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Boltzmann Equation in Bounded Domains Yan Guo Division of Applied Mathematics, Brown University

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Boltzmann Equation in Bounded Domains Yan Guo Division of Applied Mathematics, Brown University Boltzmann Equation in Bounded Domains Yan Guo Division of Applied Mathematics, Brown University 1. Introduction Dynamics of gases governed by the Boltzmann equation (1872): ∂tF + v ·∇xF = Q(F, F)(1) where F (t, x, v) ≥ 0 is the distribution function for the gas particles at time t ≥ 0, position x ∈ Ω, and v ∈ R3. The collision operator: (hard-sphere) Q(F1,F2)= |v − u|F1(u )F2(v )| cos θ|dωdu R3 S2 − |v − u|F1(u)F2(v)| cos θ|dωdu R3 S2 ≡ Qgain(F1,F2) − Qloss(F1,F2), (2) where u = u +(v − u) · ω, v = v − (v − u) · ω, cos θ = (u − v) · ω/|u − v|. 1 The Collision Invariants: Q(F, F)dv = vQ(F, F)dv = |v|2Q(F, F)dv =0, R3 R3 R3 which formally implies that mass, momentum and energy conservation: F (t, x, v)dvdx = F (0,x,v)dvdx, R3 R3 vF(t, x, v)dvdx = vF(0,x,v)dvdx, R3 R3 |v|2F (t, x, v)dvdx = |v|2F (0,x,v)dvdx. R3 R3 The H-Theorem: Q(F,F)lnFdv ≤ 0, R3 which implies the entropy inequality (time irreversible:) d 1 F ln dvdx ≥ 0 dt R3 F and solutions formally tend to a Maxwellian e−|v|2/2. 2 Particles to Fluids: Newtonian Particle System → Boltzmann → Fluids (Eu- ler, Navier-Stokes). It should be pointed out that at the particle level, the system is time reversible. The first limit (the Boltzmann- Grad limit) is quite interesting even intuitively (Landford 70’s). The second limit forms a foundation of derivation of many important fluid equations (Hilbert Program). 1.1 Mathematical Theories: There have been explosive literature in mathematical study for the Boltzmann equation. Homogeneous Solutions Carleman 1930’s Solutions Near a Maxwellian. Grad’s study of lin- earized Boltzmann operator in 1950’s. Ukai’s global solu- tions (1974), by linear decay estimate for the linear Boltz- mann equation. Solutions Near Vacuum. Illner and Shinbrot (1984), the dispersion of R3 F (t, x, v)dv for the free transport equation ∂tF + v ·∇xF = 0 in the whole space. Renormalized Solutions. DiPerna and Lions’ global renormalized (some type of the weak) solutions with large amplitude 1989. 3 Averaging Lemma: (Golse, Lions, Perthame, Sentis 1988) If F ∈ L2(R × R3 × R3) satisfy ∂ F + v ·∇ F = G ∈ L2(R × R3 × R3) t x 1 ∈ 2 × 3 then R3 F (t, x, v)χ(v)dv H (R R ). One of the most exciting accumulative results further along this direction: Diffusive limit of Boltzmann to the incompressible Navier-Stokes equations (Golse and Saint- Raymond 2004): renormalized solutions convergence to Leray-Hopf solutions of the Navier-Stokes equations. Open problems: Uniqueness and regularity of such renor- malized solutions. 1.2 Nonlinear Energy Method: There have been renewed interest recently to study so- lutions near Maxwellian with a new energy method. Landau (Fokker-Planck) Equation (1936): I v ⊗ v ∇ · − [F (v)∇ F (v) − F (v)∇ F (v)]dv v | | | |3 v v R3 v v for Coulomb interaction in charges particles (plasma). Renormalized solutions with ‘defect measure’ can be con- structed. Linear decay for the Ukai approach is difficult. 4 Guo (2002) global classical solutions near Maxwellians. Vlasov-Maxwell-Boltzmann System: Charged par- ticles interact with self-consistent electromagnetic field. Renormalized solutions can be constructed in the absence of the magnetic effect. Linear decay for the Ukai approach is hard to treat the unbounded nonlinearity. Guo (2003) global classical solutions near Maxwelians. Positivity and Stability of Shock Profiles. Liu and Yu (2004). √ In terms of F = μ + μf, the Boltzmann equation can be rewritten as {∂t + v ·∇+ L} f =Γ(f,f) where the standard linear Boltzmann operator is given by 1 √ √ Lf ≡ νf − Kf = −√ {Q(μ, μf)+Q( μf, μ)}, μ (3) and 1 √ √ Γ(f ,f )=√ Q( μf , μf ) ≡ Γ (f ,f )−Γ (f ,f ). 1 2 μ 1 2 gain 1 2 loss 1 2 (4) The collision invariant and H−Theorem implies that L ≥ 0, and 5 √ √ √ ker L = {a(t, x) μ, b(t, x) · v μ, c(t, x)|v|2 μ}. Here for fixed (t, x), the standard projection P onto the hydrodynamic (macroscopic) part is given by 2 Pf = {af (t, x)+bf (t, x) · v + cf (t, x)|v| } μ(v). The energy identity f ×{∂t + v ·∇+ L} f = f × Γ(f,f) leads to 1 d ||f||2 + Lf, f = fΓ(f,f). 