Some $$L^p$$ L p estimates for rotationally invariant systems of viscous conservation laws

Abstract

We study the Cauchy problem for a system of one-dimensional viscous conservation laws with rotational invariance, \(\displaystyle \mathbf {u}_{t} + \left[ \;\!|\mathbf {u}|^{2}\mathbf {u}\;\!\right] _{x} = \mathbf {u}_{xx}\),with the aim of deriving some a priori estimates for various \(L^{p}\) norms of its solutions through a direct method.