About Maxwell Boltzmann Relation.

Transcription

1 About Maxwell Boltzmann Equation. Retired Paris Report on an ongoing project with Toan Nguyen and several coworkers: Nicolas Besse, Irene Gamba, Francois Golse, Claudia Negulescu and Rémi Sentis. June 4, 2016

2 Figure: Left: Ludwig Boltzmann ( ); right: James Clerk Maxwell( ).

3 What is the Maxwell Boltzmann Equation? I In R d x R d v Assumption: Interactions between Electrons and Ions only due to electro magnetic forces with external given Magnetic Field B. t f + + v x f + + (E + v B) v f + = η + C + (f + ), St t f + v x f (E + v B) v f = η C (f ). f ± (t, x, v) = f ± (t, x, v 2(v n) n(x)) Specular reflection if λ 2 φ = f + (x, v, t)dv f (x, v, t)dv, n φ = 0, on. R v R v The two kernels C ± satisfies the standard hypothesis: Conservation of mass momentum and energy and H theorem: Boltzmann, BGK, Fokker Planck...Other collision operator.???

4 What is the Maxwell Boltzmann Equation? II With a Korn type hypothesis on : f (x, v, t) ( β(t) 2π ) d v 2 e β(t)( 2 2 φ(x,t)) f = e β(t)φ(x,t) v 2, 2 f (x, v, t) dx d β(t) λ 2 φ + e β(t)φ(x,t) = f + n φ = 0 on, t f + + v x f + x φ v f + = η + C + (f + ). β(t) is the inverse of the temperature. What beta? f dx = m 0d β(t)

5 Want I can do, what I want to do Try to justify the scaling Prove a convergence theorem under an extra regularity assumption Give a complete treatement of the so called reduced ions problem. Give a Arnold type stability result for the asymptotic solutions.

6 Convergence to the Ions problem Consider in a domain which satisfies the Korn hypothesis, the problem t f ɛ + + v x f ɛ + x φ ɛ v f ɛ + = 0 St ɛ t f ɛ + v x f ɛ + x φ ɛ v f ɛ = η (ɛ)c (f ɛ ), λ 2 φ ɛ = f ɛ + f ɛ With if the following boundary conditions f ɛ ±(t, x, v) = f ɛ ±(t, x, v 2(v n)) n(x) Specular reflection λ 2 φ ɛ = f ɛ + f ɛ n φ ɛ = 0 Neumann boundary condition

7 Theorem Assume that for ɛ 0 one has lim inf(st ɛ ) 1 η ɛ =, and lim sup lim ɛ 0 η ɛ < + while for t (0, T ) the distributions f ɛ ±(x, v, t) remains uniformly bounded in a convenient class (specified below) then: (f, ɛ f+) ɛ (( β(t) 2π ) d v 2 e β(t)( 2 2 φ(x,t)), f + (x, v, t)) λ 2 φ + e β(t)φ(x,t) = f + (t) n φ = 0 on, t f + + v x f + x φ v f + = η + C + (f + ). and β(t) is uniquely determined by the conservation of energy: v 2 2 f +(t) dx + m 0d β(t) x φ(x, t) 2 dx = E 0.

8 About β In the above asymptotic β which is the inverse of the temperature is independent of x. In some approximations it may also be taken independent of t and this will be justified below by a global in time stability result. One may wonder at getting a electron temperature which is constant with respect to the space variable. But recall we deal here with a modelling ence to the scale of the Debye length (for instance some tens or hundreds of Debye lengths) and at this scale it is natural that the electron temperature is constant even if it is not the case at a much larger scale. Hence consideration of the variation of the temperature may appear on some more macroscopic models as described below

9 Remark about hyperbolic scaling and others... In the above situation with the hyperbolic scaling St ɛ = O(1) and lim η (ɛ) = t f ɛ + + v x f ɛ + x φ ɛ v f ɛ + = 0 St ɛ t f ɛ + v x f ɛ + x φ ɛ v f ɛ = η (ɛ)c (f ) ɛ, λ 2 φ ɛ = f+(x, ɛ v, t)dv f (x, ɛ v, t)dv R v R v a formal limit, which can be fully justified for small deviation of an absolute Maxwellian and for a short time is the classical compressible Euler Poisson system for Electrons coupled with the Vlasov equation for Ions.

