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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 “14th 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 Linearization Main Result Formulation of the problem The equations 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 Linearization Main Result Formulation of the problem The equations 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 Linearization Main Result Formulation of the problem The equations 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. ∂tη + (v, ∇η) = 0

• n Γ(t).

(1)

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations Goal of the work

For technical simplicity: Ω±(t) are unbounded in R3 Γ(t) has the form of a graph: Γ(t) := {x1 = ϕ(t, x′)}, x′ = (x2, x3) Ω+(t) := {x1 > ϕ(t, x′)}, Ω−(t) := {x1 < ϕ(t, x′)}. With the choice η := x1 − ϕ(t, x′) equation (1) becomes ∂tϕ = (v, N)

• n Γ(t),

where N = ∇η = (1, −∂2ϕ, −∂3ϕ).

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations Goal of the work

x′ x1

x1 = ϕ(t, x′)

Γ(t) Ω+(t) Ω−(t) Plasma Vacuum

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations 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 Linearization Main Result Formulation of the problem The equations 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      ∂tv + (v · ∇)v − (H · ∇)H + ∇q = 0 , ∂tH + (v · ∇)H − (H · ∇)v = 0 , div v = 0 . (2) with v = (v1, v2, v3) velocity field H = (H1, H2, H3) magnetic field p pressure, q = p + 1

2|H|2 total pressure

(for simplicity the density ρ ≡ 1)

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations 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

• n the initial data W|t=0 = W0 for the Cauchy problem in the whole

space R3.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations 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) ∂tE, where c is the speed of light and E is the electric field.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations Goal of the work

On Γ(t): the plasma and the vacuum magnetic fields are related by: ∂tϕ = (v, N), [q] = 0, (H, N) = 0, (H, N) = 0, (4) where N = (1, −∂2ϕ, −∂3ϕ), 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 Linearization Main Result Formulation of the problem The equations Goal of the work

On Γ(t): the plasma and the vacuum magnetic field are related by: ∂tϕ = (v, N), [q] = 0, (H, N) = 0, (H, N) = 0, where N = (1, −∂2ϕ, −∂3ϕ), 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 Linearization Main Result Formulation of the problem The equations Goal of the work

On Γ(t): the plasma and the vacuum magnetic field are related by: ∂tϕ = (v, N), [q] = 0, (H, N) = 0, (H, N) = 0, where N = (1, −∂2ϕ, −∂3ϕ), 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 Linearization Main Result Formulation of the problem The equations Goal of the work

On Γ(t): the plasma and the vacuum magnetic field are related by: ∂tϕ = (v, N), [q] = 0, (H, N) = 0, (H, N) = 0, where N = (1, −∂2ϕ, −∂3ϕ), 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 Linearization Main Result Formulation of the problem The equations Goal of the work

On Γ(t): the plasma and the vacuum magnetic field are related by: ∂tϕ = (v, N), [q] = 0, (H, N) = 0, (H, N) = 0, where N = (1, −∂2ϕ, −∂3ϕ), 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 Linearization Main Result Formulation of the problem The equations Goal of the work

To summarize: we consider    Incompressible (MHD) in Ω+(t) pre-Maxwell in Ω−(t) Boundary conditions

• n Γ(t)

It is supplemented with initial conditions W(0, x) = W0(x) , H(0, x) = H0(x) , x ∈ Ω±(0) , ϕ(0, x′) = ϕ0(x′) , x′ ∈ Γ(0).

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations 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 Linearization Main Result Formulation of the problem The equations 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

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations Goal of the work

Known results

Trakhinin [J. Differential Equations, 2010]: A priori estimate in the anisotropic Sobolev space H1

∗ for the linearized

Plasma-Vacuum interface problem in compressible MHD, under a ”stability condition” of ”non collinearity” of the magnetic fields |H × H| ≥ δ > 0,

• n Γ(t);

(5) Secchi & Trakhinin [2012]: Well posedness in H1

∗ of the linearized

Plasma-Vacuum interface problem in compressible MHD, under the ”stability condition” (5) Morando, Trakhinin & T. [2012] : Well posedness in Sobolev space H1 of the linearized Plasma-Vacuum interface problem in incompressible MHD, under the ”stability condition” (5).

