Scattering structure of Vlasov equations around inhomogeneous Boltzmannian states

Transcription

1 B. Després (LJLL/UPMC and IUF Scattering structure of Vlasov s around inhomogeneous Boltzmannian states B. Després (LJLL/UPMC and IUF Collisionless Boltzmann (Vlasov Equation 207 p. / 2

2 Goal Model problem : Vlasov-Poisson-Ampère d-v (x, v I R tf + v xf ε 2 E v f = 0, t > 0, (x, v I R, xe = ρ ref (x fdv, E = xϕ, t > 0, x I, R te = vfdv, t > 0, x I := [0, ]. R The a priori small scaling parameter ε is for mathematical convenience. Understand/prove linear Landau damping E(t 0 around t + inhomogeneous Boltzmannian states in the linear regime f (x, v, t = f 0(x, v + g(x, v, t, f 0(x, v = n 0(xe v 2 /2, E(x, t = E 0(x + F (x, t, E 0(x = ϕ 0(x, ϕ 0 (x + exp( v 2 /2 + ε 2 ϕ 0(xdv = ρ ref (x. R }{{} = 2π exp(ε 2 ϕ 0 (x - Morrison, Hamiltonian Description of Vlasov Dynamics : Action-Angle Variables..., Lemou, Mehats and Raphael, A new variational approach to the stability of gravitational systems, Barré, Olivetti and Yamaguchi, Landau damping and inhomogeneous reference states, 205 ( Campa, Chavanis : A dynamical stability criterion for inhomogeneous quasi-stationary states, 200. Collisionless Boltzmann (Vlasov Equation 207 p. 2 / 2

3 Set-up : simple electric potential Electric potential Sep : v 2 2 ε2 ϕ 0(x = ε 2 ϕ 0. v 0 zone + zone c zone - Trapped particles { ϕ 0 (x > 0 for 0 < x < x 0, ϕ 0(x < 0 for x 0 < x <, x tg + v xg ε 2 E 0 v g ε 2 F v f 0 = 0, tf = R vgdv, xf = R gdv, Energy identity with G(v = exp( v 2 /2 ( d g 2 dt ε 2 n dvdx + F 2 dx = 0. 0G I R I - Kruzkal-Obermann, On the stability of plasma in static equilibrium, Antonov, Remarks on the problem of stability in stellar dynamics,96. - Barré, Olivetti and Yamaguchi, Landau damping and inhomogeneous reference states, 205. Collisionless Boltzmann (Vlasov Equation 207 p. 3 / 2

4 Scattering framework Set M(x, v = n 0(xG(v, u(x, v, t = g(x,v,t U = ( u F X := L 2 (I R L 2 0(I One has tu(t = ih ε U(t ( ih ε v x + ε 2 E 0 v εvm = ε vm 0 v εm(x,v and (Hilbert space. (unbounded operator. Decomposition of operators ih ε = ih0 ε + εik ( ( ih0 ε v x + ε 2 E 0 v 0 = and εik = εvm ε v vm 0 Gauss law and averaging lemmas (Golse, ik is a compact operator. - Diperna-Lions, Global weak solutions of Vlasov-Maxwell systems, Golse-Lions-Perthame-Sentis, Regularity of the moments of the solution of a transport, Reed-Simon, Methods of modern mathematical physics : Scattering theory, Yafaev, Scattering Theory : Some Old and New Problems, 2000 Collisionless Boltzmann (Vlasov Equation 207 p. 4 / 2

5 Moments To simplify, use Hermite functions ψ n(v = (2π 4 n! 2 Hn (v G 2 (v u(t, x, v = u n(t, xψ n(v, u n = uψ ndv. n R The unknown is now U(t, = (F (t,, u 0(t,, u (t,, u 2(t,,... t. The unperturbed operator becomes ih0 ε = A xu + ε 2 E 0(xBU with A = A t = The perturbation is iku = DU with D = α n 0 (x α n 0 (x , = Dt. - Després, Symmetrization of Vlasov-Poisson s, 204. Collisionless Boltzmann (Vlasov Equation 207 p. 5 / 2

6 Eigenstructure of A v and B v Set U µ = 0 ψ 0(µ ψ (µ ψ 2(µ... X. With n=0 ψn(λψn(µ = δ(λ µ, one understands Uµ is a Dirac mass written with moments. Since vψ n(v = n + ψ n+(v + nψ n (v, one has AU µ = µu µ Similarly BU µ = µu µ. Collisionless Boltzmann (Vlasov Equation 207 p. 6 / 2

7 Eigenstructure of ih ε 0 Generalized eigenvectors X. 0 v zone + zone c zone - x Zone + : e label of characteristics µ ε e (x = 2(e + ε 2 ϕ 0(x > 0, te ε = 0 µε e (s ds, λ ε e,k = 2iπk ir te ε Ue,k,+(x ε = exp x0 µ (2iπk ε te ε e (s ds U µε e (x te ε µ ε e (x H (I N, ( Ue,k,+ ε e 2 = x0 ( µ αt e exp 2iπk ε e (s ds exp e+ε2 ϕ 0 (x. 2 t ε e Then ih ε 0 U ε e,k,+ = λ ε e,ku ε e,k,+. Zone c. Naturally Ue,k,c(x ε = 0 for x (ae ε, be ε. Mutatis mutandi x Ue,k,c(x ε = exp a ε µ (2iπk ε e e (s ds U te εµε e (x µ ε e (x + t ε e µε e (x exp ( 2iπk x t ε e a ε e µ ε e (s ds t ε e U µ ε e (x L p (I N, p < 2. Collisionless Boltzmann (Vlasov Equation 207 p. 7 / 2

