Plasma-vacuum interface problem Analysis of linearized stability Nonlinear stability Stability of the free plasma-vacuum interface Paolo Secchi Department of Mathematics Brescia University, Italy (paolo.secchi @ ing.unibs.it) Joint work with Y. Trakhinin HYP 2012, 14th International Conference on Hyperbolic Problems Padova, June 25-29, 2012 Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Analysis of linearized stability Nonlinear stability Plan 1 Plasma-vacuum interface problem Formulation of the problem Reduction to the fixed domain 2 Analysis of linearized stability Linearized stability in H 1 Proof Hyperbolic regularization Secondary symmetrization High-order energy estimate 3 Nonlinear stability Nash-Moser technique Main result Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability Ideal compressible MHD Consider the ideal compressible MHD equations: ∂ t ρ + ∇ · ( ρ v ) = 0 , ∂ t ( ρ v ) + ∇ · ( ρ v ⊗ v − H ⊗ H ) + ∇ ( p + 1 2 | H | 2 ) = 0 , ∂ t H − ∇ × ( v × H ) = 0 , � � (1) 2 ( ρ | v | 2 + | H | 2 ) ρe + 1 ∂ t � � ρv ( e + 1 2 | v | 2 ) + vp + H × ( v × H ) + ∇ · = 0 , ∇ · H = 0 , with ρ density, S entropy, v velocity field, H magnetic field, p = p ( ρ, S ) pressure (such that p ′ ρ > 0 ), e = e ( ρ, S ) internal energy. Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability The total pressure is q = p + 1 2 | H | 2 . In terms of U = ( q, v, H, S ) T system (1) admits the symmetrization 0 T 0 1 0 1 ρ p /ρ − ( ρ p /ρ ) H 0 q 0 ρI 3 0 3 0 v B C B C A ∂ t A + B − ( ρ p /ρ ) H T C B C 0 3 I 3 + ( ρ p /ρ ) H ⊗ H 0 H @ @ 0 T 0 T 0 1 S 0 1 0 1 ( ρ p /ρ ) v · ∇ ∇· − ( ρ p /ρ ) Hv · ∇ 0 q ∇ ρv · ∇ I 3 − H · ∇ I 3 0 v B C B C A = 0 − ( ρ p /ρ ) H T v · ∇ B C B C − H · ∇ I 3 ( I 3 + ( ρ p /ρ ) H ⊗ H ) v · ∇ 0 H @ A @ 0 T 0 T 0 v · ∇ S (2) where 0 = (0 , 0 , 0) T . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability We write system (2) as 3 � A 0 ( U ) ∂ t U + A j ( U ) ∂ j U = 0 , (3) j =1 which is symmetric hyperbolic provided the hyperbolicity condition A 0 > 0 holds: ρ > 0 , ρ p > 0 . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability Given a smooth hypersurface Γ( t ) = { x 3 = f ( t, x ′ ) } in [0 , T ] × R 3 , we denote Ω ± ( t ) = R 3 ∩ { x 3 ≷ f ( t, x ′ ) } (here x ′ = ( x 1 , x 2 )) . The plasma is governed by equations (3) in the region Ω + ( t ) = R 3 ∩ { x 3 > f ( t, x ′ ) } . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability x 3 Ω + ( t ) plasma x 3 = f ( t, x ′ ) Γ x ′ Ω − ( t ) vacuum Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability The vacuum region is Ω − ( t ) = R 3 ∩ { x 3 < f ( t, x ′ ) } , where we assume the so-called pre-Maxwell dynamics : ∇ × H = 0 , div H = 0 , (4) ∇ × E = − ∂ t H , div E = 0 , (5) H denotes the vacuum magnetic field and E the electric field. As usual in nonrelativistic MHD, we neglect the displacement current (1 /c ) ∂ t E , where c is the speed of light. From (5) the electric field E is a secondary variable that may be computed from the magnetic field H . Hence, in the vacuum only one basic variable is needed, viz. H , satisfying (4). Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability On the moving interface Γ( t ) the plasma and the vacuum magnetic fields are related by: ∂ t f = v N , [ q ] = 0 , H N = 0 , H N = 0 on Γ( t ) , (6) where v N = v · N , H N = H · N , H N = H · N , N = ( − ∂ 1 f, − ∂ 2 f, 1) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ . The interface Γ( t ) moves with the plasma. The total pressure is continuous across Γ( t ) . The magnetic field on both sides is tangent to Γ( t ) . The function f describing the interface is one unknown of the problem, i.e. this is a free boundary problem. Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability System (4) for the vacuum magnetic field H , ∇ × H = 0 , div H = 0 , (4) is elliptic. Plasma-vacuum problem (3), (4) is a coupled hyperbolic-elliptic system. In (4) time t plays the role of a parameter. Time dependence of H comes from the coupling with the plasma variables through the boundary conditions (6) at the moving front Γ( t ) . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability System (4) for the vacuum magnetic field H , ∇ × H = 0 , div H = 0 , (4) is elliptic. Plasma-vacuum problem (3), (4) is a coupled hyperbolic-elliptic system. In (4) time t plays the role of a parameter. Time dependence of H comes from the coupling with the plasma variables through the boundary conditions (6) at the moving front Γ( t ) . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability System (3), (4), (6) is supplemented with initial conditions x ∈ Ω ± (0) , U (0 , x ) = U 0 ( x ) , H (0 , x ) = H 0 ( x ) , (7) x ′ ∈ Γ(0) , f (0 , x ′ ) = f 0 ( x ′ ) , where div H 0 = 0 in Ω + (0) , div H 0 = 0 in Ω − (0) . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability Motivation from astrophysics: the study of stars or the solar corona Image by Luc Viatour. From Yohkoh satellite (Courtesy by JAXA) Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability Other motivation: the study of magnetic confinement USSR stamp 1987 Tunnel at Monte Carlo (Courtesy by JET) Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability A toroidal plasma configuration: (a) surrounded by a perfectly conducting wall; (b) isolated from a wall by a vacuum region. Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability The stability condition Our goal is to prove the solvability of (3), (4), (6), (7) under the stability condition | H × H| > 0 on [0 , T ] × Γ , (8) i.e. the magnetic fields on the two sides of the free-boundary are not collinear. x 3 plasma H θ 0 < θ < π H x 2 vacuum x 1 Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability Reduction to the fixed domain x 3 y 3 Ω + ( t ) Ω + Γ( t ) ˜ Φ − 1 ( t, · ) Γ = { y 3 = 0 } x ′ y ′ Ω − ( t ) Ω − Change of variables ˜ Φ( t, · ) : y = ( y ′ , y 3 ) → x = ( x ′ , x 3 ) such that x ′ = y ′ , x 3 = ˜ Φ( t, y ) , ˜ ∂ y 3 ˜ Φ( t, y ′ , 0) = f ( t, x ′ ) , Φ( t, y ) > 0 . Paolo Secchi Plasma-vacuum interface
Plasma-vacuum interface problem Formulation of the problem Analysis of linearized stability Reduction to the fixed domain Nonlinear stability We write again x instead of y . Possible choice: Φ( t, x ′ , x 3 ) = x 3 + f ( t, x ′ ) ˜ [Majda, Proc. AMS 1983], [M´ etivier, 2003] for uniformly stable shocks. We consider a different change of variables, inspired from [Lannes, JAMS 2005]. Paolo Secchi Plasma-vacuum interface
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