Hyperbolic Systems of Conservation Laws

Transcription

1 Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 30

2 Continuous dependence on initial data u t +f(u) x = 0 u(0,x) = ū(x) Given two solutions u,v, estimate the difference u(t) v(t) L 1 Standard approach: set w = u v, show that hence w(t) e Ct w(0) d dt w(t) C w u(0) v(0) u(t) v(t) Works for Lipschitz solutions, not in the presence of shocks Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 2 / 30

3 For two solutions u, v of a hyperbolic system containing shocks, the L 1 distance can increase rapidly during short time intervals t jumps in u jumps in v t 2 t 1 u(t) v(t) L 1 x 0 t t 1 2 t Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 3 / 30

4 The scalar conservation law u t +f(u) x = 0 Contraction property (A.I. Volpert, 1967, S. Kruzhkov, 1970) u(t) v(t) L 1 u(0) v(0) L 1 for all t 0 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 4 / 30

5 The L 1 distance between continuous solutions remains constant wave speed u(0) v(0) x u(t) v(t) x Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 5 / 30

6 The L 1 distance decreases when a shock in one solution crosses the graph of the other solution wave speed u(0) v(0) x u(t) v(t) x Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 6 / 30

7 Linear Hyperbolic Systems u t +Au x = 0 u R n u L 1. = u(x) dx Right, left eigenvectors: l i A = λ i l i Ar i = λ i r i. n Equivalent norm: u A = l i u L 1 u,v solutions = w = u v satisfies w t +Aw x = 0 i=1 w(t) A = w(0) A for all t R Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 7 / 30

8 Continuous dependence on the initial data u t +f(u) x = 0 u(0,x) = ū(x) A domain of integrable functions with small total variation: D. = cl {u L 1 (R;R n ); u is piecewise constant, V(u)+C 0 Q(u) < δ 0 } Theorem. For every ū D, front tracking approximations converge to a unique limit solution u : [0, [ D. The map (ū,t) u(t, ). = S t ū is a uniformly Lipschitz semigroup, i.e.: S 0 ū = ū, S s (S t ū) = S s+t ū Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 8 / 30

9 Proof: by a homotopy method v(t) z θ(t) v(0) θ u (0) u(0) z θ(0) u(t) u θ(t) 2 2 systems : A. Bressan and R. M. Colombo, 1994 n n systems : A. Bressan, G. Crasta, B. Piccoli, 1996 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 9 / 30

10 Alternative methof: a non-linear distance functional (A. Bressan, T.P.Liu, T. Yang, 1998) Construct a functional Φ : D D R + such that Φ is equivalent to the L 1 distance 1 C v u L 1 Φ(u,v) C v u L 1 Φ is non-increasing in time, along couples of solutions Φ(u(t),v(t)) Φ(u(s),v(s)) 0 s t Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 10 / 30

11 Construction of the distance functional For each x R, decompose the jump u(x),v(x) as a concatenation of shocks: v(x) = S n ( qn(x) ) S 1 ( q1(x) )( u(x) ) λ 1 ω3= v(x) q λ 3 3 ω2 ω 1 q 1 σ α xα v u ω = u(x) 0 x x α q 1(x),...,q n(x) = scalar components of the jump Remark. In the linear case u t +Au x = 0, one has q i(x) = l i (v(x) u(x)) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 11 / 30

12 Φ(u,v) = n i=1 q i (x) W i (x)dx Weight functions: W i (x). = 1+κ 1 [total strength of waves in u and in v which approach the i-wave q i (x) ] +κ 2 [wave interaction potentials of u and of v ]. [ ] = 1+κ 1 A i (x)+κ 2 Q(u)+Q(v) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 12 / 30

13 Uniqueness of Solutions u t +f(u) x = 0 u(0,x) = ū(x) Question: Do all entropy-admissible weak solutions coincide with the ones obtained by front tracking approximations? Key step: estimate the distance between a weak solution t u(t) and a trajectory t S t ū of the semigroup obtained taking limits of front tracking approximations. Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 13 / 30

