THE BOLTZMANN EQUATION FOR A MULTI-SPECIES MIXTURE CLOSE TO GLOBAL EQUILIBRIUM MARC BRIANT AND ESTHER S. DAUS Abstract. We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the 3- dimensionaltorus. Theultimateaimofthisworkistoobtainexistence,uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann 6 equation in L1L∞(m), where m (1+ v k) is a polynomial weight. We prove 1 v x ∼ | | the existence of a spectral gap for the linear multi-species Boltzmann operator 0 allowing different masses, and then we establish a semigroup property thanks to 2 a new explicit coercive estimate for the Boltzmann operator. Then we develop an n L2 L∞ theory `a la Guo for the linear perturbed equation. Finally, we combine u − the latter results with a decomposition of the multi-species Boltzmann equation J in order to deal with the full equation. We emphasize that dealing with different 9 2 masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (e.g. Carleman representa- ] tion, Povzner inequality). Of important note is the fact that all methods used P and developed in this work are constructive. Moreover, they do not require any A Sobolev regularity and the L1L∞ framework is dealt with for any k > k , recov- . v x 0 h ering the optimal physical threshold of finite energy k =2 in the particular case 0 t of a multi-species hard spheres mixture with same masses. a m [ 3 v 6 Keywords: Multi-species mixture; Boltzmann equation; Spectral gap; Perturba- 2 tive theory; Convergence to equilibrium; L2 L∞ theory, Carleman representation, 3 − 0 Povzner inequality. 0 Acknowledgements: The second author wants to thank Ansgar Ju¨ngel for his . 1 valuable help. 0 6 1 Contents : v i X 1. Introduction 2 r 2. Main results 10 a (cid:0) (cid:1) 3. Spectral gap for the linear operator in L2 µ−1/2 12 v 4. L2 theory for the linear part with maxwellian weight 21 5. L∞ theory for the linear part with maxwellian weight 36 6. The full nonlinear equation in a perturbative regime 50 7. Ethical Statement 69 References 70 The first author was partly supported by the 150th Anniversary Postdoctoral Mobility Grant of the London Mathematical Society and the Division of Applied Mathematics at Brown University. ThesecondauthoracknowledgespartialsupportfromtheAustrianScienceFund(FWF),grants P24304, P27352, and W1245, and the Austrian-French Program of the Austrian Exchange Service (O¨AD). 1 2 MARC BRIANT AND ESTHER S. DAUS Introduction 1. The present work establishes existence, uniqueness, positivity and exponential trend to equilibrium for the multi-species Boltzmann equation close to equilibrium, which is used in physics and biology to model the evolution of a dilute gaseous mixture with different masses. The physically most relevant space for such a Cauchy theory is the space of density functions that only have finite mass and energy, which are the first and second moments in the velocity variable. This present article proves theresultinthespaceL1L∞(1+ v k)foranyk > k , wherek isanexplicitthreshold v x | | 0 0 depending heavily on the differences of the masses, recovering the physically optimal threshold k = 2 when all the masses of the mixture are the same and the particles 0 are approximated to be hard spheres. We are thus interested in the evolution of a dilute gas on the torus T3 composed of N different species of chemically non-reacting mono-atomic particles, which can be modeled by the following system of Boltzmann equations, stated on R+ T3 R3, × × (1.1) 1 (cid:54) i (cid:54) N, ∂ F (t,x,v)+v F (t,x,v) = Q (F)(t,x,v) t