤ p , define its entropy (resp. entropy dissipation) by p X i =1 Z Ω f i ln f i ( z ) dz, (cid:16) resp. p X i,j,l,m =1 Γ lmij Z Ω ( f l f m − f i f j ) ln f l f m f i f j ( z ) dz (cid:17) . The main result of the present paper is
Theorem 1.1
Consider a coplanar collision operator satisfying (1.3) - (1.4) , and a non-negative ingoing boundaryvalue f b with mass and entropy inflows bounded, Z ∂ Ω + i v i · n ( z ) f bi (1 + ln f bi )( z ) dσ ( z ) < + ∞ , ≤ i ≤ p. There exists a stationary renormalized solution in (cid:0) L (Ω) (cid:1) p to the boundary value problem (1.6) - (1.7) with finite mass and entropy-dissipation. i ∈ { , · · · , p } , if Γ lmij = 0 for all j , l and m , then f i equals its ingoing boundary value, andthe rest of the system can be solved separately. Such i are not present in the following discussion.Most mathematical results for stationary discrete velocity models of the Boltzmann equation havebeen obtained in one space dimension. An overview is given in [13]. Half-space problems [4] andweak shock waves [5] for discrete velocity models have also been studied. A discussion of normaldiscrete velocity models, i.e. conserving nothing but mass, momentum and energy, can be foundin [7]. In two dimensions, special classes of solutions to the Broadwell model are given in [8], [6],and [14]. The Broadwell model is a four-velocity model, with v + v = v + v = 0 and v , v orthogonal. [8] contains a detailed study of the stationary Broadwell equation in a rectangle withcomparison to a Carleman-like system, and a discussion of (in)compressibility aspects. A mainresult in [11] is the existence of continuous solutions to the two-dimensional stationary Broadwellmodel with continuous boundary data for a rectangle. The paper [2] solves that problem in an L -setting. The proof uses in an essential way the constancy of the sums f + f and f + f alongcharacteristics, which no longer holds in the present paper. For every normal model, there is apriori control of entropy dissipation, mass and entropy fluxes (’ fluxes ’ or ’ flows ’ as later on inthe same paragraph?) through the boundary. The main difficulty is to prove that for a sequenceof approximations, the limit of the gain term equals the gain term of the limit. For the continuousvelocity Boltzmann equation [12], this follows from the averaging lemma, not available for thediscrete velocity Boltzmann model. Also the argument used in [3] in the continuous velocity casefor obtaining control of entropy, hence weak L compactness of a sequence of approximations fromthe control of entropy dissipation, does not work in a discrete velocity case because the number ofvelocities is finite. It was replaced in [1] by a control of the entropy from the control of the entropyflow through the boundary, under the supplementary assumption of all velocities pointing into thesame half-space. In the present paper, we get rid of this restrictive assumption and do not haverecourse to any entropy control. Instead we use the ’one-circle per velocity’ assumption (1.4) toget weak L -compactness of the approximations. As stressed in Remark 1.1, assumption (1.4) isgeneric in the set of coplanar discrete velocity models. The strong L compactness of a sequenceof approximations is then obtained by a compactness property for the collision frequency and gainparts in the exponential form, again using the ’one-circle per velocity’ assumption (1.4).The proof starts in Section 2 from bounded approximations with damping and convolution added,written in exponential multiplier form, and solved by a fixed point argument. The damping andconvolutions are removed in Section 3 by taking limits using L -compactness of the integratedcollision frequency and gain term. The compactness is proven by the Kolmogorov-Riesz theorem(see [15], [16]). The limit of the remaining approximations is obtained in Section 4 by using againthe Kolmogorov-Riesz theorem. For the convenience of the reader, proofs of Lemmas that werealready proven in [1] are outlined in the Appendix. Denote by N ∗ = N \ { } and by a ∧ b the minimum of two real numbers a and b . Let µ α be a smoothmollifier in R with support in the ball centered at the origin of radius α . Outside the boundarythe function to be convolved with µ α is continued in the normal direction by its boundary value.Let ˜ µ k be a smooth mollifier on ∂ Ω with α = k . Denote by f kbi = (cid:16) f bi ( · ) ∧ k (cid:17) ∗ ˜ µ k , ≤ i ≤ p, k ∈ N ∗ . Lemma 2.1
