Energy and entropy in the Quasi-neutral limit.

Transcription

1 Energy and entropy in the Quasi-neutral limit. M. Hauray, in collaboration with D. Han-Kwan. Université d Aix-Marseille Porquerolles, June 2013 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

2 Outline 1 Introduction to the problem 2 The existing mathematical literature on the quasi neutral limit. 3 The stability of homogeneous equilibria in VP. 4 Strong instability and stability in the quasi-neutral limit (d 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

3 Outline 1 Introduction to the problem 2 The existing mathematical literature on the quasi neutral limit. 3 The stability of homogeneous equilibria in VP. 4 Strong instability and stability in the quasi-neutral limit (d 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

4 Outline 1 Introduction to the problem 2 The existing mathematical literature on the quasi neutral limit. 3 The stability of homogeneous equilibria in VP. 4 Strong instability and stability in the quasi-neutral limit (d 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

5 Outline 1 Introduction to the problem 2 The existing mathematical literature on the quasi neutral limit. 3 The stability of homogeneous equilibria in VP. 4 Strong instability and stability in the quasi-neutral limit (d 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

6 Introduction to the problem Section 1 Introduction to the problem M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

7 Introduction to the problem The Debye (- Hückel) length. Debye (- Hückel) length : The scale of charge separation, plasma oscillations. ε0k λ D : B T j ρ0 j Z j 2e2 Relatively small (with respect to typical length) in many physical situation. 1 2 From a course by Kip Thorne at Caltech. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

8 Introduction to the problem A quick explanation of its origin. Start with the density of e (charge Z 1) in a fixed background of ions. Write the Poisson equation on the potential Φ Φ Z epρe ρ0 q ε 0. Assume that the e are at thermal equilibrium with large temperature : ZeΦ k B T ρ epxq ρ 0 e Z eφpxq k B T ρ 0 ρ 0 Z eφpxq k B T. We end up with the linearised Poisson-Boltzman equation ε0k Φ B T Φ λ 2 ρ 0 Z 2 e 2 D Φ. ñ Φ varies at the scale λ D. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

9 Introduction to the problem More rigorously : the nondimensionalization of Vlasov-Poisson equation. Start from the Vlasov-Poisson eq. for the density f pt, x, vq of e (fixed ions background) Bf Bt v Bf Bv e BΦ m e Bx Bf Bv 0, with Φ epρe ρ0 q. ε 0 Introduce the typical scales and associated new variables without dimension (with prime) t Tt 1, x Lx 1, v V th v 1, n 0f 1 pt, x 1, v 1 q dx 1 dv 1 f pt, x, vq dxdv n 0 number of moles at size L, i.e. ρ 0 n 0. Also assume V L d th T L. This leads to the nondimensional equation Bf 1 Bt 1 v 1 Bf 1 BΦ 1 Bv 1 Bx 1 Bf 1 Bv 1 0, with λ 2 D L 2 Φ ρ1 1. Again λ 2 D ε0mev 2 th ρ 0 e 2. The important parameter is the ratio ε λ D L. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

10 Introduction to the problem The related Plasma oscillations, a.k.a. Langmuir Waves. Rewrite the previous system (for convenience) as B tf ε v B xf ε B xφ ε B v f ε 0, with ε 2 Φ ε ρ ε 1. The energy is E εrf εs : 1 v 2 f ε dxdv 2 1 rεφ εs 2 dx. 2 Decompose the current j ε ³ f εv dv in divergence free j d ε and gradient part B xj ε. The equations for J ε and εφ ε are B trεφ εs Jε ε B tj ε εφε ε 1 1 ddiv rε xφ εs b rε xφ εs f εv b v dv 2 ε Φε 2 Setting O ε J ε iεφ ε, B to ε i Oε something of order one. ε ñ Strong oscillations of period 2π ε in Φε, Jε and also ρε. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

11 Introduction to the problem Experimental observation of Langmuir Waves inionosphere. Very fast phenomena ñ Quite difficult to observe. From Kintner, Holback & all, Cornell University and Swedish inst. of space phy. Geophy. Rev. Letters Record form Freja plasma wave instrument ( alt km). M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

12 Introduction to the problem Experimental observation of Langmuir Waves in plasma. From Matlis, Downer & all, University of Texas and Michigan, Nature Phys M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

