Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws (original) (raw)

Existence of Solutions to Hyperbolic Conservation Laws with a Source

Communications in Mathematical Physics, 1997

Existence of solutions to three different systems of Eqs. (1.1), (1.2) and (1.3) coming from physically relevant models is shown, each needing a different proof which are given in Sects. 2, 3 and 4. The unifying theme is the presence of source terms and the general method of proof is vanishing viscosity together with compensated compactness. For system (1.2) entropy-entropy flux pairs of Lax type are constructed and estimates from singular perturbation theory of ODEs are used. For (1.1) and (1.3) weak entropyentropy flux pairs are constructed following the compensated compactness framework set up by Diperna [4].

Notes on Hyperbolic Conservation Laws

2009

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of the conservation equations and definition of weak solutions. 2. Shocks, Rankine-Hugoniot equations and admissibility conditions. 3. Genuinely nonlinear and linearly degenerate characteristic fields. Solution to the Riemann problem. Wave interaction estimates. 4. Weak solutions to the Cauchy problem, with initial data having small total variation. Proof of global existence via front-tracking approximations. 5. Continuous dependence of solutions w.r.t. the initial data, in the L distance. 6. Uniqueness of entropy-admissible weak solutions. 7. Approximate solutions constructed by the Glimm scheme. 8. Vanishing viscosity approximations. 9. Counter-examples to global existence, uniqueness, and continuous dependence of solutions, when some key hypotheses are removed. The survey is concluded with an Appendix, rev...

Lecture notes on hyperbolic conservation laws

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of the conservation equations and definition of weak solutions. 2. Shocks, Rankine-Hugoniot equations and admissibility conditions. 3. Genuinely nonlinear and linearly degenerate characteristic fields. Solution to the Riemann problem. Wave interaction estimates. 4. Weak solutions to the Cauchy problem, with initial data having small total variation. Proof of global existence via front-tracking approximations. 5. Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. 6. Uniqueness of entropy-admissible weak solutions. 7. Approximate solutions constructed by the Glimm scheme. 8. Vanishing viscosity approximations. 9. Counter-examples to global existence, uniqueness, and continuous dependence of solutions, when some key hypotheses are removed. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous chapters.

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.

Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws

Indiana University Mathematics Journal, 1999

The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm. We then consider a sequence of exact or approximate solutions u ν , converging to a solution u in L 1. The convergence of the wave-fronts of u ν to the corresponding fronts of u is studied, proving a structural stability result in a neighborhood of each point in the t-x plane.

ON QUANTITATIVE COMPACTNESS ESTIMATES FOR HYPERBOLIC CONSERVATION LAWS

We are concerned with the compactness in L 1 loc of the semigroup (St) t≥0 of entropy weak solutions generated by hyperbolic conservation laws in one space dimension. This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov ε-entropy of the image through the mapping St of bounded sets in L 1 ∩ L ∞ , both in the case of scalar and of systems of conservation laws. As suggested by Lax [16], these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.