Hyperbolic Systems of Conservation Laws I - Basic Concepts Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 27
The Scalar Conservation Law u t +f(u) = 0 u : conserved quantity f(u) : flu d dt b a u(t,) d = b u t (t,)d = b a a f ( u(t,) ) d = f ( u(t,a) ) f ( u(t,b) ) = [inflow at a] [outflow at b]. u a b ξ Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 2 / 27
Eample: Traffic Flow ρ = density of cars a b d dt b a ρ(t,)d = [flu of cars entering at a] [flu of cars eiting at b] flu: f(t,) =[number of cars crossing the point per unit time] = [density] [velocity] t ρ+ [ ] ρv(ρ) = 0 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 3 / 27
Weak solutions u a b ξ conservation equation: u t +f(u) = 0 quasilinear form: u t +a(u)u = 0 a(u) = f (u) Conservation equation remains meaningful for u = u(t, ) discontinuous, in distributional sense: {uφt +f(u)φ } ddt = 0 for all φ C 1 c Need only : u,f(u) locally integrable Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 4 / 27
Convergence u t +f(u) = 0 Assume: u n is a solution for n 1, u n u, f(u n ) f(u) in L 1 loc then {uφt } {un } +f(u)φ ddt = lim φ t +f(u n )φ ddt = 0 n for all φ C 1 c Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 5 / 27
Systems of Conservation Laws t u 1 + f 1(u 1,...,u n ) = 0, t u n + f n(u 1,...,u n ) = 0. u t +f(u) = 0 u = (u 1,...,u n ) R n conserved quantities f = (f 1,...,f n ) : R n R n flues Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 6 / 27
Hyperbolic Systems u t +f(u) = 0 u = u(t,) R n u t +A(u)u = 0 A(u) = Df(u) The system is strictly hyperbolic if each n n matri A(u) has real distinct eigenvalues λ 1 (u) < λ 2 (u) < < λ n (u) right eigenvectors r 1 (u),...,r n (u) (column vectors) left eigenvectors l 1 (u),...,l n (u) (row vectors) Ar i = λ i r i l i A = λ i l i Choose bases so that l i r j = { 1 if i = j 0 if i j Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 7 / 27
Scalar Equation with Linear Flu u t +f(u) = 0 f(u) = λu +c u t +λu = 0 u(0,) = φ() Eplicit solution: u(t, ) = φ( λt) traveling wave with speed f (u) = λ u(0) λ t u(t) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 8 / 27
A Linear Hyperbolic System u t +Au = 0 u(0,) = φ() λ 1 < < λ n eigenvalues r 1,...,r n eigenvectors Eplicit solution: linear superposition of travelling waves u(t,) = i φ i ( λ i t)r i φ i (s) = l i φ(s) u 1 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 9 / 27
Nonlinear Effects u t +A(u)u = 0 eigenvalues depend on u = waves change shape u(0) u(t) Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 10 / 27
eigenvectors depend on u = nontrivial wave interactions t t linear nonlinear Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 11 / 27
Loss of Regularity u t +(u 2 /2) = 0 u t +uu = 0 f(u) = u 2 /2 u(0) characteristic speed: f (u) = u u(t) Global solutions only in a space of discontinuous functions Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 12 / 27
Smooth solutions - evolution of wave components u t = A(u)u λ i (u) = i-th eigenvalue u i l i (u), r i (u) = i-th eigenvectors. = l i u = [i-th component of u ] = [density of i-waves in u] u = n ur i i (u) i=1 n u t = λ i (u)ur i i (u) i=1 differentiate first equation w.r.t. t, second one w.r.t. = evolution equation for scalar components u i (u) i t +(λ i u) i = ( ) j λ k ) l i [r j,r k ] uu j>k(λ j k Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 13 / 27
( ) source terms: (λ j λ k ) l i [r j,r k ] uu j k = amount of i-waves produced by the interaction of j-waves with k-waves λ j λ k = [difference in speed] = [rate at which j-waves and k-waves cross each other] u j u k = [density of j-waves] [density of k-waves] [r j,r k ] = (Dr k )r j (Dr j )r k (Lie bracket) = [directional derivative of r k in the direction of r j ] [directional derivative of r j in the direction of r k ] l i [r j,r k ] = i-th component of the Lie bracket [r j,r k ] along the basis of eigenvectors {r 1,...,r n } Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 14 / 27
Shock solutions u t +f(u) = 0 u(t,) = { u if < λt u + if > λt is a weak solution if and only if λ [u + u ] = f(u + ) f(u ) Rankine - Hugoniot equations [speed of the shock] [jump in the state] = [jump in the flu] Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 15 / 27
