Plasma-Vacuum interface problem Linearization Main Result Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD Paola Trebeschi Department of Mathematics, University of Brescia (paola.trebeschi @ ing.unibs.it) Joint work with A. Morando, Y. Trakhinin “ 14 th International Conference on Hyperbolic Problems: Theory, Numerics, Applications”, Padova, June 25-29, 2012 Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Linearization Main Result Plan 1 Plasma-Vacuum interface problem Formulation of the problem The equations Goal of the work 2 Linearization Reduction to the fixed domain Linearized problem 3 Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Plan 1 Plasma-Vacuum interface problem Formulation of the problem The equations Goal of the work 2 Linearization Reduction to the fixed domain Linearized problem 3 Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Plan 1 Plasma-Vacuum interface problem Formulation of the problem The equations Goal of the work 2 Linearization Reduction to the fixed domain Linearized problem 3 Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Plasma-Vacuum interface It is a free boundary value problem. In the classical Plasma-Vacuum interface: the plasma is confined inside a perfectly conducting rigid wall and isolated from it by a vacuum region. Ω + ( t ) := Plasma region Ω − ( t ) := Vacuum region Γ( t ) := Boundary of Ω + ( t ) = { η ( t, x ) = 0 } : the Interface between plasma and vacuum. It is to be determined and moves with the velocity of plasma particles at the boundary, i.e. ∂ t η + ( v, ∇ η ) = 0 on Γ( t ) . (1) Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work For technical simplicity: Ω ± ( t ) are unbounded in R 3 Γ( t ) has the form of a graph: Γ( t ) := { x 1 = ϕ ( t, x ′ ) } , x ′ = ( x 2 , x 3 ) Ω + ( t ) := { x 1 > ϕ ( t, x ′ ) } , Ω − ( t ) := { x 1 < ϕ ( t, x ′ ) } . With the choice η := x 1 − ϕ ( t, x ′ ) equation (1) becomes ∂ t ϕ = ( v, N ) on Γ( t ) , where N = ∇ η = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) . Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work x 1 Ω + ( t ) Plasma Γ( t ) x 1 = ϕ ( t, x ′ ) x ′ Ω − ( t ) Vacuum Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Plan 1 Plasma-Vacuum interface problem Formulation of the problem The equations Goal of the work 2 Linearization Reduction to the fixed domain Linearized problem 3 Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work In Ω + ( t ) : we consider the equations of ideal incompressible magneto-hydrodynamics (MHD), i.e., the equations governing the motion of a perfectly conducting inviscid incompressible plasma. In the case of homogeneous plasma the equations, in a dimensionless form, are ∂ t v + ( v · ∇ ) v − ( H · ∇ ) H + ∇ q = 0 , (2) ∂ t H + ( v · ∇ ) H − ( H · ∇ ) v = 0 , div v = 0 . with v = ( v 1 , v 2 , v 3 ) velocity field H = ( H 1 , H 2 , H 3 ) magnetic field 2 | H | 2 total pressure p pressure, q = p + 1 (for simplicity the density ρ ≡ 1 ) Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work As the unknown we fix the vector U := ( q, W ) , with W = ( v, H ) . System (2) is supplemented by the divergence constraint div H = 0 on the initial data W | t =0 = W 0 for the Cauchy problem in the whole space R 3 . Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work In Ω − ( t ) : we consider the elliptic (div-curl) system ∇ × H = 0 , div H = 0 , (3) H denotes the vacuum magnetic field. This system describes the so-called pre-Maxwell dynamics . That is, as usual in nonrelativistic MHD, we neglect the displacement current (1 /c ) ∂ t E , where c is the speed of light and E is the electric field. Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work On Γ( t ) : the plasma and the vacuum magnetic fields are related by: ∂ t ϕ = ( v, N ) , [ q ] = 0 , ( H, N ) = 0 , ( H , N ) = 0 , (4) where N = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ is the jump of the total pressure across the interface Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work On Γ( t ) : the plasma and the vacuum magnetic field are related by: ∂ t ϕ = ( v, N ) , [ q ] = 0 , ( H, N ) = 0 , ( H , N ) = 0 , where N = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ is the jump of the total pressure across the interface • The interface Γ( t ) moves with the plasma velocity. Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work On Γ( t ) : the plasma and the vacuum magnetic field are related by: ∂ t ϕ = ( v, N ) , [ q ] = 0 , ( H, N ) = 0 , ( H , N ) = 0 , where N = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ is the jump of the total pressure across the interface • The interface Γ( t ) moves with the plasma velocity. • The total pressure is continuous across Γ( t ) . Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work On Γ( t ) : the plasma and the vacuum magnetic field are related by: ∂ t ϕ = ( v, N ) , [ q ] = 0 , ( H, N ) = 0 , ( H , N ) = 0 , where N = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ is the jump of the total pressure across the interface • The interface Γ( t ) moves with the plasma velocity. • The total pressure is continuous across Γ( t ) . • The magnetic field on both sides is tangent to Γ( t ) . Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work On Γ( t ) : the plasma and the vacuum magnetic field are related by: ∂ t ϕ = ( v, N ) , [ q ] = 0 , ( H, N ) = 0 , ( H , N ) = 0 , where N = (1 , − ∂ 2 ϕ, − ∂ 3 ϕ ) , and [ q ] = q | Γ − 1 2 |H| 2 | Γ is the jump of the total pressure across the interface • The interface Γ( t ) moves with the plasma velocity. • The total pressure is continuous across Γ( t ) . • The magnetic field on both sides is tangent to Γ( t ) . The function ϕ describing the interface is one of the unknowns of the problem, i.e. this is a free boundary problem . Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work To summarize: we consider in Ω + ( t ) Incompressible (MHD) in Ω − ( t ) pre-Maxwell on Γ ( t ) Boundary conditions It is supplemented with initial conditions x ∈ Ω ± (0) , W (0 , x ) = W 0 ( x ) , H (0 , x ) = H 0 ( x ) , x ′ ∈ Γ(0) . ϕ (0 , x ′ ) = ϕ 0 ( x ′ ) , Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Plan 1 Plasma-Vacuum interface problem Formulation of the problem The equations Goal of the work 2 Linearization Reduction to the fixed domain Linearized problem 3 Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part Paola Trebeschi Incompressible MHD
Plasma-Vacuum interface problem Formulation of the problem Linearization The equations Main Result Goal of the work Goal: Linearization of the Plasma-Vacuum interface problem around a non constant piecewise smooth reference state. Well posedness of the linearized Plasma-Vacuum interface problem. This is the first step in order to study the well posedness of the non linear Plasma-Vacuum problem. Paola Trebeschi Incompressible MHD
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