Energy and entropy in the Quasi-neutral limit. M. Hauray, in collaboration with D. Han-Kwan. Universit´ e d’Aix-Marseille Porquerolles, June 2013 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 1 / 25
Outline Introduction to the problem 1 The existing mathematical literature on the quasi neutral limit. 2 The stability of homogeneous equilibria in VP. 3 Strong instability and stability in the quasi-neutral limit ( d ✏ 1 q . 4 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 2 / 25
Outline Introduction to the problem 1 The existing mathematical literature on the quasi neutral limit. 2 The stability of homogeneous equilibria in VP. 3 Strong instability and stability in the quasi-neutral limit ( d ✏ 1 q . 4 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 2 / 25
Outline Introduction to the problem 1 The existing mathematical literature on the quasi neutral limit. 2 The stability of homogeneous equilibria in VP. 3 Strong instability and stability in the quasi-neutral limit ( d ✏ 1 q . 4 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 2 / 25
Outline Introduction to the problem 1 The existing mathematical literature on the quasi neutral limit. 2 The stability of homogeneous equilibria in VP. 3 Strong instability and stability in the quasi-neutral limit ( d ✏ 1 q . 4 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 2 / 25
Introduction to the problem Section 1 Introduction to the problem M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 3 / 25
Introduction to the problem The Debye (- H¨ uckel) length. Debye (- H¨ uckel) length : The scale of ”charge separation”, plasma oscillations. ✡ 1 ✂ ε 0 k B T 2 λ D : ✏ j ρ 0 j Z 2 ➦ j e 2 Relatively small ( with respect to typical length ) in many physical situation. From a course by Kip Thorne at Caltech. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 3 / 25
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 e ♣ ρ e ✁ ρ 0 q . ε 0 Assume that the e ✁ are at thermal equilibrium with large temperature : Ze Φ ➔➔ k B T Z e Φ ♣ x q ✓ ρ 0 � ρ 0 Z e Φ ♣ x q ρ e ♣ x q ✏ ρ 0 e kB T . k B T We end up with the linearised Poisson-Boltzman equation ✁ ε 0 k B T ✠ Φ ✏ λ ✁ 2 ∆Φ ✏ D Φ . ρ 0 Z 2 e 2 ñ Φ varies at the scale λ D . M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 4 / 25
Introduction to the problem More rigorously : the nondimensionalization of Vlasov-Poisson equation. Start from the Vlasov-Poisson eq. for the density f ♣ t , x , v q of e ✁ (fixed ions background) ❇ f ❇ t � v ☎ ❇ f ❇ v � e ❇ Φ ❇ x ☎ ❇ f ❇ v ✏ 0 , m e ∆Φ ✏ e ♣ ρ e ✁ ρ 0 q with . ε 0 Introduce the typical scales and associated new variables without dimension ( with prime ) n 0 f ✶ ♣ t , x ✶ , v ✶ q dx ✶ dv ✶ ✏ f ♣ t , x , v q dxdv t ✏ Tt ✶ , x ✏ Lx ✶ , v ✏ V th v ✶ , n 0 number of moles at size L , i.e. ρ 0 ✏ n 0 L d . Also assume V th T ✏ L . This leads to the nondimensional equation ❇ f ✶ ❇ t ✶ � v ✶ ☎ ❇ f ✶ ❇ v ✶ � ❇ Φ ✶ ❇ x ✶ ☎ ❇ f ✶ ❇ v ✶ ✏ 0 , λ 2 L 2 ∆Φ ✏ ρ ✶ ✁ 1 . D with D ✏ ε 0 m e V 2 The important parameter is the ratio ε ✏ λ D Again λ 2 th . . ρ 0 e 2 L M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 5 / 25