For any α > and k ∈ N ∗ there is a solution F α,k ∈ ( L (Ω)) p to αF α,ki + v i · ∇ F α,ki = p X j,l,m =1 Γ lmij (cid:16) F α,kl F α,kl k F α,km ∗ µ α F α,km ∗ µ α k − F α,ki F α,ki k F α,kj ∗ µ α F α,kj ∗ µ α k (cid:17) , (2.1) F α,ki ( z + i ( z )) = f kbi ( z + i ( z )) , ≤ i ≤ p. (2.2)Proof of Lemma 2.1.The main lines of the proof are given in the Appendix. Let k > F α,k , obtained in Section 2, is bounded by a multiple of k . Therefore ( F α,k ) α ∈ ]0 , is weakly compact in ( L (Ω)) p . Denote by F k the limit for the weaktopology in ( L (Ω)) p of a converging subsequence when α →
0. Let us prove that for a subsequence,the convergence is strong in ( L (Ω)) p . Lemma 3.1
There is a sequence ( α q ) q ∈ N tending to zero when q → + ∞ , such that ( F α q ,k ) q ∈ N strongly converges to F k in ( L (Ω)) p when q → + ∞ . Proof of Lemma 3.1The proof is outlined in the Appendix.Denote by Q + ki = p X j,l,m =1 Γ lmij F kl F kl k F km F km k , ν ki = (cid:16) p X j,l,m =1 Γ lmij (cid:17) F kj (1 + F ki k )(1 + F kj k ) , ≤ i ≤ p. (3.1) Lemma 3.2 F k is a non-negative continuous solution to v i · ∇ F ki = Q + ki − F ki ν ki , (3.2) F ki ( z + i ( z )) = f kbi ( z + i ( z )) , ≤ i ≤ p. (3.3) Solutions ( F k ) k ∈ N ∗ to (3.2) - (3.3) have mass and entropy dissipation bounded from above uniformlywith respect to k . Moreover their outgoing flows at the boundary are controlled as follows, p X i =1 Z ∂ Ω − i ,F ki ≤ k | v i · n ( Z ) | F ki ln F ki ( Z ) dσ ( Z ) + ln k Z ∂ Ω − i ,F ki ≥ k | v i · n ( Z ) | F ki dσ ( Z ) ≤ c b . (3.4)Proof of Lemma 3.2The proof is outlined in the Appendix. Proposition 3.1
The sequence ( F k ) k ∈ N ∗ is weakly compact in L . j ∈ { , · · · , p } , the sequence (( F kj ) /∂ Ω − j ) k ∈ N ∗ is weakly compactin L | v j · n | ( ∂ Ω − j ). Moreover, ( F kj ) k ∈ N ∗ is uniformly bounded in L (Ω) and F kj ( z ) ≤ F kj ( z + s − j ( z ) v j ) exp (cid:16) Z s − j ( z ) − s + j ( z ) ν kj ( z + sv j ) ds (cid:17) , z ∈ Ω . Consequently, a sufficient condition for proving the weak compactness in L (Ω) of ( F kj ) k ∈ N ∗ , is theuniform boundedness on ∂ Ω + j of the sequence (cid:0) R s − j ( Z )0 F ki ( Z + sv j ) ds (cid:1) k ∈ N ∗ . A crucial remark is that Q ki = Q kj , so that Z s − j ( Z )0 F ki ( Z + sv j ) ds = Z s − j ( Z )0 f kbi ( z + i ( Z + sv j )) ds + Z s − j ( Z )0 Z − s + i ( Z + sv j ) Q ki ( Z + sv j + rv i ) drds = Z s − j ( Z )0 f kbi ( z + i ( Z + sv j )) ds + Z s − j ( Z )0 Z − s + i ( Z + sv j ) Q kj ( Z + sv j + rv i ) drds = Z s − j ( Z )0 f kbi ( z + i ( Z + sv j )) ds + Z s − j ( Z )0 Z − s + i ( Z + sv j ) v j · ∇ F kj ( Z + sv j + rv i ) drds = Z s − j ( Z )0 f kbi ( z + i ( Z + sv j )) ds + Z ω j ( Z ) v j · ∇ F kj ( z ) dσ ( z ) , Z ∈ ∂ Ω + j , (3.5)where ω j ( Z ) ⊂ Ω is defined as follows.Either (
Z, z − j ( Z )) ∈ ∂ Ω + i × ∂ Ω + i and ω j ( Z ) is the subset of Ω with boundary the union of [ Z, z − j ( Z )]and the boundary arc (cid:0) Z, z − j ( Z ) (cid:1) ∩ ∂ Ω + i .Or Z ∈ ∂ Ω + i and z − j ( Z ) / ∈ ∂ Ω + i and ω j ( Z ) is the subset of Ω with boundary the union of [ Z, z − j ( Z )],[ z − j ( Z ) , z + i ( z − j ( Z ))] and the boundary arc (cid:0) Z, z + i ( z − j ( Z )) (cid:1) ∩ ∂ Ω + i .Or Z / ∈ ∂ Ω + i and z − j ( Z ) ∈ ∂ Ω + i and ω j ( Z ) is the subset of Ω with boundary the union of [ Z, z − j ( Z )],[ Z, z + i ( Z )] and the boundary arc (cid:0) z − j ( Z ) , z + i ( Z ) (cid:1) ∩ ∂ Ω + i .Or Z / ∈ ∂ Ω + i and z − j ( Z ) / ∈ ∂ Ω + i and ω j ( Z ) is the subset of Ω with boundary the union of [ Z, z − j ( Z )],[ z − j ( Z ) , z + i ( z − j ( Z ))], the boundary arc (cid:0) z + i ( z − j ( Z )) , z + i ( Z ) (cid:1) ∩ ∂ Ω + i and [ z + i ( Z ) , Z ].And so, denoting by m the outward normal to ∂ω j ( Z ), Z s − j ( Z )0 F ki ( Z + sv j ) ds ≤ c Z ∂ Ω + i v i · n ( z ) f bi ( z ) dσ ( z ) + Z ∂ω j ( Z ) v j · m ( z ) F kj ( z ) dσ ( z ) , Z ∈ ∂ Ω + j . (3.6)Notice that by definition of the j - characteristics [ Z, z − j ( Z )], v j · m ( z ) = 0 for z ∈ [ Z, z − j ( Z )], sothat Z [ Z,z − j ( Z )] v j · m ( z ) F kj ( z ) dσ ( z ) = 0 . (3.7)6n the first case where ω j ( Z ) is the subset of Ω with boundary the union of [ Z, z − j ( Z )] and theboundary arc (cid:0) Z, z − j ( Z ) (cid:1) ∩ ∂ Ω + i , (3.7) and the boundedness of the mass flows through the boundary,imply that Z(cid:0)
Z,z − j ( Z ) (cid:1) ∩ ∂ Ω + i v j · m F kj ( z ) dσ ( z ) ≤ c b . (3.8)In the second case, the supplementary term R [ z − j ( Z ) ,z + i ( z − j ( Z ))] v j · m ( z ) F kj ( z ) dσ ( z ) is bounded fromabove by some c b . Indeed, Z [ z − j ( Z ) ,z + i ( z − j ( Z ))] v j · m ( z ) F kj ( z ) dσ ( z )= − Z ( A j ,B j ) v j · m ( z ) F kj ( z ) dσ ( z ) + Z ¯ ω ( Z ) v j · ∇ F kj ( z ) dz = − Z ( A j ,B j ) v j · m ( z ) F kj ( z ) dσ ( z ) + Z ¯ ω ( Z ) v i · ∇ F ki ( z ) dz, where ( A j , B j ) is the boundary arc ( z − j ( Z ) , z + i ( z − j ( Z ))) included in (cid:0) ∂ω j ( Z ) (cid:1) c , and ¯ ω ( Z ) is thesubdomain of Ω with the union [ z − j ( Z ) , z + i ( z − j ( Z ))] ∪ ( A j , B j ) as boundary. It follows from v i · m = 0on [ z − j ( Z ) , z + i ( z − j ( Z ))] that Z [ z − j ( Z ) ,z + i ( z − j ( Z ))] v j · m ( z ) F kj ( z ) dσ ( z )= − Z ( A j ,B j ) v j · n ( z ) F kj ( z ) dσ ( z ) + Z ( A j ,B j ) v i · n ( z ) F ki ( z ) dσ ( z ) ≤ c b . The third and fourth cases can be controlled analogously.
This section contains the proof of Theorem 1.1. The main part is a proof of strong L compactnessof ( F k ) k ∈ N ∗ , based on two compactness lemmas for integrated collision frequency and gain term.Recall the exponential multiplier form for the approximations ( F k ) k ∈ N ∗ , F ki ( z ) = f kbi ( z + i ( z )) e − R s + i ( z )0 ν ki ( z + i ( z )+ sv i ) ds + Z s + i ( z )0 Q + ki ( z + i ( z ) + sv i ) e − R s + i ( z ) s ν ki ( F k )( z + i ( z )+ rv i ) dr ds, a.a. z ∈ Ω , ≤ i ≤ p, (4.1)where ν ki and Q + ki are defined in (3.1). An i -characteristics is a segment of points [ Z − s + i ( Z ) v i , Z ],where Z ∈ ∂ Ω − i . By the strict convexity of Ω, there are for every i ∈ { , · · · p } two points of ∂ Ω,denoted by ˜ Z i and ¯ Z i such that z + i ( ˜ Z i ) = z − i ( ˜ Z i ) and z + i ( ¯ Z i ) = z − i ( ¯ Z i ) . i,j,l,m Γ lmij . Lemma 4.1
For k ∈ N ∗ , i ∈ { , ..., p } and ǫ > , there is a subset Ω kiǫ of i -characteristics of Ω with measuresmaller than c b ǫ , such that for any z ∈ Ω \ Ω kiǫ , F ki ( z ) ≤ ǫ exp (cid:0) Γ ǫ (cid:1) , Z s − i ( z ) − s + i ( z ) ν ki ( z + sv i ) ds ≤ Γ ǫ . (4.2)Proof of Lemma ?? .All along the proof, c b denotes a constant that may vary from line to line but is independent of k and ǫ . It follows from the exponential form of F ki that F ki ( z ) ≤ F ki ( z + s − i ( z ) v i ) e R s − i ( z ) − s + i ( z ) ν ki ( z + rv i ) dr ≤ F ki ( z + s − i ( z ) v i ) e Γ R s − i ( z ) − s + i ( z ) F kj ( z + rv i ) dr , z ∈ Ω . (4.3)The boundedness of the mass flow of ( F ki ) k ∈ N ∗ across ∂ Ω − i is Z ∂ Ω − i | v i · n ( Z ) | F ki ( Z ) dσ ( Z ) ≤ c b , k ∈ N ∗ . (4.4)By the strict convexity of Ω, there are for every i ∈ { , · · · p } two points of ∂ Ω, denoted by ˜ Z i and¯ Z i such that v i · n ( ˜ Z i ) = v i · n ( ¯ Z i ) = 0 . Let ˜ l i (resp. ¯ l i ) be the largest boundary arc included in ∂ Ω − i with one end point ˜ Z i (resp. ¯ Z i ) suchthat − ǫ ≤ v i · n ( Z ) ≤ , Z ∈ ˜ l i ∪ ¯ l i . (4.5)It follows from (4.4)-(4.5) that the measure of the set { Z ∈ ∂ Ω − i ∩ ˜ l ci ∩ ¯ l ci ; F ki ( Z ) > ǫ } is smaller than c b ǫ . The boundedness of the mass of ( F kj ) k ∈ N ∗ can be written Z Ω F kj ( z ) dz = Z ∂ Ω − i | v i · n ( Z ) | (cid:16) Z − s + i ( Z ) F kj ( Z + rv i ) dr (cid:17) dσ ( Z ) ≤ c b . Hence the measure of the set { Z ∈ ∂ Ω − i ∩ ˜ l ci ∩ ¯ l ci ; Z − s + i ( Z ) F kj ( Z + rv i ) dr > ǫ } , is smaller than c b ǫ . Consequently, the measure of the set of Z ∈ ∂ Ω − i outside of which F ki ( Z ) ≤ ǫ and Z − s − i ( Z ) F kj ( Z + rv i ) dr ≤ ǫ ,