8 Representation of the identity operator I d Let U X. The spectral decomposition holds U = I d U = U e 0e 0 + z with the Plancherel identity U 2 = (U, e z k Z k Z e I ε z e I ε z ( U, U ε e,k,z U ε e,k,z t ε e de ( U, U ε e,k,z 2 t ε e de. The spectral measure t ε e is explicitly provided. Collisionless Boltzmann (Vlasov Equation 207 p. 8 / 2

9 Eigenstructure of ih ε Start from Ve,k,z(x ε = Ue,k,z(x ε + ae,k,z(xe ε 0 + z p Z Plug in ih ε V = λ ε e,k,zv and rearrange z p Z s I ε z b ε s,p,z e,k,z s I ε z b ε s,p,z e,k,z U ε s,p,z (xtε s ds. ( λ ε s,p λε e,k U ε ( s,p,z tε s ds = αε exp ε 2 ϕ 0 /2 a ε e,k,z e 2 ( ( +α α λε e,k aε e,k,z (x ε e ε2 ϕ 0 /2 ε U e,k,z e 2 ε z p Z s I ε z b ε s,p,z e,k,z ( e ε2 ϕ 0 /2 ε U s,p,z e 2 Crux : use the representation of the identity b ε s,p,z e,k,z ( λ ε ( ( s,p λε e,k = ε α exp ε 2 ϕ 0 /2 a ε e,k,z e 2, U ε s,p,z, α λε e,k aε e,k,z ε z p 0 s I ε z b ε s,p,z e,k,z t ε s ds e 0. ( ( exp ε 2 ϕ 0 /2 U ε s,p,z e 2 t ε ( ( s ds = ε exp ε 2 ϕ 0 /2 U ε e, Collisionless Boltzmann (Vlasov Equation 207 p. 9 / 2

10 Keystone : solve the Strong form : for k 0, z {+,, c} and e I ε z, solve the integral for a = a ε e,k,z L 2 0(I α λε e,ka ε 2 z p 0 P.V. ( αae 2,exp(ε 2 ϕ 0 /2U ε s,p,z ( exp ( ε 2 ϕ λ ε s,p 0/2 U ε λε s,p,z e2 t ε s ds e,k = ε ( exp ( ε 2 ϕ 0/2 Ue,k,z ε e 2. Variational form : find a L 2 0(I such that (a, b + ε 2 p 0 P.V. dλ p 2 π 2 R λ iµ/2πp +ε 2 p>0 P.V. na,b,p ε (λ dλ = εl(b, b p 2 π 2 R λ iµ/2πp µ L2 0(I m ε a,b,p (λ where λ = /te ε, the kernel is ma,b,p(λ ε = ( a, v ε ( s ε ( λ,p,sign(λ v ε s ε ( λ,p,sign(λ, b exp( s ε ( λ (s ε ( λ ( with vs,p,+(x ε x0 µ = exp 2iπp ε s (t dt. t ε s Trick : P.V. R λf (λ λ µ dλ = µp.v. f (λ R λ µ dλ + R f (λdλ. Trick 2 : g L (R g L 2 (R g L 2 (R Collisionless Boltzmann (Vlasov Equation 207 p. 0 / 2

11 Well-posedness of the transformation e t e := t ε e Issue is the region of trapped particles, i.e. the second branch te e first branch s(λ second branch /B first branch ϕ 0 second branch sc(λ B λ ϕ + ϕ ϕ e Elementary calculations show that d t e de 2 = d be de x 0 dx = e ψ(x 0 where ψ(x = ϕ 0(x + max y ϕ 0(y 0. 4u 4 e ( ψ u 2 (ψ 3 (ψ (eu 2 du > 0 - ψ > 0 for HMF model. Collisionless Boltzmann (Vlasov Equation 207 p. / 2

12 Statement of the claim Usual assumptions + specific assumptions Boltzmanian hypothesis, 2 one region of trapped particles, 3 the time needed for particles to travel along characteristics is a monotone function of the characteristic label sup x 0,x 0, ( d 2 dx 2 ϕ + 0 ϕ0(x < 0 4 the initial data has zero mean value along the characteristics curves of the transport operator v x ε 2 E 0 v. There exists a constant ε such that for all 0 < ε < ε, then the electric field tends to zero in strong norm lim t F (t L (I = 0. Moreover if ϕ 0 < 2π then the ion density ρ ref is positive (and so the solution is physically admissible. - If ϕ 0 0, one solves the Fourier mode per Fourier mode and explicitly constructs a modified Möller wave operator. It is equal to the Morrisson integro-differential transform. - The small parameter ε is only for mathematical convenience. My feeling is it should be removed. - Preprint available. Collisionless Boltzmann (Vlasov Equation 207 p. 2 / 2