Hyperbolic Systems of Conservation Laws in One Space Dimension II - Solutions to the Cauchy problem Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1
Global Solutions to the Cauchy Problem u t + f(u) = u(, ) = ū() Construct a sequence of approimate solutions u m Show that (a subsequence) converges: u m u in L 1 loc = u is a weak solution u u 1 2 u m To prevent oscillations, one needs an a-priori bound on the total variation (J. Glimm, 1965) 2
Building block: the Riemann Problem u t + f(u) = u(, ) = { u if < u + if > B. Riemann 186: 2 2 system of isentropic gas dynamics P. La 1957: n n systems (+ special assumptions) T. P. Liu 1975 n n systems (generic case) S. Bianchini 23 (vanishing viscosity limit for general hyperbolic systems, possibly non-conservative) t ω ω 2 1 ω = u ω = u + 3 invariant w.r.t. rescaling symmetry: u θ (t, ). = u(θt, θ) for all θ > 3
Riemann Problem for Linear Systems u t + Au = u(, ) = { u if < u + if > t / t = λ 2 / t = λ 1 ω 1 ω 2 / t = λ 3 ω = u ω = u + 3 u + u = n c j r j (sum of eigenvectors of A) j=1 intermediate states : ω i. = u + j i c j r j i-th jump: ω i ω i 1 = c i r i travels with speed λ i 4
Scalar Conservation Law u t + f(u) = u IR CASE 1: Linear flu. f(u) = λu. Jump travels with speed λ (contact discontinuity) f (u) = λ u + u(t) λ t u 5
CASE 2: the flu f is conve, so that u f (u) is increasing. u + > u = centered rarefaction wave f (u) u + u(t) f (u) t u u + < u = stable shock f (u) u + u λ t u(t) λ = f(u + ) f(u ) + u u 6
A class of nonlinear hyperbolic systems u t + f(u) = A(u) = Df(u) A(u)r i (u) = λ i (u)r i (u) Assumption (H) (P.La, 1957): Each i-th characteristic field is either genuinely nonlinear, so that λ i r i > for all u or linearly degenerate, so that λ i r i = for all u 7
genuinely nonlinear = characteristic speed λ i (u) is strictly increasing along integral curves of the eigenvectors r i linearly degenerate = characteristic speed λ i (u) is constant along integral curves of the eigenvectors r i u 2 r 1 r 2 u u 1 R 2 (σ)(u ) 8
Shock and Rarefaction curves u t + f(u) = A(u) = Df(u) i-rarefaction curve through u : σ R i (σ)(u ) = integral curve of the field of eigenvectors r i through u du dσ = r i(u), u() = u i-shock curve through u : σ S i (σ)(u ) = set of points u connected to u by an i-shock, so that u u is an i-eigenvector of the averaged matri A(u, u ) S i R i u r i (u ) 9
Elementary waves u t + f(u) = u(, ) = { u if < u + if > CASE 1 (Centered rarefaction wave). Let the i-th field be genuinely nonlinear. Assume u + = R i (σ)(u ) for some σ >, so that λ i (u + ) > λ i (u ). Then u(t, ) = { u if < tλ i (u ), R i (s)(u ) if = tλ i (s), s [, σ], u + if > tλ i (u + ), is a weak solution (piecewise smooth) of the Riemann problem. 1
Indeed, the previous construction achieves: u is constant along rays {/t = constant}, hence tu t + u = λ i (u(t, )) = /t u t, u are parallel to r i (u), hence they are i-eigenvectors of A(u) Therefore, for λ i (u ) < /t < λ i (u + ) one has = u t + t u = u t + λ i (u)u = u t + A(u)u 11
A centered rarefaction wave u 2 u u + u(t) t t = λ (u ) i t = λ (u + i ) u 1 u = u 12
