Multi-dimensional scalar conservation laws with unbounded integrable initial data

Abstract : 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. Notations. When 1 ≤ p ≤ ∞, the natural norm in L p (R n) is denoted · p , and the conjugate exponent of p is p. The total space-time dimension is d = 1 + n and the coordinates are x = (t, y). In the space of test functions, D + (R 1+n) is the cone of functions which take non-negative values. The partial derivative with respect to the coordinate y j is ∂ j , while the time derivative is ∂ t. Various finite positive constants that depend only the dimension, but not upon the solutions of our PDE, are denoted c d ; they usually differ from one inequality to another one. * U.M.P.A., UMR CNRS–ENSL # 5669. 46 allée d'Italie,
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Denis Serre. Multi-dimensional scalar conservation laws with unbounded integrable initial data. 2018. ⟨ensl-01837034v1⟩

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