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We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.
The one-dimensional viscous conservation law is considered on the whole line $$ u_t + f(u)_x=eps u_{xx},quad (x,t)inRRtimesoverline{RP},quad eps>0, $$ subject to positive measure initial data. The flux $fin C^1(RR)$ is assumed to satisfy a
The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{tge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers
We consider two discrete completely integrable evolutions: the Toda Lattice and the Ablowitz-Ladik system. The principal thrust of the paper is the development of microscopic conservation laws that witness the conservation of the perturbation determi
We show the convergence of the zero relaxation limit in systems of $2 times 2$ hyperbolic conservation laws with stochastic initial data. Precisely, solutions converge to a solution of the local equilibrium approximation as the relaxation time tends
We prove the decay of the L 2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.