8s bounded by c b ǫ . Together with (4.3), this implies that the measure of the complement of the setof Z ∈ ∂ Ω − i , such that F ki ( z ) ≤ ǫ exp (cid:0) Γ ǫ (cid:1) and Z s − i ( z ) − s + i ( z ) ν ki ( z + rv i ) dr ≤ Γ ǫ for z = Z − sv i , ≤ s ≤ s + i ( Z ), is bounded by c b ǫ . With it c b ǫ is a bound for the measure of thecomplement, denoted by Ω kiǫ , of the set of i -characteristics in Ω such that for all points z on the i -characteristics, F ki ( z ) ≤ Γ ǫ exp (cid:0) ǫ (cid:1) and Z s − i ( z ) − s + i ( z ) ν ki ( z + rv i ) dr ≤ Γ ǫ . This ends the proof of the lemma.Given i ∈ { , ..., p } and ǫ as in Lemma 4.1, let χ kiǫ denote the characteristic function of the comple-ment of Ω kiǫ . The following lemma proves the compactness in L (Ω) of the k -sequence of integratedcollision frequencies. Lemma 4.2
The sequences (cid:16) Z s + i ( z )0 ν ki ( z + i ( z ) + sv i ) ds (cid:17) k ∈ N ∗ , ≤ i ≤ p, are strongly compact in L (Ω) . Proof of Lemma 4.2.Let 1 ≤ i ≤ p . The uniform bound for the mass of ( F k ) proven in Lemma 3.2, implies that (cid:16) Z Ω Z s + i ( z )0 ν ki ( z + i ( z ) + sv i ) dsdz (cid:17) k ∈ N ∗ is uniformly bounded with respect to k . By the Kolmogorov-Riesz theorem ([15], [16]), the compact-ness of (cid:16) R s + i ( z )0 ν ki ( z + i ( z ) + sv i ) ds (cid:17) k ∈ N ∗ will follow from its translational equi-continuity in L (Ω).For j ∈ { , · · · , p } such that Γ lmij = 0 for some ( l, m ) ∈ { , · · · , p } , the sequence (cid:16) Z s + i ( z )0 F kj ( z + i ( z ) + sv i ) ds (cid:17) k ∈ N ∗ (4.6)is translationally equi-continuous in the v i -direction. Indeed, s + i ( z + hv i ) = s + i ( z ) + h and z + i ( z + hv i ) = z + i ( z ) , z ∈ Ω , so that, denoting by I (0 , h ) the interval with endpoints 0 and h and using the uniform bound onthe mass of ( F kj ) k ∈ N ∗ , Z | Z s + i ( z + hv i )0 F kj ( z + i ( z + hv i ) + sv i ) ds − Z s + i ( z )0 F kj ( z + i ( z ) + sv i ) ds | dz = Z Z s ∈ I (0 ,h ) F kj ( z + sv i ) dsdz ≤ c | h | . v j -direction.Integrating F kj ( z + i ( z + hv j ) + sv i ) (resp. F kj ( z + i ( z ) + sv i )) along its v j -characteristics, it holds that | Z s + i ( z + hv j )0 F kj ( z + i ( z + hv j ) + sv i ) ds − Z s + i ( z )0 F kj ( z + i ( z ) + sv i ) ds |≤| A kij ( z, h ) | + | B kij ( z, h ) | , (4.7)where A kij ( z, h ) = Z s + i ( z + hv j )0 f kbj (cid:16) z + j (cid:0) z + i ( z + hv j ) + sv i (cid:1)(cid:17) ds − Z s + i ( z )0 f kbj (cid:0) z + j ( z + i ( z ) + sv i ) (cid:1) ds, (4.8)and B kij ( z, h ) = Z s + i ( z + hv j )0 Z − s + j ( z + i ( z + hv j )+ sv i ) Q kj ( z + i ( z + hv j ) + sv i + rv j ) drds − Z s + i ( z )0 Z − s + j ( z + i ( z )+ sv i ) Q kj ( z + i ( z ) + sv i + rv j ) drds. First, A kij ( z, h ) = Z ( z + i ( z ) ,z + i ( z + hv j )) f kbj ( Z ) dZ, so thatlim h → Z Ω | A kij ( z, h ) | dz = 0 , (4.9)uniformly with respect to k . Moreover, taking into account that Q j = Q i = v i · ∇ F i , B kij ( z, h ) = Z s + i ( z + hv j )0 Z − s + j ( z + i ( z + hv j )+ sv i ) v i · ∇ F ki ( z + i ( z + hv j ) + sv i + rv j ) drds − Z s + i ( z )0 Z − s + j ( z + i ( z )+ sv i ) v i · ∇ F ki ( z + i ( z ) + sv i + rv j ) drds. (4.10)Notice that the domain of integration of v i · ∇ F ki ( z + i ( z ) + sv i + rv j ) in the previous integral isthe subset of Ω with boundary [ z + j ( z ) , z ], [ z, z + i ( z )] and one of the boundary arcs ( z + j ( z ) , z + i ( z )).Consequently, B kij ( z, h ) is the integral of v i · ∇ F ki on the subset of Ω with boundary[ z + i ( z ) , z ] ∪ [ z, z + hv j ] ∪ [ z + hv j , z + i ( z + hv j )] ∪ the boundary arc ( z + i ( z + hv j ) , z + i ( z )) . Denote by n the outward normal to this domain. It follows from the Green formula together withthe properties v i · n ( z ) = 0 , z ∈ [ z + i ( z ) , z ] ∪ [ z + hv j , z + i ( z + hv j )] ,v j · n ( z ) = 0 , z ∈ [ z, z + hv j ] , and v i · ∇ F ki = v j · ∇ F kj , that | B kij ( z, h ) | ≤ Z ( z + i ( z + hv j ) ,z + i ( z )) | v i · n ( Z ) | F ki ( Z ) dσ ( Z ) . h → Z Ω | B kij ( z, h ) | dz = 0 , uniformly with respect to k. (4.11) Lemma 4.3
Up to a subsequence ( F k ) k ∈ N ∗ strongly converges in L (Ω) to a non-negative renormalized solutionwith finite mass and entropy-dissipation of (1.6) - (1.7) . Proof of Lemma 4.3.Let i ∈ { , · · · , p } and ǫ > l, m ) such that Γ lmij = 0, thesequence (cid:16) χ kiǫ ( z ) Z s + i ( z )0 F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) e − R s + i ( z ) s ν ki ( z + i ( z )+ rv i ) dr ds (cid:17) k ∈ N ∗ , (4.12)is strongly compact in L (Ω). Split the domain of integration in ( z, s ) ∈ Ω × [0 , s + i ( z )] into the setswhere F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) > J F ki F kj (1 + F ki k )(1 + F kj k ) ( z + i ( z ) + sv i ) , (cid:0) resp. F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) < J F ki F kj (1 + F ki k )(1 + F kj k ) ( z + i ( z ) + sv i )and F kj ( z + i ( z ) + sv i ) > J (cid:1) , where the integrals are arbitrarily small for J (resp. J ) large enough, and the remaining domain, X := { ( z, s ) ∈ Ω × [0 , s + i ( z )]; F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) < J F ki F kj (1 + F ki k )(1 + F kj k ) ( z + i ( z ) + sv i )and F kj ( z + i ( z ) + sv i ) < J } , where (cid:16) χ kiǫ ( z ) F kl F km (1+ F klk )(1+
F kmk ) ( z + i ( z ) + sv i ) (cid:17) k ∈ N ∗ is uniformly bounded with respect to k . Let us provethe L uniform equi-continuity on this domain of (cid:16) χ kiǫ ( z ) Z s + i ( z )0 F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) ds (cid:17) k ∈ N ∗ . We can also restrict to a domain where both F kl ( z + i ( z ) + sv i ) and F km ( z + i ( z ) + sv i ) are bounded,since Z ( z,s ) ∈ X ; s ∈ [0 ,s + i ( z )] ,F kl ( z + i ( z )+ sv i ) ≥ Λ χ kiǫ ( z ) F kl F km (1 + F kl k )(1 + F km k ) ( z + i ( z ) + sv i ) dsdz ≤ J J e ǫ ǫ | { ( z, s ) ∈ X ; s ∈ [0 , s + i ( z )] , F kl ( z + i ( z ) + sv i ) ≥ Λ } | , F kl > Λ tends to zero when Λ → + ∞ . And so, we have reducedthe problem to proving the L uniform equi-continuity in the v l direction of (cid:16) Z s + i ( z )0 χ F kl ( z + i ( z )+ sv i ) < Λ F kl F kl k ( z + i ( z ) + sv i ) ds (cid:17) k ∈ N ∗ . This is done by performing the change of variables s → t in Z s + i ( z + hv l )0 χ F kl ( z + i ( z + hv l )+ sv i ) < Λ F kl F kl k ( z + i ( z + hv l ) + sv i ) ds, where t is the unique real number such that z + i ( z + hv l ) + tv i − (cid:0) z + i ( z ) + sv i (cid:1) is parallel to v l . This gives a difference of F kl between two end points belonging to a same l -characteristics, which equals an integral of the Q l ( F k ) collision term. It can be estimated as earlier,giving an L estimate of order | h | uniformly with respect to k ∈ N ∗ . Equicontinuity and L compactness follow.It follows from the previous L compactness of (4.12) and Lemma 4.2 that the sequence ( χ kiǫ F ki ) k ∈ N ∗ is compact in L (Ω). For a converging subsequence of ( χ kiǫ F k ) k ∈ N ∗ , the limit depends on ǫ . Choosea decreasing sequence ( ǫ q ) with lim q →∞ ǫ q = 0, and a diagonal subsequence in k with χ kiǫ q converging in k for all q , and increasing with q . By Lemma 3.1, ( F k ) weakly converges in L up to a subsequencestill denoted by ( F k ), to some F ∈ L (Ω). Split F k − F into χ kiǫ q ( F ki − F i ) + (1 − χ kiǫ q ) F ki − (1 − χ kiǫ q ) F i , ≤ i ≤ p. Using that R Ω kiǫq F ki and R Ω kiǫq F i are arbitrarily small for ǫ q