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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Nonlinear stability of compressible vortex sheets in two space dimensions J.-F. Coulombel (Lille) P. Secchi (Brescia) CNRS, and Team SIMPAF of INRIA

SLIDE 1

The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion

Nonlinear stability of compressible vortex sheets in two space dimensions J.-F. Coulombel (Lille)

P. Secchi (Brescia)

CNRS, and Team SIMPAF of INRIA Futurs

HYP2006, Lyon, July 17th

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Euler equations of isentropic gas dynamics

We consider a compressible inviscid fluid described by: its density ρ(t, x) ∈ R+, its velocity field u(t, x) ∈ Rd, whose evolution is governed by the isentropic Euler equations:

∂tρ + ∇x · ρ u = 0 ,

∂tρ u + ∇x · ρ u ⊗ u + ∇x p(ρ) = 0 , where t ≥ 0 is the time variable, x ∈ Rd is the space variable, p is the pressure law.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Hyperbolicity, smooth solutions

If p′(ρ) > 0, the Euler equations form a symmetrizable hyperbolic system (convex entropy). This allows to solve (locally) the Cauchy problem: Existence, uniqueness of smooth solutions (in the space C([0, T]; Hs(Rd)), s > 1 + d/2). [Kato, 1975]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Hyperbolicity, smooth solutions

If p′(ρ) > 0, the Euler equations form a symmetrizable hyperbolic system (convex entropy). This allows to solve (locally) the Cauchy problem: Existence, uniqueness of smooth solutions (in the space C([0, T]; Hs(Rd)), s > 1 + d/2). [Kato, 1975] Blow-up of smooth solutions. [Sideris, 1985]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Rankine-Hugoniot jump conditions

A function: (ρ, u) =

(ρ+, u+)(t, x)

if xd > ϕ(t, y), (ρ−, u−)(t, x) if xd < ϕ(t, y), is a weak solution if

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Rankine-Hugoniot jump conditions

A function: (ρ, u) =

(ρ+, u+)(t, x)

if xd > ϕ(t, y), (ρ−, u−)(t, x) if xd < ϕ(t, y), is a weak solution if it solves the Euler equations away from the interface {xd = ϕ(t, y)}, and

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Rankine-Hugoniot jump conditions

A function: (ρ, u) =

(ρ+, u+)(t, x)

if xd > ϕ(t, y), (ρ−, u−)(t, x) if xd < ϕ(t, y), is a weak solution if it solves the Euler equations away from the interface {xd = ϕ(t, y)}, and the Rankine-Hugoniot jump conditions hold: ρ+ (u+ · n − σ) = ρ− (u− · n − σ) = j , j (u+ − u−) + (p(ρ+) − p(ρ−)) n = 0 . Free boundary problem !

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Existence results

Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Existence results

Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [M´ etivier, 1986]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Existence results

Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [M´ etivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Existence results

Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [M´ etivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989] Existence of sonic waves. [M´ etivier, 1991] [Sabl´ e-Tougeron, 1993]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion The equations Smooth solutions Piecewise smooth solutions

Existence results

Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [M´ etivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989] Existence of sonic waves. [M´ etivier, 1991] [Sabl´ e-Tougeron, 1993] Existence of one small shock wave. [Francheteau-M´ etivier, 2000]

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Jump conditions for a contact discontinuity

In the case j = 0, there is no mass transfer across the discontinuity. The Rankine-Hugoniot jump conditions reduce to: ∂tϕ = u+ · n = u− · n , p(ρ+) = p(ρ−) .

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Jump conditions for a contact discontinuity

In the case j = 0, there is no mass transfer across the discontinuity. The Rankine-Hugoniot jump conditions reduce to: ∂tϕ = u+ · n = u− · n , p(ρ+) = p(ρ−) . In this case, the weak solution is a contact discontinuity (associated with a linearly degenerate field). The front {xd = ϕ(t, y)} is characteristic with respect to either side. Jump of tangential velocity ⇒ vortex sheet.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear spectral stability (Landau, Miles...)

Consider a piecewise constant vortex sheet: (ρ, u) =

(ρ, v, 0)

if xd > 0, (ρ, −v, 0) if xd < 0, and linearize the Euler equations, and jump conditions around this solution.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear spectral stability (Landau, Miles...)

Consider a piecewise constant vortex sheet: (ρ, u) =

(ρ, v, 0)

if xd > 0, (ρ, −v, 0) if xd < 0, and linearize the Euler equations, and jump conditions around this solution. If d = 3, the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear spectral stability (Landau, Miles...)