10 St ɛ = o(1) and η(ɛ) t f + + v x f + x φ v f ɛ + = 0 St ɛ t ρ + (ρ u ) = 0 St ɛ t (ρ u ) + (ρ u u ) + (ρ θ ) ρ φ = 0 ( St ɛ t ρ ( u 2 + d ) 2 2 θ ) + ( u (ρ 2 u + d + 2 )) θ ρ u φ = λ 2 φ + ρ = f + (x, v, t) This does not leads to a closed system for the Ions as it is done by the Boltzmann Maxwell relation. It still requires a Euler Poisson evolution equation for the electrons.

11 Remark 3 Letting St ɛ 0 leads to: t f + + v x f + x φ v f ɛ + = 0 (ρ u ) = 0 (ρ u u ) + (ρ θ ) ρ φ = 0 (ρ ( u 2 + d ) 2 2 θ ) + ( u (ρ 2 u + d + 2 )) θ ρ u φ = λ 2 φ + ρ = f + (x, v, t) In 1d it is a shock profile but in more than 1d it is not well posed.

12 Changing u into St ɛũ St ɛ 0 formally lead to an ill posed system similar to the Sone equations for Ghost effect. t f + + v x f + x φ v f + = 0 λ 2 φ = f + ρ, t ρ + x ρ ũ = 0 x φ 1 x (ρ θ ) = 0, ρ d 2 tρ θ + d + 2 x ρ ũθ x φũ = 0. 2

13 Maxwell Boltzmann relation and θ x dependent for isentropic Euler Poisson equation. t f + + v x f + x φ v f+ ɛ = 0 t ρ + (ρ u ) = 0 t (ρ u ) + (ρ u u ) + (ρ θ ) ρ φ = 0 Taking in account the smallness of the mass of the electrons. t u + u u + 1 ( P(ρ ) φ) = 0 m e φ + P 1 (φ) = f + θ (x, t) = S(ρ (x, t))... = e S ρ 1 3 (x, t) References: Goudon-Jungel-Peng... Grenier -Toan...

14 Conclusion of the remark The large time behavior for the Euler Poisson is not only impossible to analyse (appearance of singularities and wild solutions later) but also unappropriate for the closure. (However the 0 mass limit of the isentropic electron equation could be considered). A law mach number asymptotic u = St ɛ ũ with St ɛ 0 leads also to an ill posed problem What is important is to have at the same time the relations lim inf(st ɛ ) 1 η ɛ = and lim sup lim ɛ 0 η ɛ < + Compared to the Landau damping a very different regime.

15 Proof of the soft theorem" Conservation of Mass Assume that two kernels C ± satisfies the standard hypothesis: Conservation of mass momentum and energy and H theorem then for smooth solutions one has with well prepared initial data: C ± (f ± ) = 0 d f ± (x, t)dx = 0 dt f (x, t)dx = f + (x, t)dx = M 0 And the elliptic problem (with Neumann or periodic boundary data) φ = f + f n φ = 0, φ(x)dx = 0 is well posed.

16 Proof of the soft theorem" Conservation of Energy Use C ± (f ± ) v 2 2 = 0 and obtain formally d dt ( ( v 2 2 (f + f + ) xφ(x, t) 2 )dx = 0 ( v 2 2 (f + f + ) xφ(x, t) 2 )dx = E 0 For the Ions problem with the Boltzmann Maxwell Relation f (x, v, t) ( β(t) 2π ) d v 2 e β(t)( 2 m 0 d β(t) + 1 x φ(x, t) 2 dx = E φ(x,t)) v 2 2 f + dx.