small enough, leads to the strong L convergence of ( F k ) to F , up to a subsequence.Let us prove that F is a renormalized solution of (1.6)-(1.7). Start from a renormalized formulationfor χ kiǫ F ki , − Z ∂ Ω − i ϕ i χ kiǫ ln (cid:0) F ki (cid:1) ( Z ) v i · n ( Z ) dσ ( Z ) − Z ∂ Ω + i ϕ i χ kiǫ ln (cid:0) f kbi (cid:1) ( Z ) v i · n ( Z ) dσ ( Z ) − Z Ω χ kiǫ ln (cid:0) F ki (cid:1) v i · ∇ ϕ i ( z ) dz = Z Ω ϕ i χ kiǫ F ki p X j,l,m =1 Γ lmij (cid:16) F kl F kl k F km F km k − F ki F ki k F kj F kj k (cid:17) dz, (4.13)for test functions ϕ ∈ ( C (Ω)) p . Use the strong L convergence of the sequence ( F k ) k ∈ N ∗ to F topass to the limit in the left hand side of (4.13) when k → + ∞ . This gives in the limit for the lefthand side − Z ∂ Ω − i ϕ i ln (cid:0) F i (cid:1) ( Z ) v i · n ( Z ) dσ ( Z ) − Z ∂ Ω + i ϕ i ln (cid:0) f bi (cid:1) ( Z ) v i · n ( Z ) dσ ( Z ) − Z Ω ln (cid:0) F i (cid:1) v i ∇ ϕ i ( z ) dz. k → + ∞ in the right hand side of (4.13), given η > A η of Ω with | A cη | < η , such that up to a subsequence, ( F k ) uniformly converges to F on A η and F ∈ L ∞ ( A η ). Passing to the limit when k → + ∞ on A η is straightforward. Moreover,lim η → Z A cη ϕ i F i Q − i ( F )( z ) dz = 0 and lim η → Z A cη ϕ i χ kiǫ F ki ν ki ( z ) dz = 0 , uniformly with respect to k , since F i F i ≤ , F ki (1 + F ki )(1 + F ki k )(1 + F kj k ) ≤ , and lim η → Z A cη F kj = 0 , uniformly with respect to k . The passage to the limit in the loss term follows.The passage to the limit in the gain term can be done as follows. The uniform boundedness of theentropy production term of ( F k ) proven in Lemma 3.2, implies that for any γ > Z A cη | ϕ i | χ kiǫ F ki p X j,l,m =1 Γ lmij F kl F kl k F km F km k ( z ) dz ≤ c ln γ + cγ Z A cη F ki ν ki ( z ) dz. Take first γ large, then η small. It follows that the right hand side of (4.13) converges to Z Ω ϕ i Q + i ( F )1 + F i ( z ) dz − Z Ω ϕ i Q − i ( F )1 + F i ( z ) dz, when k → + ∞ . Consequently, F satisfies (1.6)-(1.7) in renormalized form. Remark 4.2
The methods of this paper can also be used to expand the present existence result from generic to gen-eral normal families of collisions and to prove existence of renormalized solutions in an evolutionaryframe. That is work in progress.
This section outlines proofs of results from [1] that are used in the present paper.Proof of Lemma 2.1.Take α > c α = 1 α p X i =1 Z ∂ Ω + i ( n ( z ) · v i ) f bi ( z ) dσ ( z ) , K α = { f ∈ (cid:0) L (Ω) (cid:1) p ; p X i =1 Z Ω f i ( z ) dz ≤ c α } . (5.1)Let T be the map defined on K α by T ( f ) = F , where F = ( F i ) ≤ i ≤ p is the solution of αF i + v i · ∇ F i = p X j,l,m =1 Γ lmij (cid:16) F l F l k f m ∗ µ α f m ∗ µ α k − F i F i k f j ∗ µ α f j ∗ µ α k (cid:17) , (5.2) F i ( z + i ( z )) = f kbi ( z + i ( z )) . (5.3)13 = T ( f ) can be obtained as the limit in ( L (Ω)) p of the monotone sequence ( F q ) q ∈ N defined by F = 0 and αF q +1 i + v i · ∇ F q +1 i = p X j,l,m =1 Γ lmij (cid:16) F ql F ql k f m ∗ µ α f m ∗ µ α k − F q +1 i F qi k f j ∗ µ α f j ∗ µ α k (cid:17) , (5.4) F q +1 i ( z + i ( z )) = f kbi ( z + i ( z )) , q ∈ N . (5.5)The solution of (5.2)-(5.3) is unique in the set of non-negative functions. Indeed, let G = ( G i ) ≤ i ≤ p be a non-negative solution of (5.2)-(5.3). It follows by induction that ∀ q ∈ N , F qi ≤ G i , ≤ i ≤ p. (5.6)Consequently, F i ≤ G i , ≤ i ≤ p. (5.7)Moreover, subtracting the partial differential equations satisfied by G i from the partial differentialequations satisfied by F i , 1 ≤ i ≤ p , and integrating the resulting equation on Ω, it results α p X i =1 Z Ω ( G i − F i )( z ) dz + p X i =1 Z ∂ Ω − i | n ( z ) · v i | ( G i − F i )( z ) dσ ( z ) = 0 . (5.8)It results from (5.7)-(5.8) that G = F .The map T is continuous and compact in the L -norm topology (cf [1]). Hence by the Schauderfixed point theorem, there is a fixed point for T , i.e. a solution F α,k to (2.1)-(2.2).Proof of Lemma 3.1Consider the