Linear stability of contact discontinuities for the nonisentropic - PDF document

Linear stability of contact discontinuities for the nonisentropic Euler equations in two space dimensions Alessandro Morando, Paola Trebeschi Department of Mathematics University of Brescia Lausanne, September 19-22, 2006

1. The Euler equations We study the nonisentropic Euler equations of a perfect polytropic ideal gas in the plane R 2 : ∂ t p + u · ∇ p + γp ∇ · u = 0 , ρ ( ∂ t u + ( u · ∇ ) u ) + ∇ p = 0 , (1) ∂ t S + u · ∇ S = 0 . x = ( x 1 , x 2 ) ∈ R 2 , p = p ( t, x ) ∈ R the pressure, u ( t, x ) ∈ R 2 the velocity, S = S ( t, x ) ∈ R the entropy, ρ is the density, obeying the constitutive law 1 γ e − S ρ ( p, S ) = Ap γ with A > 0 given, γ > 1 adiabatic number. Problem: Stability of “contact discontinuities” Result: Under a “supersonic” condition (that precludes violent instability) we prove an ENERGY ESTIMATE for the linearized problem. 1

2. Discontinuities 2.1. Contact discontinuities Let Γ := { x 2 = ϕ ( t, x 1 ) } be a smooth hypersurphace and � ( p + , u + , S + ) if x 2 > ϕ ( t, x 1 ) ( p, u , S ) := ( p − , u − , S − ) if x 2 < ϕ ( t, x 1 ) , a smooth function on either side of Γ. Definition 1. ( p, u , S ) is a contact discontinuity solution of (1) if it is a classical solution of (1) on both sides of Γ and satisfies the Rankine-Hugoniot jump conditions at Γ : ∂ t ϕ = u + · ν = u − · ν , p + = p − , where ν := ( − ∂ x 1 ϕ, 1) is a (space) normal vector to Γ. These conditions yield that • the normal velocity and pressure are continuous across the interface Γ, • the only jumps experimented by the solution con- cern the tangential velocity and the entropy. Thus, a contact discontinuity is a vortex sheet . 2

2.2. Planar contact discontinuities A planar contact discontinuity is a piecewise constant solution to (1) � ( p r , u r , S r ) , if x 2 > σt + nx 1 , ( p, u , S ) = ( p l , u l , S l ) , if x 2 < σt + nx 1 . u r,l = ( v r,l , u r,l ) T are fixed vectors in R 2 , p r,l > 0 , S r,l , σ, n are fixed real numbers. • The previous quantities are related by the Rankine- Hugoniot jump conditions σ + v r n − u r = 0 , σ + v l n − u l = 0 , p r = p l =: p. • Without loss of generality, we may assume σ = u r = u l = 0 , n = 0 , v r + v l = 0 ( v r � = 0) . This corresponds to the following U r,l = ( p, v r,l , 0 , S r,l ) T , with v r + v l = 0 . 3

2.3. Stability of nonisentropic planar contact dis- continuities in dimension d = 2 , 3 (Fejer-Miles, 1958-1963 and Coulombel-Morando 2004) Consider a planar contact discontinuity U r,l = ( p, v r,l , 0 , S r,l ) T with v r + v l = 0 ( v r,l are the tangential velocities) and linearize the Euler equations and the jump conditions around this solution. • if d = 3, the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability ; � � 3 / 2 c 2 / 3 + c 2 / 3 • if d = 2 and | v r − v l | < 1 the linearized r l 2 2 equations do not satisfy the Lopatinskii condition ⇒ violent instability ; � � 3 / 2 • if d = 2 and | v r − v l | c 2 / 3 + c 2 / 3 > 1 the linearized r l 2 2 equations satisfy the weak Lopatinskii condition ⇒ weak stability , � p ′ ( ρ ) is the sound speed and c r , c l are the con- c ( ρ ) := stant values of c ( ρ ) in both sides of Γ. • In any case, the uniform Kreiss-Lopatinskii condition is never satisfied. 4