14 An error estimate Theorem. Let S : D [0, [ D be a Lipschitz semigroup satisfying S tu S sv L u v +L t s Then, for every Lipschitz continuous map w : [0,T] D one has w(t) S w(0) { T w(t +h) S h w(t) } L lim inf dt T h 0+ h = L 0 T 0 [instantaneous error rate at time t] dt w(t) w(t+h) w(t) S h w(t) S T t h w(t+h) S T t w(t) S w(0) T w(0) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 14 / 30

15 Instantaneous error rate for a front tracking approximation S S R R τ +h NP τ x ( τ) α x 1 w(τ +h) Sh w(τ) h L 1 = O(1) σ α 2 +O(1) α R α NP σ α Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 15 / 30

16 Characterization of semigroup trajectories Question: among all weak solutions of u t +f(u) x = 0 which ones are obtained as limits of front tracking approximations? Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 16 / 30

17 1. Comparison with solutions to a Riemann problem Fix (τ,ξ). Define U = U (τ,ξ) as the solution of the Riemann problem w t +f(w) x = 0, w(τ,x) = { u +. = u(τ,ξ+) if x > ξ u. = u(τ,ξ ) if x < ξ Then for every ˆλ > 0 we expect 1 ξ+hˆλ lim h 0+ h u(τ +h, x) U (τ,ξ)(τ +h, x) dx = 0 ξ hˆλ (E1) u t ξ u( τ, x) x u + τ Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 17 / 30 ξ x

18 2. Comparison with solutions to a linear hyperbolic problem Fix (τ,ξ). Choose ˆλ > 0 larger than all wave speeds. Define U = U (τ,ξ) as the solution of the linear Cauchy problem w t +Ãwx = 0 with frozen coefficients: Ã =. A ( u(τ,ξ) ) w(τ,x) = u(τ,x) τ + h I h τ a ξ b Then, for a < ξ < b and h > 0 we expect ( 1 b ˆλh 2 h u(τ+h, x) U (τ+h, x) dx = O(1) Tot.Var.{u(τ, ); ]a, b[}) (E2) a+ˆλh Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 18 / 30

19 Theorem (A.Bressan, 1994) Let u : [0,T] D be Lipschitz continuous w.r.t. the L 1 distance. Then u is a weak solution to the system of conservation laws u t +f(u) x = 0 obtained as limit of front tracking approximations if and only if the estimates (E1)-(E2) are satisfied for a.e. τ [0,T], at every ξ R. u(τ +h) S h u(τ) (E1) + (E2) = lim L 1 h 0+ h = 0 Hence, by error estimate: u(t) = S t u(0) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 19 / 30

20 u( τ,x) τ + h τ { } Proof. Insert points x i such that Tot.Var. u(τ); ]x i 1, x i [ < ε x x x i 1 i 1 h u(τ + h,x) S hu(τ)(x) dx = i 1 xi +ˆλh dx + h x i ˆλh i 1 xi ˆλh dx h x i 1 +ˆλh = i A i + i B i estimate (E1) implies A i 0 as h 0 { } estimate (E2) implies B i ε Tot.Var. u(τ); ]x i 1, x i [ i { } B i ε Tot.Var. u(τ); R Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 20 / 30

21 Uniqueness of weak solutions u t +f(u) x = 0 u(0,x) = ū(x) introduce a suitable set of admissibility + regularity assumptions show that these assumptions imply the estimates (E1)+(E2) = u(t) = S t ū for all t 0 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 21 / 30

22 A set of assumptions (A1) (Conservation Equations) u : [0,T] D is continuous w.r.t. the L 1 distance. The initial condition u(0, x) = ū(x) holds. Moreover, u is a weak solution: {uϕ t +f(u)ϕ x } dxdt = 0 for all ϕ C 1 c (A2) (Admissibility Conditions) u satisfies the Lax admissibility conditions at each point (τ, ξ) of approximate jump. Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 22 / 30