17 Proof of the soft theorem" Stationnary solution Theorem Under a Korn Hypothesis any local Maxwellian solution of the equation: is of the following form: f (x, v) = ρ(x)( β(x) 2π ) d v u(x) 2 e β(x)( 2 2 {E, f } = v x f + x φ v f = 0 f (x, v) = ( β 2π ) d 2 e β( v 2 2 φ(x))

18 For any smooth vector u : R d with u n = 0 one has u + ut 2 L 2 () K() u 2 L 2 () (1) is generally valid. It holds for the flat torus T d and in dimension 2 and 3 if has no axis of symmetry. Cf. Proposition 13. of Desvillettes and Villani. Invent. Math. 159 (2005) and Desvillettes and Villani ESAIM: Control, Optimisation and Calculus of Variations June 2002, Vol. 8.

19 Proof II f is a Maxwellian f (x, v) = ρ(x)( β(x) 2π ) d v u(x) 2 e β(x)( 2 ) 2 ( x f = f ( xρ(x) d x β ρ(x) 2 β ) + β(x) xu (v u(x)) v u(x) 2 2 x β β ) v f (x, v, t) = β(v u))f. (B v) v f = βb v (v u)f = β(b u)(v u)f

20 Proof III Identification of the Terms of Order 3, 2, 1, 0 in (v u). Order 3 (v u) v u 2 x β = 0, x u+ x ut Order 2 (v u) x u (v u) 2 = 0 With Korn inequality u = 0, x ρ(x,) Order 1 (v u) ( ρ(x) β( φ + B u)) = 0 Order 0 u x log ρ(x) = 0. With β = 0 and u = 0 (u constant and tangent to the boundary) only remain the equations: log(ρ(x)) = β(φ(x) f (x, v) = ( β 2π ) d 2 e β( v 2 2 φ(x)) with φ changed into φ + (log constante)/β

21 Determination of β through energy conservation Theorem Let R d x be a bounded domain and E > 0. Fix a nonnegative ion density I (x) L 2 () with finite mass m 0. Then, there exists a unique solution (β, φ) to the following elliptic problem: φ + e βφ = I (x), φ n = 0 (2) together with the mass and energy constraints m 0 d β + 1 x φ(x, t) 2 dx = E 2 Below Sentis direct proof. Idea The energy is an increasing function of the temperature.

22 For each fixed β > 0, the elliptic problem has a unique solution φ β H 2 () with total mass m 0 and energy" E(β) = m 0d β + 1 x φ β (x, t) 2 dx 2 and derivative solution of e βφβ β φ β + β β φ β = φ β, β φ β n = 0 Hence β E(β) = m 0d β 2 + x φ β x β φ β dx = m 0d β 2 φ β x β φ β dx = m 0d β 2 e βφβ ( x β φ β ) 2 dx + β β φ β x β φ β dx = ( m 0d β 2 + e βφβ ( x β φ β ) 2 dx + β x β φ β 2 dx) 0

23 End of determination of β For the existence 1 One has E(β) > (m 0 d)β 2 lim β 0 E(β) = 2 From the elliptic equation for φ β, ( φ β 2 dx = I (x)φ β (x) e βφβ φ β) dx (x)φ {φ 0}(I β (x) e βφβ φ β )dx 1 e βφβ βφ β dx. β β {φ β 0} x 0 e x x (I (x)φ β (x) e βφβ φ β )dx I L 2 φ β L 2 β φ β 2 L 1 2 2β I 2 L 2 {φ β 0} x 0 xe x e 1 1 e βφβ βφ β dx e 1. β β {φ β 0} Therefore lim β E(β) = 0 and existence of β follows.