13 Introduction to the problem Heuristic on the Quasi-neutral limit ε Ñ 0. Neglect the problem of the plasma oscillations. Very formally, the expected limit is B tf v B xf B xφ B v f 0, with ρ 1. Using ε 0 in the equation for J ε and Π ε, we get very formally (false) Φ : 1 ddiv f εv b v dv. This is correct only if ρp0q 1 and Jp0q 0, i.e. well prepared case. The previous neutral Vlasov system is very singular. We known only A Cauchy-Kowalevsky type result : local in time existence for analytic initial data [Bossy, Fontbana, Jabin, Jabir in CPDE 13]. Same analytic setting, but with a plasma seen as a superposition of fuilds [Grenier, CPDE 96]. Similar result but in H s for (very) particular initial data [Besse, ARMA 11] [Bardos, Besse, Work in progress]. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

14 The existing mathematical literature on the quasi neutral limit. Section 2 The existing mathematical literature on the quasi neutral limit. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

15 The existing mathematical literature on the quasi neutral limit. Early results in the 90 by Grenier (and Brenier) Defect measures used in [Brenier, Grenier, CRAS 94] and [Grenier, CPDE 95] : The 2 first moments will satisfy the expected equation with defect measures in the r.h.s. Deep result with the fluid point of view [Grenier, CPDE 96]. Write the plasma as a collection of many fluids (µ some measure) f εpt, x, vq ρ ε θpt, xqδ v ε θ pt,xqpvq µpdθq. The family pρ θ, v θ q θ satifies coupled Euler-Poisson B tρ ε θ divpρ ε θvθ ε q 0, B tvθ ε pvθ ε qvθ ε V, V ε ρ ε θµpdθq 1 The expected limit model : coupled incompressible Euler equation : B tρ θ divpρ θ v θ q 0, B tv θ pv θ qv θ p, ρ ε θµpdθq 1 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

16 The existing mathematical literature on the quasi neutral limit. Grenier: : convergence after filtration of the Plasma oscillations. Theorem (Grenier, CPDE 96) Assume that the family pρ ɛ θ, v ɛ θq ɛ,θ satisfies uniform H s estimates (s large). εv εp0q Ñ V 0 and j ε Ñ ν 0 J 0 with div ν 0 0. Then ρ ε θ, v ε θ J ε converges towards solution of the expected coupled inc. Euler equation, with a corrector J ɛ defined by J ε pt, xq Re e i ε t Upt, xq, and U is solution of U 0 J 0 iv 0, B tu ρ θ v θ µpdθq U 0 Contains almost everything but the formalism is unusual Not simple to pass from f formalism to the superposition of plasma. To summarize, Convergence possible only Under good a priori estimates. After filtration of the Plasma oscillation. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

17 The existing mathematical literature on the quasi neutral limit. Later results : The Quasi-neutral and zero temperature limit. Zero temperature limit : Assume that for some vpt, xq f εp0, x, vq á δ j0 pxqpvq i.e. v j 0pxq 2 f εp0, x, vq dxdv. Theorem We denote j 0 ν 0 J 0, with div j 0 0, and V 0 lim εv εñ0 ɛp0q p 1 ρ ɛp0q 1 q in H 1. ε First result in well prepared case [Brenier, CPDE 00] : Assume that J 0 0 and V 0 0. Then j ε converges weakly towards a dissipative solution to the inc. Euler equation with initial data ν 0. Based on the use of the modulated energy Eu ε ptq 1 v upt, xq 2 f ε dxdv 2 1 ε V ε 2 dx 2 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

18 The existing mathematical literature on the quasi neutral limit. Another quasi-neutral and zero temperature limit by Masmoudi. Theorem Next result in the ill-prepared case [Masmoudi, CPDE 01 ] Assume that ν is a sufficiently smooth (in some H s ) solution of the inc. Euler eq. with initial data ν 0. Define U by Up0q J 0 iv 0 and B tu ν U 0. Define Eν ε ptq 1 v νpt, xq Re e i t 2 ε Upt, xq f pt, x, vq dxdv 2 1 ε Vεpt, xq Im e i ε t 2 Upt, xq dx 2 Then if E ε ν p0q Ñ 0, we have E ɛ νptq Ñ 0 for any t 0. Compatible with Grenier s result (and more or less included in it). Based on a control of the increase of E ε ν. E ε ν ptq C tpe ɛ νp0q εq. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