CASE 2 (Shock or contact discontinuity). Assume that u + = S i (σ)(u ) for some i, σ. Let λ = λ i (u, u + ) be the shock speed. Then the function u(t, ) = { u if < λt, u + if > λt, is a weak solution to the Riemann problem. In the genuinely nonlinear case, this shock is admissible (La condition, Liu condition) iff σ <. t = t λ u = u u = u + 13
Solution to a 2 2 Riemann problem R 2 1 rarefaction t 2 shock u 2 ω 1 R 1 u = ω 1 u u = u u = u+ u + S 1 u 1 S 2 14
Solution of the general Riemann problem (P. La, 1957) u t + f(u) = u(, ) = { u if < u + if > Find states u = ω, ω 1,... ω n such that ω = u ω n = u + and every couple ω i 1, ω i are connected by an elementary wave (shock or rarefaction) either ω i = R i (σ i )(ω i 1 ) σ i or ω i = S i (σ i )(ω i 1 ) σ i < 15
General solution of the Riemann problem: concatenation of elementary waves t ω ω 2 1 ω = u ω = u + 3 16
Assume: u + u small Implicit function theorem = eistence, uniqueness of the intermediate states ω, ω 1,..., ω n Ψ i (σ)(u) = { Ri (σ)(u) if σ S i (σ)(u) if σ < S i R i u r i (u ) (σ 1, σ 2,..., σ n ) Ψ n (σ n ) Ψ 2 (σ 2 ) Ψ 1 (σ 1 )(u ) Jacobian matri at the origin: J =. ( r 1 (u ) r 2 (u ) always has full rank ) r n (u ) 17
Global solution to the Cauchy problem u t + f(u) =, u(, ) = ū() Theorem (Glimm 1965). Assume: system is strictly hyperbolic each characteristic field is either linearly degenerate or genuinely nonlinear Then there eists a constant δ > such that, for every initial condition ū L 1 (IR; IR n ) with Tot.Var.(ū) δ, the Cauchy problem has an entropy admissible weak solution u = u(t, ) defined for all t. 18
Construction of a sequence of approimate solutions: by piecing together solutions of Riemann problems - on a fied grid in t- plane (Glimm scheme) t 2 t * * θ = 1/3 2 t * * * θ = 1/2 1 2 4 - at points where fronts interact (front tracking) t 4 t t 3 α β t 2 t 1 σ σ 19
Piecewise constant approimate solution to a Riemann problem replace centered rarefaction waves with piecewise constant rarefaction fans u(t) ω =ω i 1 i, ω i,1 ω i,2 ω i,3 R i ω =ω i i,4 t _ ω i 1 i,j ω i u t 2,1 1,1 ω 2 3,j ω 1 ω ω 3 2
Front Tracking Approimations t t 4 t 3 α β t 2 t 1 σ σ Approimate the initial data ū with a piecewise constant function. Construct a piecewise constant approimate solution to each Riemann problem at t = at each time t j where two fronts interact, construct a piecewise constant approimate solution to the new Riemann problem... NEED TO CHECK: - total variation remains small - number of wave fronts remains finite 21
Interaction Estimates GOAL: estimate the strengths of the waves in the solution of a Riemann problem, depending on the strengths of the two interacting waves σ, σ σ i σ j σ k u l σ u m σ" u r Incoming: a j-wave of strength σ and an i-wave of strength σ Outgoing: waves of strengths σ 1,..., σ n. Then σ i σ + σ j σ + σ k = O(1) σ σ k i,j 22
Incoming: two i-waves of strengths σ and σ Outgoing: waves of strengths σ 1,..., σ n. Then σ i σ σ + k i σ k = O(1) σ σ ( σ + σ ) σ k σ i σ σ" u l m u u r 23
Glimm Functionals Total strength of waves: V (t). = α σ α Wave interaction potential: Q(t) =. σ α σ β (α,β) A A =. couples of approaching wave fronts t σ σ β α τ σ α σ β σ γ 24
Changes in V, Q at time τ when the fronts σ α, σ β interact: V (τ) = O(1) σ α σ β Q(τ) = σ α σ β + O(1) V (τ ) σ α σ β Choosing a constant C large enough, the map t V (t) + C Q(t) is nonincreasing, as long as V remains small Total variation initially small = global BV bounds Tot.Var.{u(t, )} V (t) V () + C Q() Front tracking approimations can be constructed for all t 25