24 Uniform regularity assumptions: URA In the ions equation there is no non linear relaxation term the solution remain f ɛ + remain bounded in any L p ( R d x ). The energy remains bounded. Therefore the only hypothesis needed is the convergence of this energy For almost every t (0, T ) lim v 2 ɛ 0 2 f +(t) dx ɛ v 2 2 f ɛ +(t) dx For the electrons because the role of the collision operator is essential the situation is more subtle and one has to assume the convergence of f ɛ almost everywhere, and the uniform integrability ( a strong assumption?): f ɛ Ce δ v m

25 End of proof of the soft theorem With η (ɛ) > 0 and bounded, lim(st ɛ ) 1 η ɛ start from t f ɛ + + v x f ɛ + x φ ɛ v f ɛ + = 0 St ɛ t f ɛ + v x f ɛ + x φ ɛ v f ɛ = η ɛ C (f ɛ ) λ 2 φ ɛ = f ɛ +(t) f ɛ (3a) (3b) (3c) the corresponding standard estimates and the Uniform regularity assumption. f+(t) dx ɛ = f (t) dx ɛ = m 0 ; (4a) 0 f+(x, ɛ v, t) = sup f+(x, ɛ v, 0) x,v v 2 2 f ɛ +(t) + v 2 2 f ɛ (t) which imply in particular the uniform bound: φ ɛ (t) W 2, d+2, tφ ɛ (t) 2 W 1, d+2 C 0 2 (4b) x φ ɛ 2 dx = E 0 (4c)

26 Then one deduces that (f ɛ, f ɛ +, φ ɛ ) converge (in a weak sense ) to (f ɛ, f ɛ +, φ ɛ ) and that (f ɛ +, φ ɛ ) are solution of the system: t f ɛ + + v x f ɛ + v ( x φ ɛ f ɛ +) = 0 (5) λ 2 φ ɛ = f ɛ + f ɛ (6) Multiply (3b ) by log f ɛ and integrate over R d v (0, T ) to obtain: T 0 C (f ) ɛ log f dxdt ɛ 0 Stɛ η ɛ ( f ɛ log f (x, ɛ 0)dx f ɛ log f ɛ (x, T )dx) 0 Then with the uniform regularity hypothesis and the H theorem f ɛ is a local Maxwellian M ρ,u,θ. (7)

27 Letting ɛ 0 in the equation one obtains: St ɛ t f ɛ + v x f ɛ + x φ ɛ ɛ v f ɛ = η ɛ C (f ɛ ) (8) v x M ρ,u,θ + x φ ɛ v M ρ,u,θ = lim ɛ 0 v x f ɛ + x φ ɛ v f ɛ = lim ɛ 0 (η ɛ C (f ɛ ) St ɛ t f ɛ ) = 0. (9) Under the uniform regularity hypothesis the last term of (9) converges to 0 (at least in a weak sense) and this implies that M ρ,u,θ = ( β(t) 2π ) d v 2 e β(t)( 2 2 φ(x,t)), (10)

28 Eventually taking the limit in the energy conservation equation: v 2 2 f +(t) ɛ + v 2 2 f (t) ɛ + 1 x φ ɛ 2 dx = E 0 (11) 2 one reaches the reduced Ions system: m 0 d β(t) + 1 x φ ɛ (x, t) 2 v 2 dx = E f ɛ + dx λ 2 φ ɛ + e β(t)φɛ = f ɛ + St + t f ɛ + v x f ɛ x φ ɛ v f ɛ = 0. (12)

29 About Unconditionnal Statements I The biggest issue is the stability " of the solution of the electron equation: St ɛ t f ɛ + v x f ɛ + x φ ɛ v f ɛ = η (ɛ)c (f ɛ ) (13) The simultaneous presence of a Boltzmann type collision term that will relax to equilibrium and of large time asymptotic with the relation lim St ɛ 1 η (ɛ) = seems compulsory and in agreement with the physical scalings. Even for the genuine Boltzmann equation (no φ ) there is no general results and perturbations near an absolute maxwellian, because they involve the large time behaviour would handle only the trivial case φ = 0. Moreover in the above derivation the conservation of energy is a crucial factor.