i x i i ∀ ·∇ with initial data 1 (cid:54) i (cid:54) N, (x,v) T3 R3, F (0,x,v) = F (x,v). i 0,i ∀ ∀ ∈ × Note that the distribution function of the system is given by the vector F = (F ,...,F ), with F describing the ith species at time t, position x and velocity 1 N i v. The Boltzmann operator Q(F) = (Q (F),...,Q (F)) is given for all i by 1 N N (cid:88) Q (F) = Q (F ,F ), i ij i j j=1 where Q describes interactions between particles of either the same (i = j) or of ij different (i = j) species and are local in time and space. (cid:54) (cid:90) (cid:104) (cid:105) Q (F ,F )(v) = B ( v v ,cosθ) F(cid:48)F(cid:48)∗ F F∗ dv dσ, ij i j ij | − ∗| i j − i j ∗ R3×S2 whereweusedtheshorthandsF(cid:48) = F (v(cid:48)),F = F (v),F(cid:48)∗ = F (v(cid:48))andF∗ = F (v ). i i i i j j ∗ j j ∗  1  v(cid:48) = (m v +m v +m v v σ)  mi +mj i j ∗ j| − ∗| (cid:28) v v (cid:29) ∗ , and cosθ = − ,σ . 1  v∗(cid:48) = mi +mj (miv +mjv∗ −mi|v −v∗|σ) |v −v∗| Note that these expressions imply that we deal with gases where only binary elastic collisions occur (the mass m of all molecules of species i remains the same, since i there is no reaction). Indeed, v(cid:48) and v(cid:48) are the velocities of two molecules of species ∗ i and j before collision giving post-collisional velocities v and v respectively, with ∗ conservation of momentum and kinetic energy: m v +m v = m v(cid:48) +m v(cid:48), i j ∗ i j ∗ (1.2) 1 1 1 1 m v 2 + m v 2 = m v(cid:48) 2 + m v(cid:48) 2. 2 i| | 2 j| ∗| 2 i| | 2 j| ∗| MULTI-SPECIES BOLTZMANN EQUATION 3 The collision kernels B are nonnegative, moreover they contain all the informa- ij tion about the interaction between two particles and are determined by physics. We mention at this point that one can derive this type of equations from Newtonian mechanics at least formally in the case of single species [11][12]. The rigorous va- lidity of the mono-species Boltzmann equation from Newtonian laws is known for short times (Landford’s theorem [28] or more recently [17][33]). 1.1. The perturbative regime and its motivation. Using the standard changes of variables (v,v ) (v(cid:48),v(cid:48)) and (v,v ) (v ,v) (note the lack of symmetry be- ∗ (cid:55)→ ∗ ∗ (cid:55)→ ∗ tween v(cid:48) and v(cid:48) compared to v for the second transformation due to different masses) ∗ together with the symmetries of the collision operators (see [11][12][37] among oth- ers and [14][13] and in particular [7] for multi-species specifically), we recover the following weak forms: (cid:90) (cid:90) (cid:90) Q (F ,F )(v)ψ (v)dv = B ( v v ,cos(θ))F F∗(ψ(cid:48) ψ ) dσdvdv ij i j i ij | − ∗| i j i − i ∗ R3 R6 S2 and (1.3) (cid:90) (cid:90) Q (F ,F )(v)ψ (v)dv + Q (F ,F )(v)ψ (v)dv = ij i j i ji j i j R3 R3 (cid:90) (cid:90) 1 (cid:0) (cid:1)(cid:0) (cid:1) B ( v v ,cos(θ)) F(cid:48)F∗ F F∗ ψ(cid:48) +ψ(cid:48)∗ ψ ψ∗ dσdvdv . − 2 ij | − ∗| i j − i j i j − i − j ∗ R6 S2 Thus N (cid:90) (cid:88) (1.4) Q (F ,F )(v)ψ (v)dv = 0 ij i j i R3 i,j=1 if and only if ψ(v) belongs to Span(cid:8)e ,...,e ,v m,v m,v m, v 2m(cid:9), where e 1 N 1 2 3 k | | stands for the kth unit vector in RN and m = (m ,...,m ). The fact that we need 1 N to sum over i has interesting consequences and implies a fundamental difference compared with the single-species Boltzmann equation. In particular it implies con- servation of the total number density c of each species, of the total momentum of ∞,i the gas ρ u and its total energy 3ρ θ /2: ∞ ∞ ∞ ∞ (cid:90) t (cid:62) 0, c = F (t,x,v)dxdv (1 (cid:54) i (cid:54) N) ∞,i i ∀ T3×R3 N (cid:90) 1 (cid:88) u = m vF (t,x,v)dxdv (1.5) ∞ i i ρ ∞ T3×R3 i=1 N (cid:90) 1 (cid:88) 2 θ = m v u F (t,x,v)dxdv, ∞ i ∞ i 3ρ | − | ∞ T3×R3 i=1 (cid:80)N where ρ = m c is the global density of the gas. Note that this already ∞ i=1 i ∞,i shows intricate interactions between each species and the total mixture itself. The operator Q = (Q ,...,Q ) also satisfies a multi-species version of the clas- 1 N sical H-theorem [14] which implies that any local equilibrium, i.e. any function 4 MARC BRIANT AND ESTHER S. DAUS F = (F ,...,F ) being the maximum of the Boltzmann entropy, has the form of a 1 N local Maxwellian, that is (cid:34) (cid:35) (cid:18) (cid:19)3/2 2 m v u (t,x) 1 (cid:54) i (cid:54) N, F (t,x,v) = c (t,x) i exp m | − loc | . i loc,i i ∀ 2πk θ (t,x) − 2k θ (t,x) B loc B loc Here k is the Boltzmann constant and, denoting the total local mass density by B (cid:80)N ρ = m c , we used the following local definitions loc i=1 i loc,i (cid:90) 1 (cid:54) i (cid:54) N, c (t,x) = F (t,x,v)dv, loc,i i ∀ R3 N (cid:90) N (cid:90) 1 (cid:88) 1 (cid:88) 2 u (t,x) = m vF dv, θ (t,x) = m v u F dv. loc i i loc i loc i ρ 3ρ | − | loc R3 loc R3 i=1 i=1 On the torus, this multi-species H-theorem also implies that the global equilib- rium, i.e. a stationary solution F to (1.1), associated to the initial data F (x,v) = 0 (F ,...,F ) is uniquely given by the global Maxwellian 0,1 0,N (cid:34) (cid:35) (cid:18) (cid:19)3/2 2 m v u 1 (cid:54) i (cid:54) N, F (t,x,v) = F (v) = c i exp m | − ∞| . i i ∞,i i ∀ 2πk θ − 2k θ B ∞ B ∞ By translating and rescaling the coordinate system we can always assume that u = ∞ 0 and k θ = 1 so that the only global equilibrium is the normalized Maxwellian B ∞ (1.6) µ = (µi)1(cid:54)i(cid:54)N with µi(v) = c∞,i(cid:16)m2πi(cid:17)3/2e−mi|v2|2. The aim of the present article is to construct a Cauchy theory for the multi- species Boltzmann equation (1.1) around the global equilibrium µ. In other terms we study the existence, uniqueness and exponential decay of solutions of the form F (t,x,v) = µ (v)+f (t,x,v) for all i. i i i Under this perturbative regime, the Cauchy problem amounts to solving the per- turbed multi-species Boltzmann system of equations (1.7) ∂ f +v f = L(f)+Q(f), t x ·∇ or equivalently in the non-vectorial form 1 (cid:54) i (cid:54) N, ∂ f +v f = L (f)+Q (f), t i x i i i ∀ ·∇ where f = (f ,...,f ) and the operator L = (L ,...,L ) is the linear Boltzmann 1 N 1 N operator given for all 1 (cid:54) i (cid:54) N by N (cid:88) L (f) = L (f ,f ), i ij i j j=1 with L (f ,f ) = Q (µ ,f )+Q (f ,µ ). ij i j ij i j ij i j Since we are looking for solutions F preserving individual mass, total momentum and total energy (1.5) we have the equivalent perturbed conservation laws for f = MULTI-SPECIES BOLTZMANN EQUATION 5 F µ which are given by − (cid:90) t (cid:62) 0, 0 = f (t,x,v)dxdv (1 (cid:54) i (cid:54) N) i ∀ T3×R3 N (cid:90) (cid:88) 0 = m vf (t,x,v)dxdv (1.8) i i T3×R3 i=1 N (cid:90) (cid:88) 2 0 = m v f (t,x,v)dxdv. i i | | T3×R3 i=1 1.2. Notations and assumptions on the collision kernel. First, to avoid any confusion, vectors and vector-valued operators in RN will be denoted by a bold symbol, whereas their components by the same indexed symbol. For instance, W represents the vector or vector-valued operator (W ,...,W ). 