23 (A3) (Tame Oscillation Condition) For some constants C, ˆλ the following holds. For every point x R and every t,h > 0 one has u(t +h,x) u(t,x) { } C Tot.Var. u(t, ); [x ˆλh, x + ˆλh] (A4) (Bounded Variation Condition) There exists δ > 0 such that, for every space-like curve { t = τ(x) } with dτ/dx δ a.e., the function x u ( τ(x),x ) has locally bounded variation. t+h γ t a x b Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 23 / 30

24 Uniqueness results u t +f(u) x = 0 u(0,x) = ū(x) Theorem. Let the system be strictly hyperbolic, with each characteristic field either linearly degenerate or genuinely nonlinear. Every weak solution u = u(t,x) obtained as limit of front tracking approximations satisfies all conditions (A1) (A4). If u : [0,T] D satisfies (A1),(A2),(A3), then u(t) = S t ū for t 0. The same is true if u satisfies (A1), (A2), (A4). (A.B. & P. LeFloch, 1997) (A. B. & P. Goatin, 1999) (A.B. & M. Lewicka, 2000) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 24 / 30

25 Vanishing viscosity approximations Goal: show that the weak solutions of the hyperbolic system u t +f(u) x = 0 which satisfy suitable admissibility conditions are precisely the limits of solutions to the parabolic system letting the viscosity ε 0+ u ε t +f(uε ) x = εu ε xx Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 25 / 30

26 u Alberto piecewise Bressan smooth: (Penn State) Singular Hyperbolic Perturbation Systems of Conservation techniques Laws can be used 26 / 30 Three basic cases: 1 - Smooth solutions Let u = u(t,x) be a smooth solution of the hyperbolic system u t +A(u)u x = 0 u(0,x) = ū(x) Then a standard perturbation argument yields: u = lim ε 0+ u ε, where u ε t +A(u ε )u ε x = εu ε xx u ε (0,x) = ū(x) Remark. The conclusion is far from obvious if u contains jumps. ε u u x

27 2. Scalar conservation laws u ε t +f (u ε )u ε x = εu ε xx Total variation of viscous solutions is non-increasing in time (u ε x) t +[f (u ε )u ε x] x = ε(u ε x) xx Uniform BV bound = a subsequence converges strongly in L 1 loc (A. I. Volpert 1968, S. Kruzhkov 1970) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 27 / 30

28 3 - Linear hyperbolic systems u t +Au x = 0 u(0,x) = ū(x) u ε t +Auε x = εu ε xx u(0,x) = ū(x) l i A = λ i l i Ar i = λ i r i i = 1,...,n Solve componentwise: u i. = li u (u i ) t +λ i (u i ) x = 0 u i (0,x) = l i ū(x) = limit of scalar viscous approximations u ε i = l i u ε (u ε i ) t +λ i (u ε i ) x = ε(u ε i ) xx u i (0,x) = l i ū(x) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 28 / 30

29 The general case Theorem. (S. Bianchini, A. Bressan, Annals of Math. 2005). Consider a strictly hyperbolic system with viscosity u t +A(u)u x = εu xx u(0,x) = ū(x). (CP) If Tot.Var.{ū} is sufficiently small, then (CP) admits a unique solution u ε (t, ) = S ε t ū, defined for all t 0. Moreover, we have the estimates Tot.Var. { S ε tū } C Tot.Var.{ū}, S ε t ū S ε t v L 1 L ū v L 1 (BV bounds) (L 1 stability) (Convergence) As ε 0, the solutions u ε converge to the trajectories of a semigroup S such that St ū S t v L 1 L ū v L 1 t 0. Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 29 / 30

30 These vanishing viscosity limits can be regarded as the unique viscosity solutions of the hyperbolic Cauchy problem u t +A(u)u x = 0 u(0,x) = ū(x). In the conservative case A(u) = Df(u), the viscosity solutions are weak solutions of u t +f(u) x = 0 u(0,x) = ū(x) satisfying the Liu entropy conditions. Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 30 / 30