Derivation of the Rankine - Hugoniot Equations 0 = = = } {uφ t +f(u)φ ddt = div (uφ, f(u)φ) ddt Ω + Ω n + vds + n vds Ω + Ω [ ] λ(u + u ) (f(u + ) f(u )) φ(t,λt)dt. Ω + u = u+ =λt v. = ( ) uφ, f(u)φ Supp φ Ω n n + u = u Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 16 / 27 t
Alternative formulation: λ(u + u ) = f(u + ) f(u ) = 1 0 Df ( θu + +(1 θ)u ) (u + u ) dθ = A(u +,u ) (u + u ) A(u,v). = 1 0 Df ( θu +(1 θ)v ) dθ = [averaged Jacobian matri] The Rankine-Hugoniot conditions hold if and only if λ(u + u ) = A(u +,u )(u + u ) The jump u + u is an eigenvector of the averaged matri A(u +,u ) The speed λ coincides with the corresponding eigenvalue Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 17 / 27
The Rankine-Hugoniot condition for the scalar conservation law u t +f(u) = 0 u λ f f (u) + u u + u u λ = f(u+ ) f(u ) u + u = 1 u + u + u f (s)ds u [speed of the shock] = [slope of secant line through u,u + on the graph of f] = [average of the characteristic speeds between u and u + ] Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 18 / 27
Points of Approimate Jump The function u = u(t,) has an approimate jump at a point (τ,ξ) if there eists states u u + and a speed λ such that, calling there holds: U(t,) =. { u if < λt u + if > λt 1 τ+ρ ξ+ρ u(t,) U(t τ, ξ) ddt = 0 (2) ρ 2 lim ρ 0+ τ ρ ξ ρ t. = λ u τ + u ξ Theorem. If u is a weak solution to the system of conservation laws u t +f(u) = 0 then the Rankine-Hugoniot equations hold at each point of approimate jump. Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 19 / 27
Construction of Shock Curves Problem: Given u R n, find the states u + R n which, for some speed λ, satisfy the Rankine - Hugoniot equations λ(u + u ) = f(u + ) f(u ) = A(u,u + )(u + u ) Alternative formulation: Fi i {1,...,n}. The jump u + u is a (right) i-eigenvector of the averaged matri A(u,u + ) if and only if it is orthogonal to all (left) eigenvectors l j (u,u + ) of A(u,u + ), for j i l j (u,u + ) (u + u ) = 0 for all j i (RH i ) Implicit function theorem = for each i there eists a curve s S i (s)(u ) of points that satisfy S i u ri Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 20 / 27
Non-uniqueness of Weak solutions Eample: a Cauchy problem for Burgers equation { u t +(u 2 1 if 0 /2) = 0 u(0,) = 0 if < 0 Each α [0,1] yields a weak solution 0 if < αt/2 u α (t,) = α if αt/2 < (1+α)t/2 1 if (1+α)t/2 1 α t u = 0 = α t /2 u = α u = 1 0 0 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 21 / 27
Admissibility conditions on shocks u t +f(u) = 0 solutions should be stable w.r.t. small perturbations solutions should be limits of suitable approimations (vanishing viscosity, relaation, etc...) any conve entropy should not increase Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 22 / 27
Stability conditions: the scalar case Perturb the shock with left and right states u, u + by inserting an intermediate state u [u,u + ] Initial shock is stable [speed of jump behind] [speed of jump ahead] f(u ) f(u ) u u f(u+ ) f(u ) u + u + u _ u u* _ u u* u + Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 23 / 27
speed of a shock = slope of a secant line to the graph of f f f u u* u + u + u* u Stability conditions: when u < u + the graph of f should remain above the secant line when u > u +, the graph of f should remain below the secant line Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 24 / 27
General stability conditions Scalar case: stability holds if and only if f(u ) f(u ) u u f(u+ ) f(u ) u + u for every intermediate state u [u,u + ] f(u) f(u) u u* u + + u u * u Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 25 / 27
Vector valued case: u + = S i (σ)(u ) for some σ R. Admissibility Condition (T.P.Liu) The speed λ(σ) of the shock joining u with u + must be less or equal to the speed of every smaller shock, joining u with an intermediate state u = S i (s)(u ), s [0,σ]. λ(u,u + ) λ(u,u ) u u* u + = S ( σ) (u ) i The Liu condition singles out precisely the solutions which are limits of vanishing viscosity approimations (S.Bianchini & A.B.) u ε t +f(uε ) = εu ε u ε u as ε 0 Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 26 / 27
Admissibility Condition (P. La) A shock connecting the states u,u +, travelling with speed λ = λ i (u,u + ) is admissible if λ i (u ) λ i (u,u + ) λ i (u + ) t admissible t not admissible Geometric meaning: characteristics flow toward the shock from both sides The Liu condition implies the La condition Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 27 / 27