2dt If Lf, f ≥δ||f||2, global small solutions can be con- structed via a continuity argument. But we only know Lf, f ≥δ||{I − P}f||2, It suffices to estimate Pf in terms of {I − P}f (the microscopic part) for small solution f to the nonlinear Boltzmann equation (not true at the operator level)! In- 6 deed, we can derive the macroscopic equations as ∇xc = g1 ∂tc + ∇x · b = g2 j i ∂xib + ∂xj b = g3 ∂tb + ∇xa = g4 ∂ta = g5 where gl ∂{I − P}f+high order. We deduce that Δb ∂2{I − P}f. We hence have ellipticity for b. 2. Boundary Value Problem One basic problem in the Boltzmann study: Unique- ness, Time-decay toward e−|v|2/2, with physical bound- ary conditions in a general domain Ω. There are four basic types of boundary conditions at the boundary ∂Ω: (1) In-flow injection: in which the incoming particles are prescribed; (2) Bounce-back reflection: in which the particles bounce back at the reverse the velocity; (3) Specular reflection: in which the particle bounce back specularly; 7 (4) Diffuse reflection (stochastic): in which the incoming particles are a probability average of the outgoing par- ticles. Previous Work: In 1977, Asano and Shizuta announced that Boltzmann solutions near a Maxwellian would decay exponentially to it in a smooth bounded convex domain with specular reflection boundary conditions. Desvillettes and Villani (2005) establish almost expo- nential decay rate with large amplitude, for general col- lision kernels and general boundary conditions, provided certain a-priori strong Sobolev estimates can be verified. Difficulties: The validity of these a-priori estimates is completely open even local in time, in a bounded domain. As a matter of fact, such kind of strong Sobolev estimates may not be expected for a general non-convex domain. Characteristic Boundary: Because of the differential op- 3 erator v·∇x, the phase boundary ∂Ω×R is always char- acteristic but not uniformly characteristic at the grazing set γ0 = {(x, v):x ∈ ∂Ω, and v · n(x)=0} where n(x) being the outward normal at x. Hence it is very challenging and delicate to obtain regularity from 8 general theory of hyperbolic PDE. The complication of the geometry makes it difficult to employ spatial Fourier transforms in x. Bouncing Characteristics: particles interacting with the boundary repeatedly, analysis of such generalized trajec- tories. Strategy: develop an unified L2−L∞ theory in the near Maxwellian regime, to establish exponential decay toward 2 −|v| a normalized Maxwellian μ = e 2 , for all four basic types of the boundary conditions and rather general do- mains. Consequently, uniqueness among these solutions can be obtained. For convex domains, these solutions are shown to be continuous away from the singular grazing set γ0. 2.1 Domain and Characteristics Let Ω = {x : ξ(x) < 0} is connected, and bounded with ξ(x) smooth. The outward normal vector at ∂Ωis given by ∇ξ(x) n(x)= . (5) |∇ξ(x)| We say Ω is real analytic if ξ is real analytic in x. We 9 define Ω is strictly convex if there exists cξ > 0 such that i j 2 ∂ijξ(x)ζ ζ ≥ cξ|ζ| (6) for all x such that ξ(x) ≤ 0, and all ζ ∈ R3. We say that Ω has a rotational symmetry, if there are vectors x0 and , such that for all x ∈ ∂Ω {(x − x0) × }·n(x) ≡ 0. (7) We denote the phase boundary in the space Ω × R3 as γ = ∂Ω × R3 : 3 γ+ = {(x, v) ∈ ∂Ω × R : n(x) · v>0}, 3 γ− = {(x, v) ∈ ∂Ω × R : n(x) · v<0}, 3 γ0 = {(x, v) ∈ ∂Ω × R : n(x) · v =0}. Given (t, x, v), let [X(s),V(s)] = [X(s; t, x, v),V(s; t, x, v)] =[x+(s−t)v, v] be the trajectory (or the characteristics) for the Boltzmann equation (1): dX(s) dV (s) = V (s), =0. (8) ds ds with the initial condition: [X(t; t, x, v),V(t; t, x, v)] = [x, v]. For any (x, v) such that x ∈ Ω¯,v =0 , we de- fine its backward exit time tb(x, v) ≥ 0tobethe 10 the last moment at which the back-time straight line [X(s;0,x,v),V(s;0,x,v)] remains in Ω:¯ tb(x, v)=sup{τ ≥ 0:x − τv ∈ Ω¯}. (9) We also define xb(x, v)=x(tb)=x − tbv ∈ ∂Ω. (10) 2.2 Boundary Condition and Conservation Laws In√ terms of the standard perturbation f such that F = μ + μf, we formulate the boundary conditions as (1) The in-flow boundary condition: for (x, v) ∈ γ− f|γ− = g(t, x, v) (11) (2) The bounce-back boundary condition: for x ∈ ∂Ω, f(t, x, v)|γ− = f(t, x, −v) (12) (3) Specular reflection: for x ∈ ∂Ω, let R(x)v = v − 2(n(x) · v)n(x), (13) and f(t, x, v)|γ− = f(x, v, R(x)v) (14) 11 (4) Diffusive reflection: assume the natural normalization, cμ μ(v)|n(x) · v|dv =1, (15) v·n(x)>0 then for (x, v) ∈ γ−, f(t, x, v)|γ− = cμ μ(v) f(t, x, v ) μ(v ) v·n(x)>0 {nx · v }dv .
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