Introduction to the problem The related Plasma oscillations, a.k.a.“Langmuir Waves”. Rewrite the previous system (for convenience) as ❇ t f ε � v ☎ ❇ x f ε ✁ ❇ x Φ ε ☎ ❇ v f ε ✏ 0 , ✁ ε 2 ∆Φ ε ✏ ρ ε ✁ 1 . with The energy is E ε r f ε s : ✏ 1 ➺ v 2 f ε dxdv � 1 ➺ ✞ ✞ 2 dx . ✞ ✞ ∇ r ε Φ ε s 2 2 f ε v dv in divergence free j d ➩ Decompose the current j ε ✏ ε and gradient part ❇ x J ε . The equations for J ε and ε Φ ε are ❇ t r ε Φ ε s ✏ ✁ J ε ε ❇ t J ε ✏ ε Φ ε ✂ ✡ � 1 ➺ � ∆ ✁ 1 ddiv 2 ⑤ ε ∇ Φ ε ⑤ 2 r ε ∇ x Φ ε s ❜ r ε ∇ x Φ ε s ✁ f ε v ❜ v dv ε ❇ t O ε ✏ i Setting O ε ✏ J ε � i ε Φ ε , ε O ε � something of order one . ñ Strong oscillations of period 2 π ε in Φ ε , J ε and also ρ ε . M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 6 / 25
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 1995. Record form Freja plasma wave instrument ( alt. 1700 km). M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 7 / 25
Introduction to the problem Experimental observation of Langmuir Waves in plasma. From Matlis, Downer & all, University of Texas and Michigan, Nature Phys 2006. M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 8 / 25
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 ❇ t f � v ☎ ❇ x f ✁ ❇ x Φ ☎ ❇ 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 ε v ❜ v dv . This is correct only if ρ ♣ 0 q ✏ 1 and J ♣ 0 q ✏ 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 2013 9 / 25
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 2013 10 / 25
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) ➺ ρ ε f ε ♣ t , x , v q ✏ θ ♣ t , x q δ v ε θ ♣ t , x q ♣ v q µ ♣ d θ q . The family ♣ ρ θ , v θ q θ satifies coupled Euler-Poisson ❇ t ρ ε θ � div ♣ ρ ε θ v ε ❇ t v ε θ � ♣ v ε θ ☎ ∇ q v ε θ q ✏ 0 , θ ✏ ✁ ∇ V , ➺ ρ ε ∆ V ε ✏ θ µ ♣ d θ q ✁ 1 The expected limit model : coupled incompressible Euler equation : ❇ t ρ θ � div ♣ ρ θ v θ q ✏ 0 , ❇ t v θ � ♣ v θ ☎ ∇ q v θ ✏ ✁ ∇ p , ➺ ρ ε θ µ ♣ d θ q ✏ 1 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 10 / 25
The existing mathematical literature on the quasi neutral limit. Grenier: : convergence after filtration of the Plasma oscillations. Theorem (Grenier, CPDE ’96) Assume that θ q ǫ,θ satisfies uniform H s estimates (s large). the family ♣ ρ ǫ θ , v ǫ ε V ε ♣ 0 q 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 ǫ defined by J ε ♣ t , x q ✏ Re e i t ✏ ε U ♣ t , x q ✘ , and U is solution of ✁➺ ✠ U 0 ✏ J 0 � iV 0 , ❇ t U � ρ θ v θ µ ♣ d θ q ☎ ∇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 2013 11 / 25
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 ¯ v ♣ t , x q ➺ ⑤ v ✁ j 0 ♣ x q⑤ 2 f ε ♣ 0 , x , v q dxdv . f ε ♣ 0 , x , v q á δ j 0 ♣ x q ♣ v q i.e. We denote j 0 ✏ ν 0 � ∇ J 0 , with div j 0 ✏ 0, and ♣✏ ∆ ✁ 1 ρ ǫ ♣ 0 q ✁ 1 H 1 . V 0 ✏ lim ε Ñ 0 ε V ǫ ♣ 0 q q in ε First result in well prepared case [Brenier, CPDE ’00] : Theorem 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” u ♣ t q ✏ 1 ➺ ⑤ v ✁ u ♣ t , x q⑤ 2 f ε dxdv � 1 ➺ ⑤ ε ∇ V ε ⑤ 2 dx E ε 2 2 M. Hauray (UAM) Quasi-neutral limit Porquerolles, June 2013 12 / 25
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