Hyperbolic Systems of Conservation Laws in One Space Dimension I - Basic concepts Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1
The Scalar Conservation Law u t + f(u) x = 0 u : conserved quantity f(u) : flux d dt b a u(t, x) dx = b a u t (t, x) dx = b a f(u(t, x)) x dx = f(u(t, a)) f(u(t, b)) = [inflow at a] [outflow at b]. u a b ξ x 2
Example : Traffic Flow ρ = density of cars a b x d dt b a ρ(t, x) dx = [flux of cars entering at a] [flux of cars exiting at b] flux: f(t, x) =[number of cars crossing the point x per unit time] = [density] [velocity] t ρ + x [ρ v(ρ)] = 0 3
Weak solutions u a b ξ x conservation equation: u t + f(u) x = 0 quasilinear form: u t + a(u)u x = 0 a(u) = f (u) Conservation equation remains meaningful for u = u(t, x) discontinuous, in distributional sense: {uφ t + f(u)φ x } dxdt = 0 for all φ Cc 1 Need only : u, f(u) locally integrable 4
Convergence u t + f(u) x = 0 Assume: u n is a solution for n 1, u n u f(u n ) f(u) in L 1 loc then {uφ t + f(u)φ x } dxdt = lim n {u n φ t + f(u n )φ x } dxdt = 0 for all φ C 1 c. Hence u is a weak solution as well. 5
Systems of Conservation Laws t u 1 + x f 1(u 1,..., u n ) = 0, t u n + x f n(u 1,..., u n ) = 0. u t + f(u) x = 0 u = (u 1,..., u n ) IR n conserved quantities f = (f 1,..., f n ) : IR n IR n fluxes 6
Euler equations of gas dynamics (1755) ρ t + (ρv) x = 0 (conservation of mass) (ρv) t + (ρv 2 + p) x = 0 (conservation of momentum) (ρe) t + (ρev + pv) x = 0 (conservation of energy) ρ = mass density v = velocity E = e + v 2 /2 = energy density per unit mass (internal + kinetic) p = p(ρ, e) constitutive relation 7
Hyperbolic Systems u t + f(u) x = 0 u = u(t, x) IR n u t + A(u)u x = 0 A(u) = Df(u) The system is strictly hyperbolic if each n n matrix 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 Choose bases so that l i r j = l i A = λ i l i { 1 if i = j 0 if i j 8
Scalar Equation with Linear Flux u t + f(u) x = 0 f(u) = λu + c u t + λu x = 0 u(0, x) = φ(x) Explicit solution: u(t, x) = φ(x λt) traveling wave with speed f (u) = λ u(0) λ t u(t) x 9
A Linear Hyperbolic System u t + Au x = 0 u(0, x) = φ(x) λ 1 < < λ n eigenvalues { r1,..., r n right eigenvectors l 1,..., r n left eigenvectors Explicit solution: linear superposition of travelling waves u(t, x) = i φ i (x λ i t)r i φ i (s) = l i φ(s) u 1 u 2 x 10
Nonlinear Effects u t + A(u)u x = 0 eigenvalues depend on u = waves change shape u(0) u(t) x 11
eigenvectors depend on u = nontrivial wave interactions t t x x linear nonlinear 12
Loss of Regularity u t + f(u) x = 0 u(0, x) = φ(x) Method of characteristics yields: u ( t, x 0 + t f (φ(x 0 )) ) = φ(x 0 ) characteristic speed = f (u) u(0) u(t) Global solutions only in a space of discontinuous functions x u(t, ) BV 13
Wave Interactions u t = A(u)u x λ i (u) = i-th eigenvalue u i x l i (u), r i (u) = i-th eigenvectors. = l i u x = [i-th component of u x ] = [density of i-waves in u] u x = n u i xr i (u) u t = i=1 n λ i (u)u i xr i (u) i=1 differentiate first equation w.r.t. t, second one w.r.t. x = evolution equation for scalar components u i x (u i x ) t + (λ i u i x ) x = j>k(λ j λ k ) ( l i [r j, r k ] ) u j x uk x 14
source terms: (λ j λ k ) ( l i [r j, r k ] ) u j xu k x = 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 xu k x = [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 } 15
Shock solutions u t + f(u) x = 0 u(t, x) = { u if x < λt u + if x > 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 flux] 16