Consider a piecewise constant vortex sheet: (ρ, u) =

(ρ, v, 0)

if xd > 0, (ρ, −v, 0) if xd < 0, and linearize the Euler equations, and jump conditions around this solution. If d = 3, the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability. If d = 2, and |v| < √ 2 c(ρ), the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear spectral stability (Landau, Miles...)

Consider a piecewise constant vortex sheet: (ρ, u) =

(ρ, v, 0)

if xd > 0, (ρ, −v, 0) if xd < 0, and linearize the Euler equations, and jump conditions around this solution. If d = 3, the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability. If d = 2, and |v| < √ 2 c(ρ), the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability. If d = 2, and |v| > √ 2 c(ρ), the linearized equations satisfy the weak Lopatinskii condition ⇒ weak stability.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Main result

Theorem

Let d = 2, and consider a piecewise constant weakly stable vortex

sheet. Let T > 0, and µ ≥ 6. Consider initial data (ρ0, u0), ϕ0 that

are perturbations in Hµ+15/2(R2

+) × Hµ+8(R) of the piecewise

constant vortex sheet. If the perturbations are small, and if the compatibility conditions hold, then there exists a contact discontinuity on [0, T] with initial data (ρ0, u0), ϕ0. The solution belongs to Hµ(]0, T[×R2

+) × Hµ+1(]0, T[×R).

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Linear stability result

Under the assumptions of the Theorem, consider the linearized equations around a small perturbation of the piecewise constant vortex

sheet. Assume that the Rankine-Hugoniot jump conditions are

satisfied by this perturbation.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear stability result

Under the assumptions of the Theorem, consider the linearized equations around a small perturbation of the piecewise constant vortex

sheet. Assume that the Rankine-Hugoniot jump conditions are

satisfied by this perturbation. Then the linearized equations LV = f , B(V, ψ) = g , satisfy an a priori tame estimate: V Hm(]0,T[×R2

+) + ψHm+1(]0,T[×R)

≤ C

fHm+1(]0,T[×R2

+) + gHm+1(]0,T[×R)

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Linear stability result

Under the assumptions of the Theorem, consider the linearized equations around a small perturbation of the piecewise constant vortex

sheet. Assume that the Rankine-Hugoniot jump conditions are

satisfied by this perturbation. Then the linearized equations LV = f , B(V, ψ) = g , satisfy an a priori tame estimate: V Hm(]0,T[×R2

+) + ψHm+1(]0,T[×R)

≤ C

fHm+1(]0,T[×R2

+) + gHm+1(]0,T[×R)

No loss of normal regularity.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

SLIDE 29

The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Plan

1 The Cauchy problem for the Euler equations

The equations Smooth solutions Piecewise smooth solutions

2 Stability of contact discontinuities

Main result Linear stability Nonlinear stability

3 Conclusion

Nash-Moser iteration

To solve the nonlinear equations, we use a Nash-Moser iteration where we force the Rankine-Hugoniot jump conditions at each step:

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Nash-Moser iteration

To solve the nonlinear equations, we use a Nash-Moser iteration where we force the Rankine-Hugoniot jump conditions at each step: Start from an approximate solution.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Nash-Moser iteration

To solve the nonlinear equations, we use a Nash-Moser iteration where we force the Rankine-Hugoniot jump conditions at each step: Start from an approximate solution. Regularize the coefficients, and force the Rankine-Hugoniot conditions.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Nash-Moser iteration

To solve the nonlinear equations, we use a Nash-Moser iteration where we force the Rankine-Hugoniot jump conditions at each step: Start from an approximate solution. Regularize the coefficients, and force the Rankine-Hugoniot conditions. Solve the linearized equations, for well-chosen source terms.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Main result Linear stability Nonlinear stability

Nash-Moser iteration

To solve the nonlinear equations, we use a Nash-Moser iteration where we force the Rankine-Hugoniot jump conditions at each step: Start from an approximate solution. Regularize the coefficients, and force the Rankine-Hugoniot conditions. Solve the linearized equations, for well-chosen source terms. Regularize the new coefficients, and force the Rankine-Hugoniot conditions etc.

J.-F. Coulombel (Lille), P. Secchi (Brescia)

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The Cauchy problem for the Euler equations Stability of contact discontinuities Conclusion Related problems