Nonlinear stability of compressible vortex sheets in two space dimensions

Transcription

1 of compressible vortex sheets in two space dimensions J.-F. Coulombel (Lille) P. Secchi (Brescia) CNRS, and Team SIMPAF of INRIA Futurs Evolution Equations 2006, Mons, August 29th

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3 Plan Related problems

4 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) R d, whose evolution is governed by the isentropic Euler equations: { t ρ + x ρ u = 0, t ρ u + x ρ u u + x p(ρ) = 0, where t 0 is the time variable, x R d is the space variable, p is the pressure law.

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6 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 ]; H s (R d )), s > 1 + d/2). [Kato, 1975]

7 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 ]; H s (R d )), s > 1 + d/2). [Kato, 1975] Blow-up of smooth solutions. [Sideris, 1985]

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9 Rankine-Hugoniot jump conditions A function: (ρ, u) = { (ρ +, u + )(t, x) if x d > ϕ(t, y), (ρ, u )(t, x) if x d < ϕ(t, y), is a weak solution if

10 Rankine-Hugoniot jump conditions A function: (ρ, u) = { (ρ +, u + )(t, x) if x d > ϕ(t, y), (ρ, u )(t, x) if x d < ϕ(t, y), is a weak solution if it solves the Euler equations away from the interface {x d = ϕ(t, y)}, and

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

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

13 Existence results Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [Métivier, 1986]

14 Existence results Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [Métivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989]

15 Existence results Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [Métivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989] Existence of sonic waves. [Métivier, 1991] [Sablé-Tougeron, 1993]

16 Existence results Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] Existence of two uniformly stable shock waves. [Métivier, 1986] Existence of one rarefaction wave. [Alinhac, 1989] Existence of sonic waves. [Métivier, 1991] [Sablé-Tougeron, 1993] Existence of one small shock wave. [Francheteau-Métivier, 2000]

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18 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: t ϕ = u + n = u n, p(ρ + ) = p(ρ ).

19 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: t ϕ = 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 {x d = ϕ(t, y)} is characteristic with respect to either side. Jump of tangential velocity vortex sheet.

20 Linear spectral stability (Landau, Miles...) Consider a piecewise constant vortex sheet: { (ρ, v, 0) if x d > 0, (ρ, u) = (ρ, v, 0) if x d < 0, and linearize the Euler equations, and jump conditions around this solution.

21 Linear spectral stability (Landau, Miles...) Consider a piecewise constant vortex sheet: { (ρ, v, 0) if x d > 0, (ρ, u) = (ρ, v, 0) if x d < 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.

22 Linear spectral stability (Landau, Miles...) Consider a piecewise constant vortex sheet: { (ρ, v, 0) if x d > 0, (ρ, u) = (ρ, v, 0) if x d < 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.

23 Linear spectral stability (Landau, Miles...) Consider a piecewise constant vortex sheet: { (ρ, v, 0) if x d > 0, (ρ, u) = (ρ, v, 0) if x d < 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.

24 Theorem Let d = 2, and consider a piecewise constant weakly stable vortex sheet. Let T > 0, and µ 6. Consider initial data (ρ 0, u 0 ), ϕ 0 that are perturbations in H µ+15/2 (R 2 +) H µ+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 (ρ 0, u 0 ), ϕ 0. The solution belongs to H µ (]0, T [ R 2 +) H µ+1 (]0, T [ R).

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26 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.

27 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 satisfy an a priori tame estimate: LV = f, B(V, ψ) = g, V H m (]0,T [ R 2 + ) + ψ H m+1 (]0,T [ R) [ ] C f H m+1 (]0,T [ R 2 + ) + g H m+1 (]0,T [ R).

28 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 satisfy an a priori tame estimate: LV = f, B(V, ψ) = g, V H m (]0,T [ R 2 + ) + ψ H m+1 (]0,T [ R) [ ] C f H m+1 (]0,T [ R 2 + ) + g H m+1 (]0,T [ R). Also tame dependence on the coefficients.

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30 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:

31 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.

32 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.

33 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.

34 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.

35 Plan Related problems Related problems

36 Related problems Related problems Related problems can be handled with the same approach: Weakly stable shocks for the isentropic Euler equations. Uniform/weak stability criterion [Majda, 1983]. The discontinuity is noncharacteristic, and the weak Lopatinskii condition is satisfied.

37 Related problems Related problems Related problems can be handled with the same approach: Weakly stable shocks for the isentropic Euler equations. Uniform/weak stability criterion [Majda, 1983]. The discontinuity is noncharacteristic, and the weak Lopatinskii condition is satisfied. Isothermal liquid-vapor phase transitions. [Benzoni, 1998]. These are undercompressive shocks (additional jump condition). The discontinuity is noncharacteristic, and the weak Lopatinskii condition is satisfied (surface waves).

38 Related problems Related problems Other works: Contact discontinuities for the nonisentropic Euler equations. Work in progress by Morando-Secchi-Trebeschi. Uniqueness of contact discontinuities. Work almost finished! Contact discontinuities in MHD [Trakhinin, 2005]. Dissipative symmetrizers approach.