Journal ArticleDOI
Rotational degeneracy of hyperbolic systems of conservation laws
TLDR
In this paper, the authors study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy, and they give a unified presentation of examples from continuum mechanics.Abstract:
The purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy. In this situation hyperbolicity is necessarily non-strict, so that the elementary waves have interesting patterns. The discussion is centered around a theorem on existence and uniqueness of solutions of the Riemann problem. We give a unified presentation of examples from continuum mechanicsread more
Citations
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MonographDOI
Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves
Denis Serre,I. N. Sneddon +1 more
TL;DR: In this paper, the author sets up the foundations of the modern theory of conservation laws, describing the physical models and mathematical methods, leading to the Glimm scheme, and then takes the reader to the current state of knowledge in the subject.
Book ChapterDOI
Chapter 4 – Notes on Hyperbolic Systems of Conservation Laws and Transport Equations
TL;DR: In this article, Ambrosio's renormalization Theorem 4 1.4.5.6.1.1 Theorem 5 1.2.3 Theorem 6 1.3.
Journal ArticleDOI
Well-Posedness for a Class of Hyperbolic Systems of Conservation Laws in Several Space Dimensions
TL;DR: In this article, a system of conservation laws in several space dimensions whose nonlinearity is due only to the modulus of the solution is considered, using standard methods from DiPerna-Lions theory.
Journal ArticleDOI
Nonuniqueness of solutions of Riemann problems
TL;DR: In this paper, a general mechanism for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws was proposed, which occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU 1 and U 2 such that a traveling wave leads fromU 1 to U 2 and another leads from U 2 to U 1.
Journal ArticleDOI
Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws
TL;DR: In this paper, it was shown that certain non-classical shock waves in a rotationally invariant system of viscous conservation laws posses nonlinear large-time stability against sufficiently small perturbations.
References
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Journal ArticleDOI
Formation of singularities in one‐dimensional nonlinear wave propagation
TL;DR: In this paper, it was shown that if the system (1) is "genuinely nonlinear" in a sense defined below, and if the initial data are "sufficiently small" (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0.