We consider di erent situations of blowup in sup norm for hyper bolic equations For scalar conservation laws with a source the asymptotic pro le of the solution close to a blowup point is described in detail Based on an example of Je rey we next show how blowup for ordinary di erential equations can be used to construct examples of blowup for systems of hy perbolic equations Finally we outline ...

Relaxation is a phenomenon which appears in a wide variety of physical situations. In gas dynamics, it occurs when the gas isn’t in thermodynamic equilibrium. In elasticity, it is usually referred to as a fading memory mechanism. Hyperbolic conservation laws with relaxation are also served as discrete kinetic models. Relaxation approximations to scalar conservation laws can be considered for pu...

We study the limiting behavior of systems of hyperbolic conservation laws with stii relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown t...

Abstract. We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup....

Abstract. In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin s...

We study the regularity of discontinuous entropy solutions to scalar hyperbolic conservation laws with uniformly convex fluxes posed as initial value problems on R. For positive α we show that if the initial data has bounded variation and the flux is smooth enough then the solution u( · , t) is in the Besov space Bα σ (L σ) where σ = 1/(α + 1) whenever the initial data is in this space. As a co...

A general idea for solving hyperbolic systems of conservation laws is to use a local relaxation approximation. The motivation is to have a simple discrete velocity kinetic type relaxation regularization which approximates the original system with a small dissipative correction. In this paper we extend the previous approach to systems of nonlinear parabolic equations. The corresponding relaxatio...

We are concerned with the compactness in Lloc 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 L1 ∩ L∞, both in the case of scalar and of systems of conservatio...

In this paper, we study semi-discrete central-upwind difference schemes with a modified multidimensional limiting process (MLP) to solve two-dimensional hyperbolic systems of conservation laws. In general, high-order central difference schemes for conservation laws involve no Riemann solvers or characteristic decompositions but have a tendency to smear linear discontinuities. To overcome this d...

Abstract. We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions–the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prov...