1 N We define the Euclidian scalar product in RN weighted by a vector W by N (cid:88) f,g = f g W . W i i i (cid:104) (cid:105) i=1 In the case W = 1 = (1,...,1) we may omit the index 1. Function spaces. We define the following shorthand notation (cid:113) 2 v = 1+ v . (cid:104) (cid:105) | | The convention we choose is to index the space by the name of the concerned variable, so we have for p in [1,+ ] ∞ Lp = Lp([0,T]), Lp = Lp(cid:0)R+(cid:1), Lp = Lp(cid:0)T3(cid:1), Lp = Lp(cid:0)R3(cid:1). [0,T] t x v For W = (W ,...,W ) : R3 R+ a strictly positive measurable function in v, 1 N −→ we will use the following vector-valued weighted Lebesgue spaces defined by their norms (cid:18) (cid:19)1/2 N (cid:80) 2 f = f , f = f W (v) , (cid:107) (cid:107)L2v(W) (cid:107) i(cid:107)L2v(Wi) (cid:107) i(cid:107)L2v(Wi) (cid:107) i i (cid:107)L2v i=1 (cid:18) (cid:19)1/2 (cid:80)N 2 (cid:13) (cid:13) (cid:107)f(cid:107)L2x,v(W) = i=1(cid:107)fi(cid:107)L2x,v(Wi) , (cid:107)fi(cid:107)L2x,v(Wi) = (cid:13)(cid:107)fi(cid:107)L2xWi(v)(cid:13)L2v , N (cid:80) (cid:0) (cid:1) f = f , f = sup f (x,v) W (v) , (cid:107) (cid:107)L∞x,v(W) i=1(cid:107) i(cid:107)L∞x,v(Wi) (cid:107) i(cid:107)L∞x,v(Wi) (x,v)∈T3×R3 | i | i (cid:13) (cid:13) (cid:80)N (cid:13) (cid:13) f = f , f = (cid:13)sup f (x,v) W (v)(cid:13) . (cid:107) (cid:107)L1vL∞x (W) i=1(cid:107) i(cid:107)L1vL∞x (Wi) (cid:107) i(cid:107)L1vL∞x (Wi) (cid:13)x∈T3| i | i (cid:13)L1 v NotethatL2(W)andL2 (W)areHilbertspaceswithrespecttothescalarproducts v x,v N N (cid:90) (cid:88) (cid:88) f,g = f ,g = f g W2dv, (cid:104) (cid:105)L2v(W) (cid:104) i i(cid:105)L2v(Wi) i i i R3 i=1 i=1 N N (cid:90) (cid:88) (cid:88) f,g = f ,g = f g W2dxdv. (cid:104) (cid:105)L2x,v(W) (cid:104) i i(cid:105)L2x,v(Wi) i i i T3×R3 i=1 i=1 6 MARC BRIANT AND ESTHER S. DAUS Assumptions on the collision kernel. We will use the following assumptions on the collision kernels B . ij (H1) The following symmetry holds B ( v v ,cosθ) = B ( v v ,cosθ) for 1 i,j N. ij ∗ ji ∗ | − | | − | ≤ ≤ (H2) The collision kernels decompose into the product B ( v v ,cosθ) = Φ ( v v )b (cosθ), 1 i,j N, ij ∗ ij ∗ ij | − | | − | ≤ ≤ where the functions Φ 0 are called kinetic part and b 0 angular part. ij ij ≥ ≥ This is a common assumption as it is technically more convenient and also covers a wide range of physical applications. (H3) The kinetic part has the form of hard or Maxwellian (γ = 0) potentials, i.e. Φ ( v v ) = CΦ v v γ, CΦ > 0, γ [0,1], 1 (cid:54) i,j (cid:54) N. ij | − ∗| ij| − ∗| ij ∈ ∀ (H4) For the angular part, we assume a strong form of Grad’s angular cutoff (first introduced in [19]), that is: there exist constants C , C > 0 such that for b1 b2 all 1 i,j N and θ [0,π], ≤ ≤ ∈ 0 < b (cosθ) C sinθ cosθ , b(cid:48) (cosθ) C . ij ≤ b1| || | ij ≤ b2 Furthermore, (cid:90) (cid:8) (cid:9) Cb := min inf min b (σ σ ),b (σ σ ) dσ > 0. ii 1 3 ii 2 3 3 1≤i≤Nσ1,σ2∈S2 S2 · · We emphasize here that the important cases of Maxwellian molecules (γ = 0 and b = 1) and of hard spheres (γ = b = 1) are included in our study. We shall use the standard shorthand notations (1.9) b∞ = b and l = b cos . ij (cid:107) ij(cid:107)L∞[−1,1] bij (cid:107) ◦ (cid:107)L1S2 1.3. Novelty of this article. As mentioned previously, the present work proves the existence, uniqueness, positivity and exponential trend to equilibrium for the full (cid:0) (cid:1) nonlinear multi-species Boltzmann equation (1.1) in L1L∞ v k with the explicit v x (cid:104) (cid:105) threshold k > k defined in Lemma 6.3, when the initial data F is close enough to 0 0 the global equilibrium µ. This is equivalent to solving the perturbed equation (1.7) for small f . This perturbative Cauchy theory for gaseous mixtures is completely 0 new. Moreover, one of the major contributions of the