3. Energy estimates in two space dimensions 3.1. Reformulation of the problem in a fixed do- main The interface Γ := { x 2 = ϕ ( t, x 1 ) } is unknown so that the problem is a free boundary problem . • In order to work in a fixed domain we introduce the change of variables ( τ, y 1 , y 2 ) → ( t, x 1 , x 2 ) , ( t, x 1 ) = ( τ, y 1 ) , x 2 = Φ( τ, y 1 , y 2 ) , where Φ : R 3 → R , Φ( τ, y 1 , 0) = ϕ ( τ, y 1 ) , ∂ y 2 Φ( τ, y 1 , y 2 ) ≥ κ > 0 . • We define the new unknowns ( p + ♯ , u + ♯ , S + ♯ )( τ, y 1 , y 2 ) := ( p, u , S )( τ, y 1 , Φ( τ, y 1 , y 2 )) , ( p − ♯ , u − ♯ , S − ♯ )( τ, y 1 , y 2 ) := ( p, u , S )( τ, y 1 , Φ( t, y 1 , − y 2 )) . The functions p ± ♯ , u ± ♯ , S ± ♯ are smooth on the fixed domain { y 2 > 0 } . • For convenience, we drop the ♯ index and only keep the + and − exponents. • Finally, we write ( t, x 1 , x 2 ) instead of ( τ, y 1 , y 2 ). 5

• Let us set u = ( v, u ). The existence of compressible vortex sheets amounts to prove the existence of smooth solutions of the first order system � u + − ∂ t Φ + − v + ∂ x 1 Φ + � ∂ x 2 p + ∂ t p + + v + ∂ x 1 p + + ∂ x 2 Φ + + γp + ∂ x 1 v + + γp + ∂ x 2 u + ∂ x 2 Φ + ∂ x 2 v + = 0 , ∂ x 2 Φ + − γp + ∂ x 1 Φ + � u + − ∂ t Φ + − v + ∂ x 1 Φ + � ∂ x 2 v + ∂ t v + + v + ∂ x 1 v + + ∂ x 2 Φ + ∂ x 2 Φ + ∂ x 2 p + = 0 , ρ + ∂ x 1 p + − 1 ∂ x 1 Φ + + 1 ρ + ∂ t u + + v + ∂ x 1 u + + � u + − ∂ t Φ + − v + ∂ x 1 Φ + � ∂ x 2 u + ∂ x 2 Φ + ∂ x 2 p + + 1 ∂ x 2 Φ + = 0 , ρ + � u + − ∂ t Φ + − v + ∂ x 1 Φ + � ∂ x 2 S + ∂ t S + + v + ∂ x 1 S + + ∂ x 2 Φ + = 0 , in the fixed domain { x 2 > 0 } , where Φ ± ( t, x 1 , x 2 ) := Φ( t, x 1 , ± x 2 ) (( p − , v − , u − , S − , Φ − ) must solve a similar system ) fulfilling the boundary conditions Φ + | x 2 =0 = Φ − | x 2 =0 = ϕ, ∂ t ϕ = − v + | x 2 =0 ∂ x 1 ϕ + u + | x 2 =0 = − v − | x 2 =0 ∂ x 1 ϕ + u − | x 2 =0 , p + | x 2 =0 = p − | x 2 =0 . The functions Φ ± should also satisfy ∂ x 2 Φ + ( t, x 1 , x 2 ) ≥ k, ∂ x 2 Φ − ( t, x 1 , x 2 ) ≤ − k. 6

• The equations are not sufficient to determine the un- knowns U ± := ( p ± , v ± , u ± , S ± ) and Φ ± : another equation relating Φ, U and ϕ is needed in order to close the sys- tem. We may prescribe that Φ ± solve in the domain { x 2 > 0 } the eikonal equations ∂ t Φ ± + v ± ∂ x 1 Φ ± − u ± = 0 . With this choice the boundary matrix of the system for U ± has constant rank in the whole domain { x 2 ≥ 0 } and not only on the boundary { x 2 = 0 } . • Matrix form of the system For all U = ( p, v, u, S ) T , we get the system (in the interior { x 2 > 0 } ) L ( U + , ∇ Φ + ) U + = 0 , L ( U − , ∇ Φ − ) U − = 0 , ∂ t Φ ± + v ± ∂ x 1 Φ ± − u ± = 0 where L ( U + , ∇ Φ + ) U + = ∂ t U + + A 1 ( U + ) ∂ x 1 U + + � � 1 A 2 ( U + ) − ∂ t Φ + I 4 − ∂ x 1 Φ + A 1 ( U + ) ∂ x 2 U + + ∂ x 2 Φ + with the boundary conditions (on { x 2 = 0 } ) Φ + | x 2=0 = Φ − | x 2=0 = ϕ , ∂ t ϕ = − v + | x 2=0 ∂ x 1 ϕ + u + | x 2=0 = − v − | x 2=0 ∂ x 1 ϕ + u − | x 2=0 p + | x 2=0 = p − | x 2=0 . 7

