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)
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 Related problems J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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 Related problems J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion Piecewise smooth solutions 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. J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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 Related problems J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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 ]; H s ( R d )), s > 1 + d/ 2). [Kato, 1975] J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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 ]; H s ( R d )), s > 1 + d/ 2). [Kato, 1975] Blow-up of smooth solutions. [Sideris, 1985] J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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 Related problems J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion Piecewise smooth solutions Rankine-Hugoniot jump conditions A function: � ( ρ + , u + )( t, x ) if x d > ϕ ( t, y ), ( ρ, u ) = ( ρ − , u − )( t, x ) if x d < ϕ ( t, y ), is a weak solution if J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion Piecewise smooth solutions Rankine-Hugoniot jump conditions A function: � ( ρ + , u + )( t, x ) if x d > ϕ ( t, y ), ( ρ, u ) = ( 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 J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion Piecewise smooth solutions Rankine-Hugoniot jump conditions A function: � ( ρ + , u + )( t, x ) if x d > ϕ ( t, y ), ( ρ, u ) = ( 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: 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)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion Piecewise smooth solutions Existence results Existence of one uniformly stable shock wave. [Majda, 1983] [Blokhin, 1981] J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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)
The Cauchy problem for the Euler equations The equations Stability of contact discontinuities Smooth solutions Conclusion 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)
The Cauchy problem for the Euler equations Main result Stability of contact discontinuities Linear stability Conclusion 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 Related problems J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations Main result Stability of contact discontinuities Linear stability Conclusion 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: ∂ t ϕ = u + · n = u − · n , p ( ρ + ) = p ( ρ − ) . J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations Main result Stability of contact discontinuities Linear stability Conclusion 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: ∂ 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. J.-F. Coulombel (Lille), P. Secchi (Brescia)
The Cauchy problem for the Euler equations Main result Stability of contact discontinuities Linear stability Conclusion Nonlinear stability 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. J.-F. Coulombel (Lille), P. Secchi (Brescia)
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