present article is to combine and adapt several very recent strategies, combined with new hypocoercivity estimates, in order to develop a new constructive approach that allows to deal with polynomial weights without requiring any spatial Sobolev regularity. This is new even in the mono-species case even though the final result we obtain has recently been proved for the mono-species hard sphere model [22]) (which we therefore also extend to more general hard and Maxwellian potential kernels.). Also, as a by-product, we prove explicitly that the linear operator L v gen- x − ·∇(cid:0) (cid:1) erates a strongly continuous semigroup with exponential decay both in L2 µ−1/2 x,v (cid:0) (cid:1) and in L∞ v βµ−1/2 ; such constructive and direct results on the torus are new to x,v (cid:104) (cid:105) our knowledge, even for the single-species Boltzmann equation. MULTI-SPECIES BOLTZMANN EQUATION 7 At last, we derive new estimates in order to deal with different masses and the multi-species cross-interaction operators, and we also extend recent mono-species estimates to more general collision kernels. Note that the asymmetry of the elastic collisions requires to derive a new description of Carleman’s representation of the Boltzmann operator as well as new Povzner-type inequalities suitable for this lack of symmetry. 1.4. State of the art and strategy. Very little is known about any rigorous Cauchy theory for multi-species gases with different masses. We want to mention [6], where a compactness result for the linear operator K := L + ν was proved in L2(µ−1/2). For multi-species gases with same masses, the recent work [13] proved v (cid:0) (cid:1) that the operator L has a spectral gap in L2 µ−1/2 and obtained an a priori expo- v (cid:0) (cid:1) nential convergence to equilibrium for the perturbed equation (1.7) in H1 µ−1/2 . x,v We emphasize here that [13] only studied the case of same masses m = m for all i, i j j. Onthecontrary, thesingle-speciesBoltzmannequationintheperturbativeregime around a global Maxwellian has been extensively studied over the past fifty years (see [35] for an exhaustive review). Starting with Grad [21], the Cauchy problem has (cid:0) (cid:1) (cid:0) (cid:1) been tackled in L2Hs µ−1/2 spaces [34], in Hs µ−1/2(1+ v )k [24][38] was then (cid:0)v x (cid:1) x,v | | extended to Hs µ−1/2 where an exponential trend to equilibrium has also been x,v obtained [31][25]. Recently, [22] proved existence and uniqueness for single-species (cid:0) (cid:1) Boltzmann equation in more the general spaces (Wα,1 Wα,q)Wβ,p (1+ v )k for α (cid:54) β and β and k large enough with explicit thrvesh∩olds.v Thexlatter pa|p|er thus (cid:0) (cid:1) includes L1L∞ v k . All the results presented above hold in the case of the torus v x (cid:104) (cid:105) for hard and Maxwellian potentials. We refer the reader interested in the Cauchy problem to the review [35]. All the works mentioned above involve to working in spaces with derivatives in the space variable x (we shall discuss some of the reasons later) with exponential weight. The recent breakthrough [22] gets rid of both the Sobolev regularity and the exponential weight but uses a new extension method which still requires to have (cid:0) (cid:1) a well-established linear theory in Hs µ−1/2 . x,v Our strategy can be decomposed into four main steps and we now describe each of them and their link to existing works. (cid:0) (cid:1) Step 1: Spectral gap for the linear operator in L2 µ−1/2 . It has been v known for long that the single-species