19 The stability of homogeneous equilibria in VP. Section 3 The stability of homogeneous equilibria in VP. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

20 The stability of homogeneous equilibria in VP. The Penrose criterion for existence of Growing model. In 1D. Study the linearized Vlasov equation around f pvq B tg vb xg B xv g B v f 0, ε 2 B 2 xv g g (1) Ansatz : gpt, x, vq e ikx ωt hpvq (or Use Fourier-Laplace transform). It satisfies (1) iff F i ω : k B v f v i ω k dv pεkq 2 with hpvq 1 pεkq 2 B v f v i ω k dv If exists z with Im z 0 satisfying F pzq P R, then c c F pzq F pzq k ε, ω iz ùñ Growing mode ε F p zq F pzq ùñ consider only the case Im z 0. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

21 The stability of homogeneous equilibria in VP. The Penrose criterion : a story of contour. Bv f pvq Introduce F pξq : lim F pξ iηq PV ηñ0 v ξ dv iπb v f pξq F pzq P R for some z with Im z 0 ðñ the contour F prq circles (ö) some part of R ðñ F prq cross R from below at some point. Left : Contour of a Maxwellian distribution. Right : Contour of unstable profiles. From O. Penrose, Phys of Fluids M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

22 The stability of homogeneous equilibria in VP. The Penrose criterion : A condition on local minimum. Now F pξ 0q P R ùñ ξ 0 a critical point of f. F cross R by below at ξ 0 ùñ is a local minimum. At this local minimum Re F pξ 0q 0. Definition (Penrose criterion on R.) A homogeneous profile f with sufficient regularity and moments satisfy the Penrose criterion iff there exists a local minimum ξ 0 such that Bv f pvq f pvq f pξ0q PV v ξ dv dv 0 pv ξ 0q 2 The criterion is slightly different on a torus. The contour should circle some part of tεk 2, k P N u and not R. Non-linear instability : If f satisfy the PC is symetric, then it is non-linearly unstable in H s with some weight [Guo, Strauss, ANIHP 95] One-humped profiles, with no local minima do not satisfy the criterion. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

23 The stability of homogeneous equilibria in VP. The non-linear stability for symetric profile. The so called Energy-Casimir method introduced by Arnold [Dokl. USSR 65] [IVUZM 66] may be used in VP. The adaptation to plasmas is done in [Holm, Mardsen, Ratiu, Weinstein Phy Rep 85] and [Rein, MMAS 95]. Idea : Use the invariant to construct a convex functional that is minimal at some profil f. In VP on the 1D torus, ε fixed, the invariants are : The total energy E ɛrfεs : 1 2 the total quantity of mvt : f ε v 2 ɛ dxdv B x V εrfεs 2 dx, 2 Prf εs : f εv dxdv, The integral I Q below for any smooth enough Q. (Requires strong solutions) I Q rf εs Qpf εq dxdv M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

24 The stability of homogeneous equilibria in VP. Construction of an appropriate Casimir functional. At which condition does FQ ε defined by FQrf ɛ εs : Qpf ε dxdvq E ɛrf εs admits f as critical point. Answer: possible only if Q 1 pf q v 2 2. ùñ f is radial: f pvq ϕp v 2 {2q with an injective ϕ, and Q ϕ 1. At which condition is Q and then F ε Q convex? Answer : ϕ is decreasing. The momentum invariance allows to replace v by v v 2. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

25 The stability of homogeneous equilibria in VP. Generalized entropy for a model without collision. In the previous situation : f pvq ϕp v 2 {2q and Q ϕ 1, define H Q rgs : rqpgq Qpf q Q 1 pf qpg f qs dxdv 1 Qpgq dxdv g v 2 dxdv C st 2 H Q is convex (often strictly). H Q strictly convex ùñ Non-linear stability of f in L 2. H Q is the usual relative entropy if f is a Maxwellian dist. f pvq e v 2 2T ùñ H Q pgq THpg f q T g ln g H Q is a kind of relative entropy. H Q H Q is not uniquely defined for a fixed ϕ. E pot a kind of free energy. T v 2 g 2 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

