On the system of conservation laws and its perturbation in the Besov spaces

Abstract

We prove that a system of conservation laws on $ \mathbb R^N$ is locally well posed in the `critical' Besov space $B^{\frac{N}{2} +1}_{2,1}$. The time of local existence depends only on the size of the inhomogeneous part of the initial data. We also obtain a blow-up criterion of the local solution. For the conservation system with a dissipation term added we prove global existence of solutions under the assumption of smallness of the homogeneous part of the arbitrary-sized $B^{\frac{N}{2} -1}_{2,1}$ norm of the initial data. For the proof of these results we essentially use the Littlewood-Paley decomposition of functions to derive the energy type of estimate.