approximation scheme ( f α,ρ ) ρ ∈ N of F α,k , f α, i = 0 , (5.9) αf α,ρ +1 i + v i · ∇ f α,ρ +1 i = p X j,l,m =1 Γ lmij (cid:16) F α,kl F α,kl k F α,km ∗ µ α F α,km ∗ µ α k − f α,ρ +1 i f α,ρ +1 i k f α,ρj ∗ µ α f α,ρj ∗ µ α k (cid:17) , (5.10) f α,ρ +1 i ( z + i ( z )) = f kbi ( z + i ( z )) , ≤ i ≤ p, ρ ∈ N . (5.11) f α, is obviously given in terms of F α,k . It follows from the exponential form that F α,ki ≤ f α, i , α ∈ ]0 , . Denote by S the map from R p × R p mapping ( X, Z ) into W = S ( X, Z ) ∈ R p solution to αW i + v i · ∇ W i = p X j,l,m =1 Γ lmij (cid:16) F α,kl F α,kl k F α,km ∗ µ α F α,km ∗ µ α k − W i X i k Z j ∗ µ α Z j ∗ µ α k (cid:17) ,W i ( z + i ( z )) = f kbi ( z + i ( z )) , ≤ i ≤ p. Denote by f α, , = S (0 , f α, ) , f α, ,r = S ( f α, ,r − , f α, ) ,F α,k, = S (0 , F α,k ) , F α,k,r = S ( F α,k,r − , F α,k ) , r ∈ N ∗ . f α, , i ≤ F α,k, i . Then the sequence ( f α, ,ri ) r ∈ N (resp. ( F α,k,ri ) r ∈ N ) is increasing with limit f α, i (resp. F α,ki ). Itfollows from f α, ,ri ≤ F α,k,ri , r ∈ N , that f α, i ≤ F α,ki , ≤ i ≤ p. (5.12)Let f α, , := S (0 , f α, ) , f α, ,r := S ( f α,r − , f α, ) , r ∈ N ∗ . It follows from (5.12) that f α, , i ≥ F α,k, i , ≤ i ≤ p. The sequence ( f α, ,ri ) r ∈ N is also increasing with limit f α, i and with f α, ,ri ≥ F α,k,ri . Hence f α, i ≥ F α,ki . From here by induction on ρ , it holds that f α, ρi ≤ f α, ρ +2 i ≤ F α,ki ≤ f α, ρ +3 i ≤ f α, ρ +1 i , α ∈ ]0 , , ρ ∈ N . (5.13)By induction on r , for each r the sequence ( f α, ,r ) α ∈ ]0 , is translationally equicontinuous in α . Thelimit sequence ( f α, ) α ∈ ]0 , is also translationally equicontinuous. This is so, since given ǫ > r and then h can be taken so that Z ( f α, − f α, ,r )( z ) dz < ǫ and Z | f α, ,r ( z + h ) − f α, ,r ( z ) | dz < ǫ, | h | < h . It can analogously be proven that for each ρ ∈ N , ( f α,ρ ) α ∈ ]0 , is translationally equicontinuous in α .Let ( α q ) q ∈ N be a sequence tending to zero. Take a subsequence in ( α q ) q ∈ N , still denoted by ( α q ) q ∈ N ,such that ( f α q , ) q ∈ N converges in L to some f , when q → + ∞ .Continuing by induction gives a sequence ( f ,ρ ) ρ ∈ N satisfying f , ρi ≤ f , ρ +2 i ≤ F ki ≤ f , ρ +3 i ≤ f , ρ +1 i , ρ ∈ N , (5.14) v i · ∇ f ,ρ +1 i = G i − p X j,l,m =1 Γ lmij f ,ρ +1 i f ,ρ +1 i k f ,ρj f ρj k ,f ,ρ +1 i ( z + i ( z )) = f kbi ( z + i ( z )) . Here, G ki is the weak L limit when α → p X j,l,m =1 Γ lmij F α,kl F α,kl k F α,km ∗ µ α F α,km ∗ µ α k .
15n particular, ( f , ρi ) ρ ∈ N (resp. ( f , ρ +1 i ) ρ ∈ N ) non decreasingly (resp. non increasingly) converges in L to some g i (resp. h i ) when ρ → + ∞ . The limits satisfy0 ≤ g i ≤ F ki ≤ h i ,v i · ∇ h i = G i − p X j,l,m =1 Γ lmij h i h i k g j g j k , (5.15) v i · ∇ g i = G i − p X j,l,m =1 Γ lmij g i g i k h j h j k , (5.16)( h i − g i )( z + i ( z )) = 0 . Integrating and summing gives that p X i =1 Z ∂ Ω − i | v i · n ( Z ) | ( h i − g i )( Z ) dσ ( Z ) = 0 , i.e. that g i = h i also on ∂ Ω − i . Integrating the equation satisfied by h i − g i over the part of Ω onone side of a line orthogonal to n , summing over i and using (1.4) implies that g = h on thatline, hence in all of Ω, and is equal to F ki . ( F α q ,k ) q ∈ N converges to F k in ( L (Ω)) p when q → + ∞ .Indeed, given η >
0, choose ρ big enough so that k f , ρ +1 i − f , ρ i k L < η and k f , ρ i − F ki k L < η, ≤ i ≤ p, then q big enough, so that k f α q , ρ +1 i − f , ρ +1 i k L ≤ η and k f α q , ρ i − f , ρ i k L ≤ η, q ≥ q . Then split k F α q ,ki − F ki k L as follows, k F α q ,ki − F ki k L ≤k F α q ,ki − f α, ρ i k L + k f α, ρ i − f , ρ i k L + k f , ρ i − F ki k L ≤k f α, ρ +1 i − f α, ρ i k L +2 η by (5.13) ≤k f α, ρ +1 i − f , ρ +1 i k L + k f , ρ +1 i − f , ρ i k L + k f , ρ i − f α, ρ i k L +2 η ≤ η, q ≥ q . Proof of Lemma 3.2Passing to the limit when q → + ∞ in (2.1)-(2.2) written for F α q ,k , implies that F k is a solutionin ( L (Ω)) p to (3.2)-(3.3). It remains