linear Boltzmann operator L is a self-adjoint (cid:0) (cid:1) non positive linear operator in the space L2 µ−1/2 . Moreover it has a spectral gap v λ . This has been proved in [10][19][20] with non constructive methods for hard 0 potential with cutoff and in [4][5] in the Maxwellian case. These results were made constructive in [1][30] for more general collision operators. One can easily extend (cid:0) (cid:1) this spectral gap to Sobolev spaces Hs µ−1/2 (see for instance [22] Section 4.1). v Recently, [13] proved the existence of an explicit spectral gap for the operator L for multi-species mixtures where all the masses are the same (m = m ). Our i j (cid:0) (cid:1) constructive spectral gap estimate in L2 µ−1/2 closely follows their methods that v consistinprovingthatthecross-interactionsbetweendifferentspeciesdonotperturb toomuchthespectralgapthatisknowntoexistforthediagonaloperatorL (single- ii species operators). We emphasize here that not only we adapt the methods of [13] to fit the different masses framework but we also derive estimates on the collision 8 MARC BRIANT AND ESTHER S. DAUS frequencies that allow us to get rid of their strong requirement on the collision kernels: B (cid:54) βB for all i, j. The latter assumption is indeed physically irrelevant ij ii in our framework. (cid:0) (cid:1) Step 2: L2 µ−1/2 theory for the full perturbed linear operator. The x,v next step is to prove that the existence of a spectral gap for L in the sole veloc- (cid:0) (cid:1) ity variable can be transposed to L2 µ−1/2 when one adds the skew-symmetric x,v transport operator v . In other words, we prove that G = L v generates x x − ·∇ (cid:0) (cid:1) − ·∇ a strongly continuous semigroup in L2 µ−1/2 with exponential decay. x,v Onethuswantstoderiveanexponentialdecayforsolutionstothelinearperturbed Boltzmann equation ∂ f +v f = L(f). t x ·∇ A straightforward use of the spectral gap λ of L shows for such a solution L d f 2 (cid:54) 2λ f π (f) 2 , dt (cid:107) (cid:107)L2x,v(µ−1/2) − L(cid:107) − L (cid:107)L2x,v(µ−1/2) (cid:0) (cid:1) where π stands for the orthogonal projection in L2 µ−1/2 onto the kernel of the L v operator L. This inequality exhibits the hypocoercivity of L. Roughly speaking, the (cid:0) (cid:1) exponential decay in L2 µ−1/2 would follow for solutions f if the microscopic part x,v π⊥(f) = f π (f) controls the fluid part which has the following form (see Section L − L 3) (cid:20) v 2 3m−1(cid:21) 1 (cid:54) i (cid:54) N, π (f) (t,x,v) = a (t,x)+b(t,x) v +c(t,x)| | − i m µ (v), L i i i i ∀ · 2 where a (t,x),c(t,x) R and b(t,x) R3 are the coordinates of an orthogonal basis. i ∈ ∈ The standard strategies in the case of the single-species Boltzmann equation are basedonhigherSobolevregularityeitherfromhypocoercivitymethods[31]orelliptic regularity of the coefficients a, b and c [23][25]. Roughly speaking one has [23][25] (1.10) ∆π (f) ∂2π⊥f +higher order terms, L ∼ L which can be combined with elliptic estimates to control the fluid part by the mi- croscopic part in Sobolev spaces Hs. Our main contribution to avoid involving high regularity is based on an adaptation of the recent work [15] (dealing with the single-species Boltzmann equation with diffusive boundary conditions). The key idea consists in integrating against test functions that contains a weak version of the elliptic regularity of a(t,x), b(t,x) and c(t,x). Basically, the elliptic regularity of π (f) will be recovered thanks to the transport part applied to these test functions L while on the other side L encodes