30 About Unconditionnal Statements II At the present stage two type of results are available. The existence and uniqueness result for the reduced ions system. At variance with the oldest simple construction of weak solution of Vlasov equation here the compulsory conservation of energy requires a construction preserving some regularity. An Arnold stability both for the reduced ions problem and for the full system result because it is robust under weak limit (like the standard weak strong stability of compressible or incompressible Euler equation).

31 Well posedness of the reduced ions problem Theorem (Existence of weak solutions) Assume that initial data f 0,+ L 1 L, compactly supported in v. There exists a time T > 0 so that weak solutions (f +, φ, β) to the ion problem so that f + remains compactly supported in v and satisfies the estimates: f + C(0, T ; L 1 L ( R d )), ρ + C(0, T ; L 1 L ()), the electric field E = φ C(0, T ; L ()), and β L ([0, T ]). The solution can be extended globally in time for d = 1, 2, 3. The non trivial part is the fact that the solutions remains of compact support in v. A priori estimates are given then the rest of the proof involves a classical iteration

32 1 Boundedness of β(t) d v 2 dt 2 f +(x, v, t) dvdx = φ vf + dx = φ t f + 1 d φ 2 dx = φ φ t = φ t ρ + φ t e βφ 2 dt 1 d ( v 2 2 dt 2 f +(x, v, t) + φ 2 dx) = φ t e βφ = 1 βφ t e βφ β = 1 β t (βφ 1)e βφ dx = 1 β t βφe βφ dx m 0d β 2 tβ(t) + 1 d ( v 2 2 dt 2 f +(x, v, t) + φ 2 dx) = 0 t (m o d log β(t) + βφe βφ dx) = 0!!!

33 II More Estimates on the elliptic problem 1 The Vlasov dynamic preserves the L norm of f + (x, v, t) and with m 0 d β(t) + v 2 2 f + dx + 1 x φ(x, t) 2 dx = E 0 2 one has sup t 0 ρ + (, t) L d+2 d () C and β(t) m 0d E 0 Multiplying by e (p 1)β(t)φ ( φ + e β(t)φ = f + (t) ) one obtains: (p 1)β(t) e (p 1)β(t)φ φ 2 dx + e pβ(t)φ dx f + (, t) L p e pβ(t)φ 1 p p 1 p L 1 e pβ(t)φ L p f + (, t) L p +ellipticicity E = φ L C f + (, t) 1 d L 1 f + (, t) d 1 d L

34 III The Liouville Dynamic for Ions. A standard a-priori estimate for Vlasov equation. The solution f + (x, v, t) of the ions equation is the pushforward of f + (x, v, 0) by the flow given by the equations Ẋ (s) = V (s) V (s) = x φ(x (s)) and the specular reflexion when ever X (s) meets. Hence one has V (t) V (0) C t 0 t 0 sup φ(x, s) ds x f + (, s) 1 d L 1 f + (, s) d 1 d L ds Then with v K 0 f + (x, v, 0) = 0 one has f + (x, v, 0) = 0 for v K 0 f + (x, v, t) = 0 f + (, t) L C 0 (K 0 + t 0 φ(, s) L ds) d (14)

35 Gronwall type a priori estimate: Hence for X (t) = ( f + (, t) L ) 1 d (for d = 2) a linear and for (d > 2) a non linear Gronwall estimate: f + (, t) 1 d L C 0 (K 0 + t 0 ( f + (, s) 1 d L ) d 1 ds t X C 0 + C 0 X d 1 (s)ds 0 (15) Moreover for d = 3 the estimate can be made global following the proof of Schaeffer for the classical Vlasov equation.