26 The stability of homogeneous equilibria in VP. Non-linear stability of VP via rearrangment inequality. Notation : f g if their symmetric rearrangement are equals f g. Basic idea : The Vlasov equation preserves the rearrangement : g ptq g p0q. By conservation of the total energy rg ptq g p0q s v 2 dxdv E ɛrg p0qs g p0q v 2 dxdv If g as kinetic energy close to g, they should be close. [Marchioro, Pulvirenti, MMAS 86] : Precise the later idea in dim d 2 }g g } 2 1 C rg g s v 2 dxdv C depends on }g} 8,.. ùñ Non-linear stability in L 1. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

27 Strong instability and stability in the quasi-neutral limit (d 1q. Section 4 Strong instability and stability in the quasi-neutral limit (d 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

28 Strong instability and stability in the quasi-neutral limit (d 1q. Plasma oscillations in dimension one. Assume f εp0q f 0pvq g 0ppv vq 2 q, an homogeneous profile, symetric w.r.t. v. In dim 1, j d ε is a constant. The equations on J ε and εv ε are simpler $ '& B tpεv εq Jε ε '% B tpj εq εvε 1 ε 2 Bxp?. εv εq 2 f εv 2 dv Setting as before O ε J ε iεφ ε leads to B to ε i ε Oε BxpIm Oεq 2 f εv 2 dv. Due to the fast variation of B xj ε, we cannot have f εpt, x, vq f 0pvq, but maybe f εpt, x, vq f 0pv B xj εpxqq If the later is true, then f εpt, x, vqv 2 dv 2T v B xj εpt, xq 2, M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

29 Strong instability and stability in the quasi-neutral limit (d 1q. Plasma oscillations in dimension one, part II. So that the equation for O ε may be approximated by (erase the constants) Setting U ε e i t ε Oε, it comes B to ε i ε Oε BxpIm Oεq 2 v B x Re O ε 2. B t U ε e i t ε BxpIm e i t ε Uεq 2 e i t ε v Bx Re e i t ε Uε 2 Using Im z 1 pz zq, expanding and keeping only the non-oscillating terms 2 B t U ε vb xu ε quickly oscillating terms U ε should converge towards U, solution of B t U ε vb xu 0, Up0q : lim J εñ0 εp0q iεv εp0q i lim εv εñ0 εp0q V εpt, xq V 0px vtq cos and J εpt, xq V 0px vtq sin. t ε t ε M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

30 Strong instability and stability in the quasi-neutral limit (d 1q. A rigorous stability result in the ill -prepared case. Again around f pvq ϕ v v 2, and H Q the associated Casimir functional. Assume that lim εñ0 εv εp0q V 0 P W 3,8. Use the energy Casimir method together with the filtration of the oscillations. Define the functional L O ε ptq : H Q f ε t, x, v B xv 0px vtq sin t ε 1 εb xv ɛ B xv 0px vtq cos t 2dx 2 ε Theorem (Han-Kwan, Hauray, 13) Under the above assumptions, and also E εpf εp0qq C 0, Q pf εp0qq Q 2 pf εp0qq dvdx C 0, f εp0q there is C 0, such 0, L O ε ptq e 2}Bxxx V 0} 8t L O ε p0q Cεpe 2}Bxxx V 0} 8t 1q. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

31 Strong instability and stability in the quasi-neutral limit (d 1q. Very fast instability around unstable profil. pvp εq in the variables pt 1, x 1 q t, x is pvp ε ε 1q. ùñ The possible instabilities are much faster. Notation : H s γ is the space H s with weight p1 v q γ. Theorem (Han-Kwan, Hauray 13) Assume that f is a symmetric profile, unstable in the sense of Penrose. Then, for some γ 0 and any s 0 and N P N, there exists a family of initial conditions rf εp0qs ε such that }f ɛp0q f }H s γ Cɛ N, lim sup }f εp0q f }L 2 0. εñ0 t ε ln ε γ Uses a technic introduced by Grenier (again!) for Euler and Prandtl equation [CPAM 00]. ùñ General stability is not possible, except in analytic framework. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25

32 Strong instability and stability in the quasi-neutral limit (d 1q. Conclusion Our stability results requires symmetry, but the only symmetric solutions are the homogeneous equilibria. Plasma oscillation are not damped (in our setting). No initial boundary layer. May leads to fast instabilities. Open problems : Non symmetric equilibria? Non stationary solutions??? Lot s of inspiration from [Grenier, JEDP 99]. Thanks (him and you)! M. Hauray (UAM) Quasi-neutral limit Porquerolles, June / 25