Derivation of the Rankine - Hugoniot Equations {uφt + f(u)φ x } dxdt = 0 for all φ C 1 c x u = u+ x =λt Supp φ Ω + n u = u n + t v =. ( uφ, f(u)φ ) Ω 0 = = div v dxdt = n + v ds + n v ds Ω + Ω Ω + Ω [λu + f(u + )] φ(t, λt) dt + [ λu + f(u )] φ(t, λt) dt = [λ(u + u ) (f(u + ) f(u )) ] φ(t, λt) dt. 17
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 matrix] 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 matrix A(u +, u ) The speed λ coincides with the corresponding eigenvalue 18
scalar conservation law: u t + f(u) x = 0 f(u) u λ t f (u) u + u + u x λ = f(u+ ) f(u ) u + u = 1 u + u u + u f (s) ds [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 + ] 19
Points of Approximate Jump The function u = u(t, x) has an approximate jump at a point (τ, ξ) if there exists states u u + and a speed λ such that, calling U(t, x) =. { u if x < λt, u + if x > λt, there holds lim ρ 0+ 1 ρ 2 τ+ρ ξ+ρ τ ρ ξ ρ u(t, x) U(t τ, x ξ) dxdt = 0 (2) t τ. x = λ u u + ξ x Theorem. If u is a weak solution to the system of conservation laws u t +f(u) x = 0 then the Rankine-Hugoniot equations hold at each point of approximate jump. 20
Construction of Shock Curves Problem: Given u IR n, find the states u + IR 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: Fix i {1,..., n}. The jump u + u is a (right) i-eigenvector of the averaged matrix 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 exists a curve s S i (s)(u ) of points that satisfy (RH i ) u ri S i 21
Weak solutions can be non-unique Example: a Cauchy problem for Burgers equation u t + (u 2 /2) x = 0 u(0, x) = { 1 if x 0, 0 if x < 0. Each α [0, 1] yields a weak solution u α (t, x) = { 0 if x < αt/2, α if αt/2 x < (1 + α)t/2, 1 if x (1 + α)t/2. 1 α u = 0 x= α t/2 u = α u = 1 0 x 0 x 22
Admissibility Conditions on Shocks For physical systems: a concave entropy should not decrease For general hyperbolic systems: require stability w.r.t. small perturbations 23
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 + x x 24
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 25
General Stability Conditions Scalar case: stability holds if and only if f(u ) f(u ) f(u+ ) f(u ) u u u + u f(u) f(u) u u* u + + u u * Vector valued case: u + = S i (σ)(u ) for some σ IR. u Admissibility Condition (T.P.Liu, 1976). 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* + σ i u = S ( ) (u ) 26
x = γ (t) x = α t / 2 Admissibility Condition (P. Lax, 1957) 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 + ) [Liu condition] = [Lax condition] 27
Entropy - Entropy Flux u t + f(u) x = 0 Definition. A function η : IR n IR is called an entropy, with entropy flux q : IR n IR if Dη(u) Df(u) = Dq(u) For smooth solutions u = u(t, x), this implies η(u) t + q(u) x = Dη u t + Dq u x = Dη Df u x + Dq u x = 0 = η(u) is an additional conserved quantity, with flux q(u) 28
Entropy Admissibility Condition A weak solution u of the hyperbolic system u t + f(u) x = 0 is entropy admissible if [η(u)] t + [q(u)] x 0 in the sense of distributions, for every pair (η, q), where η is a convex entropy and q is the corresponding entropy flux. {η(u)ϕ t + q(u)ϕ x } dxdt 0 ϕ C 1 c, ϕ 0 - smooth solutions conserve all entropies - solutions with shocks are admissible if they dissipate all convex entropies 29
Existence of entropy - entropy flux pairs Dη(u) Df(u) = Dq(u) ( η u 1 ) η u n f 1 u 1 f n u 1 f 1 u n f n u n = ( q u 1 ) q u n - a system of n equations for 2 unknown functions: η(u), q(u) - over-determined if n > 2 - however, some physical systems (described by several conservation laws) are endowed with natural entropies 30