to prove its continuity. Using twice its exponential form andthe continuity of f kb , this comes back to prove the continuity of Z s + i ( z )0 Z s + j ( z + i ( z )+ sv i )0 G k ( z + j ( z + i ( z ) + sv i ) + σv j ) dσds, i = j, (5.17)for given measurable bounded functions G k . The mapping( s, σ ) ∈ [0 , s + i ( z )] × [0 , s + j ( z + i ( z ) + sv i )] → Z = z + j ( z + i ( z ) + sv i ) + σv j , (5.18)16s a change of variables. Indeed, the strict convexity of Ω and the C regularity of ∂ Ω imply that z → z + i ( z ) is well-defined and C for any i ∈ { , · · · , p } . Hence the map ( s, σ ) → Z is one to one and C . Its Jacobian equals one since Z = Z i v i + Z j v j , with Z i = s − s + i ( z ) linear in s and independentof σ , and Z j = σ − s + j (cid:0) z + ( s − s + i ( z )) v i (cid:1) linear in σ . Using this change of variable leads to thecontinuity of the map defined in (5.17).In order to prove that ( F k ) k ∈ N ∗ is uniformly bounded in L (Ω), choose an orthonormal basis ( e x , e y )of R so that neither the x -direction nor the y -direction is parallel to any of v , ..., v p . Observe thatintegrating (3.2)-(3.3) over Ω and summing over i , shows that outflow of mass equals inflow. Weshall first obtain uniformly in k , an upper bound for the energy p X i =1 v i Z Ω F ki ( z ) dz. Recalling that the genericity condition (1.3) implies that all velocities are different from zero, theenergy bound implies an upper estimate for the mass. Write v i = ξ i e x + ζ i e y . Multiply the equationfor F ki with ξ i and integrate over Ω a = Ω ∩ { ( x, y ); x ≤ a } . Set S a = Ω ∩ { ( x, y ); x = a } and ∂ Ω a = ∂ Ω ∩ ¯Ω a . From (3.2)-(3.3) follows p X i =1 ξ i Z S a F ki ( a, y ) dy = p X i =1 ξ i Z ∂ Ω a ( v i · n ( Z )) F ki ( Z ) dσ ( Z ) . (5.19)For any ( x, y ) ∈ Ω let the line-segment through ( x, y ) in the x -direction (resp. y -direction) intersectthe boundary ∂ Ω at x − ( y ) < x + ( y ) (resp. y − ( x ) < y + ( x )). Denote by x − := min ( x,y ) ∈ Ω { x − ( y ) } , x +0 := max ( x,y ) ∈ Ω { x + ( y ) } . (5.20)Integrating (5.19) on a ∈ [ x − , x +0 ] gives uniformly in k , p X i =1 ξ i Z Ω F ki ( z ) dz = p X i =1 ξ i Z x +0 x − (cid:16) Z ∂ Ω a ( v i · n ( Z )) F ki ( Z ) dσ ( Z ) (cid:17) da ≤ c b , where c b only depends on the given inflow. Analogously P pi =1 ζ i R Ω F ki ( z ) dz ≤ c b . The boundednessof energy and with it mass, follows. The entropy dissipation estimate is proved as follows. Denoteby D k the entropy production term for the approximation F k , D k = X ijlm Γ lmij Z Ω ( F ki F ki k F kj F kj k − F kl F kl k F km F km k ) ln F ki F kj (1 + F kl k )(1 + F km k )(1 + F ki k )(1 + F kj k ) F kl F km ( z ) dz. Multiply (3.2) by ln F ki F kik , add the equations in i , and integrate the resulting equation on Ω. Itleads to p X i =1 Z ∂ Ω − i | v i · n ( Z ) | (cid:16) F ki ln F ki − k (1 + F ki k ) ln(1 + F ki k ) (cid:17) ( Z ) dσ ( Z ) + Z Ω D k ( z ) dz ≤ c b . k Z ∂ Ω − i | v i · n ( Z ) | ln(1 + F ki k )( Z ) dσ ( Z ) ≤ Z ∂ Ω − i | v i · n ( Z ) | F ki ( Z ) dσ ( Z ) ≤ c b . Hence p X i =1 Z ∂ Ω − i | v i · n ( Z ) | F ki ln F ki F ki k ( Z ) dσ ( Z ) + Z Ω D k ( z ) dz ≤ c b . (5.21)The uniform entropy dissipation bound holds, since x → x ln xk x is bounded from above on ]0 , + ∞ [.The control of the outgoing flows of ( F k ) k ∈ N ∗ through the boundary is now performed. It holdsthat Z ∂ Ω − i | v i · n ( Z ) | F ki ln(1 + F ki k )( Z ) dσ ( Z ) ≤ Z ∂ Ω − i ,F ki ≤ k | v i · n ( Z ) | F ki ln(1 + F ki k )( Z ) dσ ( Z )+ Z ∂ Ω − i ,F ki ≥ k | v i · n ( Z ) | F ki ln(1 + F ki k )( Z ) dσ ( Z ) ≤ ln 2 Z ∂ Ω − i | v i · n ( Z ) | F ki ( Z ) dσ ( Z ) + Z ∂ Ω − i ,F ki ≥ k | v i · n ( Z ) | F ki ln 2 F ki k ( Z ) dσ ( Z ) ≤ c b + Z ∂ Ω − i ,F ki ≥ k | v i · n ( Z ) | F ki ln F ki ( Z ) dσ ( Z ) − ln k Z ∂ Ω − i ,F ki ≥ k | v i · n ( Z ) | F ki ( Z ) dσ ( Z ) . Together with (5.21), this implies (3.4).
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