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Formulation of the problem The equations Goal of the work

Stability condition

x2 x1 x3 H H x1 = ϕ(t, x′) Γ(t) Plasma Vacuum

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

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 Linearization Main Result Reduction to the fixed domain Linearized problem

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 Linearization Main Result Reduction to the fixed domain Linearized problem

Reduction to the fixed domain

Lagrangian coordinates

• incompressible Euler equations in vacuum [Coutand & Shkoller,

JAMS 2007],

• compressible Euler equations in vacuum [Coutand, Lindblad &

Shkoller, CMP 2010], [Coutand & Shkoller, Preprint 2010]; Change of variables

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

Change of variables

Construct a global diffeomorphism of R3 Φ(t, x) := (Φ1(t, x), x′) mapping Ω±(t) and Γ(t) onto Ω± := R3 ∩ {x1 ≷ 0}, Γ := R3 ∩ {x1 = 0} x′ x1 Φ(t, ·) Γ(t) Ω+(t) Ω−(t) Plasma Vacuum x′ x1 Ω+ Ω− Γ Plasma Vacuum

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

Possible choices for Φ1: Φ1(t, x1, x′) = x1 + ϕ(t, x′) [Majda, PAMS 1983], [M´ etivier, 2003] for uniformly stable shocks. We consider a different change of variables, inspired from [Lannes, JAMS 2005] Φ1(t, x1, x′) = x1 + Ψ(t, x1, x′) where Ψ(t, x1, x′) is a suitable lifting of ϕ(t, x′).

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

We introduce the change of independent variables Φ(t, x) = (Φ1(t, x), x′) = (x1 + Ψ(t, x), x′) Define

• U(t, x) := U(t, Φ(t, x)),
• H(t, x) := H(t, Φ(t, x)).

Dropping for convenience tildes in U and H, Plasma-Vacuum problem

• n the fixed reference domains Ω± becomes

P(U, Ψ) = 0 in [0, T] × Ω+, V(H, Ψ) = 0 in [0, T] × Ω−, (6) B(U, H, ϕ) = 0

• n [0, T] × Γ,

(7) (W, H)|t=0 = (W0, H0) in Ω+ × Ω−, ϕ|t=0 = ϕ0

• n Γ

(8)

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

P(U, Ψ) := L(U, Ψ) divu

• = 0

in [0, T] × Ω+ , with L(U, Ψ) := L1(W, Ψ)W + ∇Φq

• ,

L1(W, Ψ) := ∂t + A1(W, Ψ)∂1 + A2(W)∂2 + A3(W)∂3,

• A1(W, Ψ) :=

1 ∂1Φ1

• A1(W) −

3

• k=2

Ak(W)∂kΨ − I6∂tΨ

• ,

Ak(W) :=

• vkI3

−HkI3 −HkI3 vkI3

• ,

W := v H

• ,

∇Φq := ∂1q ∂1Φ1 , − ∂2Ψ ∂1Φ1 ∂1q + ∂2q, − ∂3Ψ ∂1Φ1 ∂1q + ∂3q

• ,

u := (vn, v2∂1Φ1, v3∂1Φ1), vn := v1 − v2∂2Ψ − v3∂3Ψ

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

V(H, Ψ) := ∇ × H div h

• = 0

in [0, T] × Ω−, H := (H1∂1Φ1, Hτ2, Hτ3), h := (Hn, H2∂1Φ1, H3∂1Φ1), Hτk := H1∂kΨ + Hk, k = 2, 3, Hn := H1 − H2∂2Ψ − H3∂3Ψ and B(U, H, ϕ) :=   ∂tϕ − vN [q] HN   = 0

• n [0, T] × Γ,

[q] := q|Γ − 1 2|H|2

|Γ,

vN := (v, N) , HN := (H, N) .

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

We also did not include in our problem the equation div h = 0 in [0, T] × Ω+ (h := (Hn, H2∂1Φ1, H3∂1Φ1), with Hn = H1 − H2∂2Ψ − H3∂3Ψ) and the boundary condition HN = 0

• n [0, T] × Γ

because they are just restrictions on the initial data.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

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 Linearization Main Result Reduction to the fixed domain Linearized problem

Basic state

For T > 0, let us set Q±

T := (−∞, T] × Ω± ,

ωT := (−∞, T] × Γ . Let ( U(t, x), H(t, x), ˆ ϕ(t, x′)) be a given sufficiently smooth vector-function, respectively defined on Q+

T , Q− T , ωT , with

U = (ˆ q, ˆ v, H). Corresponding to ϕ, let the function Ψ and the diffeomorphism Φ be constructed above, such that ∂1 Φ1 ≥ 1/2 .