the control by π⊥(f). L It has to be emphasized that thanks to boundary conditions, [15] only needed the conservation of mass whereas in our case this “weak version” of estimates (1.10) strongly relies on all the conservation laws. The choice of test functions thus has to take into account the delicate interaction between each species and the total mixture we already pointed out. This leads to intricate technicalities since for each species we need to deal with different reference rates of decay m . Finally, our proof also i involves elliptic regularity in negative Sobolev spaces to deal with ∂ a, ∂ b and ∂ c. t t t (cid:0) (cid:1) Step 3: L∞ v βµ−1/2 theory for the full nonlinear equation. Thanks x,v (cid:104) (cid:105) to the first two steps we have a satisfactory L2 semigroup theory for the full linear operator. Unfortunately, as it is already the case for the single-species Boltzmann MULTI-SPECIES BOLTZMANN EQUATION 9 equation (see [11][12] or [37] for instance), the underlying L2 -norm is not an alge- x,v braic norm for the nonlinear operator Q whereas the L∞ -norm is. x,v The key idea of proving a semigroup property in L∞ is thanks to an L2 L∞ − theory “a` la Guo” [26], where the L∞-norm will be controlled by the L2-norm along the characteristics. As we shall see, each component L can be decomposed into i L = K ν where ν (f) = ν (v)f is a multiplicative operator. If we denote by i i i i i i − S (t) the semigroup generated by G = L v , we have the following implicit G x − · ∇ Duhamel representation of its ith component along the characteristics (cid:90) t S (t) = e−νi(v)t + e−νi(v)(t−s)K [S (s)] ds. G i i G 0 Following the idea of Vidav [36] and later used in [26], an iteration of the above should yield a certain compactness property. Hiding here all the cross-interactions, we end up with (cid:90) t S (t) =e−ν(v)t + e−ν(v)(t−s)Ke−ν(v)s ds G 0 (cid:90) t(cid:90) s + e−ν(v)(t−s)Ke−ν(v)(s−s1)K[S (s )] ds ds. G 1 1 0 0 We shall prove that K is compact and is a kernel operator. The first two terms will be easily estimated and the last term will be roughly of the form (cid:90) t(cid:90) s(cid:90) S (s ,x (t s)v (s s )v ,v dv dv ds ds. G 1 1 1 2 2 1 1 | − − − − | 0 0 v1,v2bounded The double integration implies that v and v are independent and we can thus 1 2 perform a change of variables which changes the integral in v into an integral over 1 T3 that we can bound thanks to the previous L2 theory. For integrability reasons, this third step actually proves that G generates a strongly continuous semigroup (cid:0) (cid:1) with exponential decay in L∞ v βµ−1/2 for β > 3/2. (cid:104) (cid:105) Our work provides two key contributions to prove the latter result. First, to prove the desired pointwise estimate for the kernel of K, we need to give a new representation of the operator in terms of the parameters (v(cid:48),v(cid:48)) instead of (v ,σ). ∗ ∗ In the single-species case, such a representation is the well-known Carleman rep- resentation [10] and requires integration onto the so-called Carleman hyperplanes v(cid:48) v,v(cid:48) v = 0. However, when particles have different masses, the lack of (cid:104) − ∗ − (cid:105) symmetry between v(cid:48) and v(cid:48) compared to v obliges us to derive new Carleman ad- ∗ missible sets (some become spheres). Second, the decay of the exponential weight differs from one species to the other. To obtain estimates that are similar to the case of single-species we exhibit the property that K mixes the exponential rate of decay among the