3.2. The linearized equations • Consider a planar contact discontinuity     p p     v r v l U r = U l = Φ r,l = ± x 2 .   ,   , 0 0 S r S l We assume that v r + v l = 0 , v r > 0. • Let us consider U r,l + ε ˙ ± x 2 + ε ˙ U ± , ψ ± where we have set ˙ u ± , ˙ S ± ), ˙ U ± = ( ˙ p ± , ˙ ψ ± is a small perturbation of the exact solution U r,l , Φ r,l . S ± ) T must solve the • Up to second order, ˙ u ± , ˙ U ± = ( ˙ p ± , ˙ system L ′ ˙ U = 0 , in { x 2 > 0 } , B ( ˙ U, ψ ) = 0 , on { x 2 = 0 } , where we have set for shortness ˙ U := ( ˙ U + , ˙ U − ) T , � � � � A 1 ( U r ) 0 A 2 ( U r ) 0 L ′ ˙ U := ∂ t ˙ ∂ x 1 ˙ ∂ x 2 ˙ U + U + U, 0 A 1 ( U l ) 0 − A 2 ( U l ) and   ( v r − v l ) ∂ x 1 ψ − ( ˙ u + − ˙ u − )   . B ( ˙ U, ψ ) := ∂ t ψ + v r ∂ x 1 ψ − ˙ u + p + − p − 8

• We have to introduce source terms; we study L ′ ˙ U = f in { x 2 > 0 } , B ( ˙ U, ψ ) = g, on { x 2 = 0 } • It is useful make another linear change of unknowns � � � � − ˙ p + γp + ˙ u + γp + ˙ p + ˙ u + v + , W 2 := 1 W 3 := 1 , W 4 := ˙ W 1 := ˙ S + , 2 c r 2 c r � � � � − ˙ p − γp + ˙ p − γp + ˙ ˙ v − , W 6 := 1 u − W 7 := 1 u − , W 8 := ˙ W 5 := ˙ S − 2 c l 2 c l • This change of unknowns W transform the system L ′ ˙ U = f in a symmetric hyperbolic form L W = f, in { x 2 > 0 } B ( W nc , ψ ) = g, on { x 2 = 0 } , with new data f, g , and L W := A 0 ∂ t W + A 1 ∂ x 1 W + A 2 ∂ x 2 W � � ∂ t ψ B ( W nc , ψ ) := MW nc + b . ∂ x 1 ψ W := ( W 1 , W 2 , W 3 , W 4 , W 5 , W 6 , W 7 , W 8 ) T , W c := ( W 1 , W 4 , W 5 , W 8 ) T , W nc := ( W 2 , W 3 , W 6 , W 7 ) T = (linear combination of p, u )     − c r − c r 0 v r − v l c l c l   ,   . M := − c r − c r 0 0 b := 1 v r − 1 1 1 − 1 0 0 9

• We want to find an L 2 a priori estimate of the solution to the linearized problem L W = f, in { x 2 > 0 } B ( W nc , ψ ) = g, on { x 2 = 0 } , in Ω := { ( t, x 1 , x 2 ) ∈ R 3 s.t. x 2 > 0 } = R 2 × R + . The boundary ∂ Ω = { x 2 = 0 } is identified to R 2 . 3.3. The functional setting • Define H s γ ( R 2 ) := { u ∈ D ′ ( R 2 ) s.t. exp( − γt ) u ∈ H s ( R 2 ) } , equipped with the norm � u � H s γ ( R 2 ) := � exp( − γt ) u � H s ( R 2 ) . • Define similarly the space H s γ (Ω). • The space L 2 ( R + ; H s γ ( R 2 ) is the space of all functions v = v ( t, x 1 , x 2 ) in Ω such that the following norm � + ∞ ||| v ||| 2 � v ( · , x 2 ) � 2 γ ) := γ ( R 2 ) dx 2 L 2 ( H s H s 0 is finite. 10