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

We assume that the basic state satisfies ∂t H +

1 ∂1 Φ1

• (

w, ∇) H − (ˆ h, ∇)ˆ v

• = 0,

div ˆ u = 0 in Q+

T ,

div ˆ h = 0 in Q−

T ,

∂t ˆ ϕ − ˆ v ˆ

N = 0,

[ˆ q] = 0,

• H ˆ

N = 0

• n ωT ,

where all the “hat” values are determined like corresponding values for (U, H, ϕ), and where ˆ u := (ˆ vˆ

n, ˆ

v2∂1 Φ1, ˆ v3∂1 Φ1),

• vˆ

n := (ˆ

v, ˆ n) , ˆ n := (1, −∂2 Ψ, −∂3 Ψ)

• w := ˆ

u − (∂t Ψ, 0, 0) . It follows that div ˆ h = 0 in Q+

T ,

• H ˆ

N = 0 on ωT

are satisfied for the basic state if they hold at t = 0.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

Linearized problem

Linearizing around the basic state we obtain the linearized problem ∂tW + 3

j=1

Aj∂jW + CW + ∇

Φq

• =

fv fH

• ,

div u = f7 , in Q+

T ,

∇ × H = χ , div h = Ξ , in Q−

T ,

∂tϕ = v ˆ

N − ˆ

v2∂2ϕ − ˆ v3∂3ϕ + ϕ∂1ˆ v ˆ

N + g1 ,

q = ( H, H) − [∂1ˆ q]ϕ + g2 , H ˆ

N = ∂2(

H2ϕ) + ∂3( H3ϕ) + g3 ,

• n ωT ,

(U, H, ϕ) = 0 , for t < 0 , where W = (v, H), A1 = A1( W, Ψ), A2,3 = A2,3( W), C = C( W) and div χ = 0 ,

• Ω− Ξ dx =
• Γ

g3 dx′ .

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Reduction to the fixed domain Linearized problem

Reduction to homogeneous data

We reduce the problem to that with homogeneous data (except fv) ∂tW + 3

j=1

Aj∂jW + CW + ∇

Φq

• =

fv

• ,

div u = 0 , in Q+

T ,

∇ × H = 0 , div h = 0 , in Q−

T ,

∂tϕ = v ˆ

N − ˆ

v2∂2ϕ − ˆ v3∂3ϕ + ϕ∂1ˆ v ˆ

N + 0 ,

q = ( H, H) − [∂1ˆ q]ϕ + 0 , H ˆ

N = ∂2(

H2ϕ) + ∂3( H3ϕ) + 0 ,

• n ωT ,

(U, H, ϕ) = 0 , for t < 0 , (9) where W = (v, H), A1 = A1( W, Ψ), A2,3 = A2,3( W), C = C( W)

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

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 Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Theorem (Morando, Trakhinin, T., 2012)

Let T > 0. Let the basic state satisfy the previous assumptions and | H × H| ≥ δ > 0

• n ωT

(δ fixed). Then ∃γ0 ≥ 1 such that ∀γ ≥ γ0 and ∀fv ∈ H1

γ(Q+ T ) vanishing for t < 0, the

problem (9) has a solution (U, H, ϕ) with (q, W, H, ϕ) ∈ ˙ H1(Q+

T ) × H1 γ(Q+ T ) × H1 γ(Q− T ) × H1 γ(ωT ) ( W := (v, H)),

γ

• ||Wγ||2

H1

γ(Q+ T ) + ||∇qγ||2

L2(Q+

T ) + ||Hγ||2

H1

γ(Q− T )

+|| (qγ, u1, γ, h1, γ, Hγ) |ωT ||2

H1/2

γ

(ωT )

• +γ2||ϕγ||2

H1

γ(ωT ) ≤ C

γ ||fv,γ||2

H1

γ(Q+ T ) ,

where we have set Gγ := e−γtG and where C = C(K, T, δ) > 0 is a constant independent of the data fv and the parameter γ. ˙ H1(Q+

T ) := {u ∈ L1 loc(Q+ T ) : ∇q ∈ L2(Q+ T )}

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

The stability condition on the basic state | H × H| ≥ δ > 0

• n ωT

allows to solve the system of the boundary conditions as an algebraic system for the gradient ∇t,x′ϕε = (∂tϕε, ∂2ϕε, ∂3ϕε) as ∇t,x′ϕε = F(U ε, nc|x1=0, Hε|x1=0, ϕε), hence, to estimate the front in terms of the trace.

x2 x1 x3

• H
• H

{x1 = 0} = Γ Plasma Vacuum

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Ideas for the proof

Well posedness in anisotropic Sobolev space H1

∗

Hyperbolic regularization Secondary symmetrization of the Vacuum part

Well posedness in Sobolev space H1 Using the divergence constraints and a current-vorticity type linearized system, we can estimate the missing normal derivatives

• f the velocity and the plasma magnetic field through conormal

derivatives and source term and prove the well posedness of the linearized problem in Sobolev spaces.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

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 Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Plasma part: we replace the incompressible MHD equations with their “compressible” counterpart, by introducing an evolution equation for the total pressure involving a small parameter ε which corresponds to the reciprocal of the sound speed in the fluid. Vacuum part: we consider a “hyperbolic” regularization of the elliptic system by introducing a new auxiliary unknown E which plays the role of the vacuum electric field, where the same small parameter of regularization ε as above is now associated with the physical parameter 1/c, being c the speed of light. New boundary conditions: we regularize the boundary condition for the total pressure and introduce two boundary conditions for the unknown E.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Plasma:

ε2

• ∂tqε − (∂t

H, Hε) − ( H, ∂tHε) + 1 ∂1 Φ1 ( ˆ w, ∇qε) − 1 ∂1 Φ1

• w, (∇

H, Hε)

• −

1 ∂1 Φ1

• w, (

H, ∇Hε)

• +

1 ∂1 Φ1 div uε = 0 , ∂tvε + 1 ∂1 Φ1

• ( ˆ

w, ∇)vε − (ˆ h, ∇)Hε + ∇

Φqε +

C1W ε = fv , ∂tHε + 1 ∂1 Φ1

• ( ˆ

w, ∇)Hε − (ˆ h, ∇)vε + C2W ε +

• H

∂1 Φ1 div uε = 0 in Q+

T ,

The above system, with ε = 1, looks like the linearized system of compressible isentropic MHD equations reduced to a dimensionless form.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Vacuum: ε ∂thε + ∇ × Eε = 0, ε ∂teε − ∇ × Hε = 0 in Q−

T ,

where Eε = (Eε

1, Eε 2, Eε 3),

Eε = (Eε

1∂1

Φ1, Eε

ˆ τ2, Eε ˆ τ3),

eε = (Eε

ˆ n, Eε 2∂1

Φ1, Eε

3∂1

Φ1), Eε

ˆ n := Eε 1 − Eε 2∂2

Ψ − Eε

3∂3

Ψ, Eε

ˆ τk := Eε 1∂k

Ψ + Eε

k, k = 2, 3 .

All the other notations for Hε (i.e. hε, Hε) are analogous. The above system, if ε = 1, coincide with the corrisponding one for the vacuum Maxwell equations.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Boundary conditions ∂tϕε = vε

ˆ N − ˆ

v2∂2ϕε − ˆ v3∂3ϕε + ϕε∂1ˆ v ˆ

N,

qε = ( H, Hε) − [∂1ˆ q]ϕε−ε( E, Eε) Eε

ˆ τ2 = ε ∂t(

H3ϕε) − ε ∂2( E1ϕε), Eε

ˆ τ3 = −ε ∂t(

H2ϕε) − ε ∂3( E1ϕε)

• n ωT ,

where E = ( E1, E2, E3) and the functions Ej will be chosen later on. The choice of the functions Ej will be crucial to make the boundary conditions dissipative.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Goal: Well posedness for the linearized compressible regularized problem:

prove the existence of the solution (it comes from [Secchi & Trakhinin, 2012] derive an a priori estimate for the ε-regularized problem with constant uniformly bounded in ε

This gives Well posedness for the linearized incompressible problem

For every ε there exists a solution (U ε, V ε) of the regularized problem. The uniform a-priori estimate of the ε-regularized problem ⇒ (U ε, V ε) ⇀ (U, V ) (up to subsequences) The weak limit (U, V ) gives a weak solution to the incompressible problem.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

To show the a priori estimate of the ε−regularized problem we need to perform the secondary symmetrization of the vacuum part

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

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 Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Let us define

• J :=

        1 ∂2 Ψ ∂1 Φ1 ∂3 Ψ ∂1 Φ1 1 ∂1 Φ1 1 ∂1 Φ1         . We consider

• J(∂thε + 1

ε∇ × Eε) + J(∂teε − 1 ε∇ × Hε) × εν + ν ∂1 Φ1 div hε = 0,

• J(∂teε − 1

ε∇ × Hε) − J(∂thε + 1 ε∇ × Eε) × εν + ν ∂1 Φ1 div eε = 0, where ν := (ν1, ν2, ν3) will be chosen in an appropriate way later on.

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

The secondary symmetrization allows (with the choice of ν and E) to treat the boundary terms in the energy estimates. ν1 = v1 = ˆ v2∂2 ˆ ϕ + ˆ v3∂3 ˆ ϕ, νk = ˆ vk, k = 2, 3 on ωT

• E := −v ×

H, v := (v1, ˆ v2, ˆ v3).

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Conclusion

Under the stability condition on the basic state | H × H| ≥ δ > 0

• n ωT

we prove the well-posedness of the linearized Plasma-Vacuum interface problem for Incompressible MHD system

x2 x1 x3

• H
• H

{x1 = 0} = Γ Plasma Vacuum

Paola Trebeschi Incompressible MHD

Plasma-Vacuum interface problem Linearization Main Result Hyperbolic regularization Secondary symmetrization of the vacuum part

Thanks!!!

Paola Trebeschi Incompressible MHD