cross-interaction between species. This enables us to close the L∞ estimate for the first two terms of the iterated Duhamel representation. Step 4: Extension to polynomial weights and L1L∞ space. To conclude v x the present study, we develop an analytic and nonlinear version of the recent work [22], also recently adapted in a nonlinear setting [8]. The main strategy is to find a decomposition of the full linear operator G into G + A. We shall prove that 1 G acts like a small perturbation of the operator G = v ν(v) and is thus 1 ν x − ·∇ − hypodissipative, and that A has a regularizing effect. The regularizing property of 10 MARC BRIANT AND ESTHER S. DAUS the operator A allows us to decompose the perturbative equation (1.7) into a system of differential equations ∂ f = G (f )+Q(f +f ,f +f ) t 1 1 1 1 2 1 2 ∂ f +v f = L(f )+A(f ) t 2 x 2 2 1 ·∇ The first equation is solved in L∞ (m) or L1L∞(m) with the initial data f thanks x,v v x 0 to the hypodissipativity of G . The regularity of A(f ) allows us to use Step 3 and 1 1 (cid:0) (cid:1) thus solve the second equation with null initial data in L∞ v βµ−1/2) . First, the x,v (cid:104) (cid:105) existence of a solution to the system having exponential decay is obtained thanks to an iterative scheme combined with new estimates on the multi-species operators G and A. Then uniqueness follows a new stability estimate in an equivalent norm 1 (proposed in [22]), that fits the dissipativity of the semigroup generated by G. Finally, positivity of the unique solution comes from a different iterative scheme. In the case of the single-species Boltzmann equation, the less regular weight m(v) one can achieve with this method is determined by the hypodissipative property of G and gives m = v k with k > 2, which is indeed obtained also in the multi-species 1 (cid:104) (cid:105) framework of same masses. In the general case of different masses, the threshold k 0 is more intricate (see Theorem 2.2), since it also depends on the different masses m . i 1.5. Organisation of the paper. The paper follows exactly the four steps de- scribed above. Section 2 gives a precise statement of the main theorems that will be proved in this work and the rest of the article is dedicated to the proof of these theorems. (cid:0) (cid:1) Section3dealswiththespectralgapofL. ThesemigrouppropertyinL2 µ−1/2 x,v (cid:0) (cid:1) istreatedinSection4. ThispropertyisthenpassedontoL∞ v βµ−1/2 inSection x,v (cid:104) (cid:105) 5. At last, we work out the Cauchy problem for the full nonlinear equation in Section 6. Main results 2. As explained in the introduction, the ultimate goal of this article is a full per- turbative Cauchy theory for the multi-species Boltzmann equation (1.1). Along the way, we shall also prove the following important results about the linear perturbed operator L v . x − ·∇ Theorem 2.1. Let the collision kernels B satisfy assumptions (H1) (H4). Then ij − the following holds. (cid:0) (cid:1) (i) The operator L is a closed self-adjoint operator in L2 µ−1/2 and there exists v λ > 0 such that L f L2(cid:0)µ−1/2(cid:1), f,L(f) (cid:54) λ f π (f) 2 ; ∀ ∈ v (cid:104) (cid:105)L2v(µ−1/2) − L(cid:107) − L (cid:107)L2v((cid:104)v(cid:105)γ/2µ−1/2) (cid:0) (cid:1) (cid:0) (cid:1) (ii) Let E = L2 µ−1/2 or E = L∞ v βµ−1/2 with β > 3/2. The linear x,v x,v (cid:104) (cid:105) perturbed operator G = L v generates a strongly continuous semigroup x − ·∇ S (t) on E and there exist C , λ > 0 such that G E E t (cid:62) 0, S (t)(Id Π ) (cid:54) C e−λEt, ∀ (cid:107) G − G (cid:107)E E