Well-posedness of the Cauchy problem for nxn systems of conservation laws (original) (raw)

Uniqueness of Weak Solutions to Systems of Conservation Laws

Archive for Rational Mechanics and Analysis, 1997

Consider a strictly hyperbolic n n system of conservation laws in one space dimension: u t + F(u) x = 0: ( ) Relying on the existence of the Standard Riemann Semigroup generated by ( ), we establish the uniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumption on the variation of u along space-like segments. 0

L² semigroup and linear stability for Riemann solutions of conservation laws

Dynamics of Partial Differential Equations, 2005

Riemann solutions for the systems of conservation laws uτ +f (u) ξ = 0 are self-similar solutions of the form u = u(ξ/τ). Using the change of variables x = ξ/τ, t = ln(τ), Riemann solutions become stationary to the system ut + (Df (u) − xI)ux = 0. For the linear variational system around the Riemann solution with n-Lax shocks, we introduce a semigroup in the Hilbert space with weighted L 2 norm. We show that (A) the region λ > −η consists of normal points only. (B) Eigenvalues of the linear system correspond to zeros of the determinant of a transcendental matrix. They lie on vertical lines in the complex plane. There can be resonance values where the response of the system to forcing terms can be arbitrarily large, see Definition 6.2. Resonance values also lie on vertical lines in the complex plane. (C) Solutions of the linear system are O(e γt) for any constant γ that is greater than the largest real parts of the eigenvalues and the coordinates of resonance lines. This work can be applied to the linear and nonlinear stability of Riemann solutions of conservation laws and the stability of nearby solutions of the Dafermos regularizations ut + (Df (u) − xI)ux = uxx.

Well-Posedness for a Class of Hyperbolic Systems of Conservation Laws in Several Space Dimensions

Communications in Partial Differential Equations, 2005

In this paper we consider a system of conservation laws in several space dimensions whose nonlinearity is due only to the modulus of the solution. This system, first considered by Keyfitz and Kranzer in one space dimension, has been recently studied by many authors. In particular, using standard methods from DiPerna-Lions theory, we improve the results obtained by the first and third author, showing existence, uniqueness and stability results in the class of functions whose modulus satisfies, in the entropy sense, a suitable scalar conservation law. In the last part of the paper we consider a conjecture on renormalizable solutions and show that this conjecture implies another one recently made by Bressan in connection with the system of Keyfitz and Kranzer.

A uniqueness condition for hyperbolic systems of conservation laws

Discrete and Continuous Dynamical Systems, 2000

Page 1. A Uniqueness Condition for Hyperbolic Systems of Conservation Laws Alberto Bressan and Marta Lewicka SISSA, Via Beirut 4, Trieste 34014, Italy. Abstract. Consider the Cauchy problem for a hyperbolic n × n system of conservation laws in one space dimension: ...

On the structure of solutions of nonlinear hyperbolic systems of conservation laws

Communications on Pure and Applied Analysis, 2011

We are concerned with entropy solutions u in L ∞ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function η and any hyperplane t = const., if u satisfies a vanishing mean oscillation property on the half balls, then η(u) has a trace H d -almost everywhere on the hyperplane. For the general case, given any set E of finite perimeter and its inner unit normal ν : ∂ * E → S d and assuming the vanishing mean oscillation property of u on the half balls, we show that the weak trace of the vector field (η(u), q(u)), defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair (η, q). We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.