Keeping finite the number of wave fronts At each interaction point, the Accurate Riemann Solver yields a solution, possibly introducing several new fronts Number of fronts can become infinite in finite time accurate Riemann solver simplified Riemann solver Need: a Simplified Riemann Solver, producing only one non-physical front 26
A sequence of approimate solutions u t + f(u) = u(, ) = ū() let ε ν as ν (u ν ) ν 1 sequence of approimate front tracking solutions initial data satisfy u ν (, ) ū L 1 ε ν all shock fronts in u ν are entropy-admissible each rarefaction front in u ν has strength ε ν at each time t, the total strength of all non-physical fronts in u ν (t, ) is ε ν 27
t S R S S NP R t 2 t 1 α
Eistence of a convergent subsequence Tot.Var.{u ν (t, )} C u ν (t) u ν (s) L 1 (t s) [total strength of all wave fronts] [maimum speed] L (t s) Helly s compactness theorem = a subsequence converges u ν u in L 1 loc 28
Claim: u = lim ν u ν is a weak solution ( ) {φ t u + φ f(u)} ddt = φ Cc 1 ], [ IR Need to show: {φt lim u ν + φ f(u ν ) } ddt = ν 29
T { } φ t (t, )u ν (t, ) + φ (t, )f(u ν (t, )) R R S S ddt ( ) R Supp.φ Γ j n α NP S R = Rarefaction front S = Shock front NP = Non physical front Assume φ(t, ) = outside the strip [, T ] IR. Define u ν (t, α ). = u ν (t, α +) u ν (t, α ) f(u ν (t, α )) =. f(u ν (t, α +)) f(u ν (t, α )). Φ ν = (φ u ν, φ f(u ν )). Use the divergence theorem on each polygonal domain Γ j where u ν is constant: ( ) = div Φ ν (t, ) ddt = Φ ν n dσ j Γ j j Γ j 3
T R R S S R Supp.φ Γ j n α NP S R = Rarefaction front S = Shock front NP = Non physical front lim sup ν = Φ ν n dσ Γ j j lim sup ν lim sup ν α S R N P [ẋα (t) u ν (t, α ) f(u ν (t, α )) ] φ(t, α (t)) {O(1) ε ν σ α + O(1) α R α N P } σ α ma φ(t, ) t, 31
The Glimm scheme u t + f(u) = u(, ) = ū() Assume: all characteristic speeds satisfy λ i (u) [, 1] This is not restrictive. If λ i (u) [ M, M], simply change coordinates: y = + Mt, τ = 2Mt Choose: a grid in the t- plane with step size t = a sequence of numbers θ 1, θ 2, θ 3,... uniformly distributed over [, 1] #{j ; 1 j N, θ j [, λ] } lim N N θ 3 θ = λ for each λ [, 1]. θ 1 2 λ 1 32
Glimm approimations Grid points : j = j, t k = k t for each k, u(t k, ) is piecewise constant, with jumps at the points j. The Riemann problems are solved eactly, for t k t < t k+1 at time t k+1 the solution is again approimated by a piecewise constant function, by a sampling technique t * * * θ = 1/3 2 t * * * θ = 1/2 1 33
Eample: shock Glimm s scheme applied to a solution containing a single U(t, ) = { u + if > λt, u if < λt. t t u=u * * * * * * * * * * u=u + Fi T >, take = t = T/N (T ) = #{j ; 1 j N, θ j [, λ] } t = #{j ; 1 j N, θ j [, λ] } N T λt as N 34
Rate of convergence for Glimm approimations Random sampling at points determined by the equidistributed sequence (θ k ) k 1 #{j ; 1 j N, θ j [, λ] } lim N N = λ for each λ [, 1]. Need fast convergence to uniform distribution. Achieved by choosing: θ 1 =.1,..., θ 759 =.957,..., θ 3922 =.2293,... Error Estimate: u Glimm (T, ) u eact (T, ) lim L 1 = ln (A.Bressan & A.Marson, 1998) 35