36 A Priori Estimate from Averaging Lemma E L and f L 1 L. t f + + v x f + = v (Ef + ). f + 2 L 2 (0,T ;L 2 ( R 3 )) f + L f + L 1 (0,T ;L 1 ( R 3 )) f + L ρ + L 1 (0,T ) and Ef + L 2 (0,T ;L 2 ( R 3 )) E L f + L 2 (0,T ;L 2 ( R 3 )). f + (t, x, v)ϕ(v) dv H 1/4 ((0, T ) ) R 3 together with the uniform bound f + (,, v)ϕ(v) dv C ϕ E L f + L R 3 H 1/4 ((0,T ) ) 2 (0,T ;L 2 ( R 3 )) for any test function ϕ(v) in C c (R 3 ).

37 Construction of a solution by iteration I Start from (β n, φ n ) solution of the elliptic problem φ n + e βnφn = f n (x, v, t) = ρ n (t) e βnφn dx = m 0 m 0 d + 1 φ n 2 dx = E 0 β n 2 v 2 2 f n(x, v, t) dx = E n (t) Then construct f n+1 by solving the linearized Vlasov equation t f n+1 + v x f n+1 x φ n v f n+1 = 0 with the same initial data f n+1 (x, v, 0) = f 0,+ (x, v) and now with density and energy (ρ n+1 (t), E n+1 (t)).

38 Construction of a solution by iteration I Modulo extraction of a subsequence (f n (x, v, t), ρ n (x, t), E(t), β(t) and φ n (x, t) weakly converge and satisfy all the above a priori estimates!!! With the averaging lemma and the fact that support of f n (x, v, t) remains bounded in velocity space the functions ρ n (x, t) and E n (t) converge almost every where. With the uniqueness of the solution of the elliptic problem (ρ(x, t), E(t)) (β(t), φ(x, t)) same is true for the sequence (β n (t), φ n (x, t))!!

39 Uniqueness of the solution for the reduced ions problem We same type of estimates one has the following Theorem (Uniqueness) Let T > 0. There exists at most one weak solution (f +, φ, β) to the ion problem with v-compactly supported initial data f 0,+ so that sup sup t [0,T ] x Remark T v f 1 L 2 (R d ) + φ(s, ) L () ds <. (16) 0 The estimate T 0 φ(s, ) L () ds <. has been established above under the only hypothesis that initial data be of compact support in v space. The hypothesis sup t [0,T ] sup x v f 1 L 2 (R d ) < would be trivial in the absence of boundary effect but up to now not so clear in the presence of specular reflection!

40 Arnold type stability results With the a-priory estimates one can get global in time non linear stability results. Such results seem useful because the estimate persist are remain valid for weak limits of solutions. In many cases the Boltzmann Maxwell relation is used with a prescribed time independent?. Hence it is convenient to validate this choice.

41 Arnold Stability for the Electrons. For the stability of the electrons with a given constant ion density and specular reflexion boundary condition t f + v x f + x φ v f = C (f ) φ + f (t) = I (x) (17) near a stationary Boltzmann-Maxwell solution, with the relative entropy [ ( f ) ] H(f f ) := f log f + f (x, v) dxdv, f one has: R 3

42 Theorem Let (f, φ) be any stationary solution : f (x, v) = ( β v 2 2π )d/2 β( e 2 φ(x)), φ + e βφ = f (18) and (f, φ) be any weak solution of the system (17) then: d dt (H(f f ) + β φ φ 2 dx) = D(f ) 2 with D(f ) := C + (f ) log f dx (19)

43 Proof. d H(f f ) + D(f ) = (1 + log f ) t f (x, v, t) dx dt = (1 + d 2 log( β 2π ) β( v 2 2 φ)) tf (x, v, t) dx = β t v 2 2 f (x, v, t) φ t f (x, v, t) dx = β ( x φ x φ) vf dx 3 = β φ) t f dx = (φ β d φ φ 2 dx 2 dt d dt (H(f f ) + φ φ 2 dx) + D(f ) 0

44 THANKS FOR ATTENTION THANKS FOR THE INVITATION