Existence of solutions for a scalar conservation law with a flux of low regularity

Abstract

We prove existence of solutions to Cauchy problem for scalar conservation laws with non-degenerate discontinuous flux $$ \partial_t u+ \hbox{div}f(t,\mathbf{x},u)=s(t,\mathbf{x},u), \quad t\geq 0, \mathbf{x}\in \mathbb{R}^d, $$ where for every $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}$, the flux $f(t,\mathbf{x},\cdot) \in \hbox{Lip}(\mathbb{R};\mathbb{R}^d)$ and $\partial_\lambda f \in L^r(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$, additionally satisfying $\max_{|\lambda| \leq M} f(\cdot,\cdot,\lambda) \in L^r(\mathbb{R}^+\times \mathbb{R}^d)$, for some $r>1$ and every $M>0$, and, for every $\lambda \in \mathbb{R}$, $\hbox{div}_{(t,\mathbf{x})} f(\cdot,\cdot,\lambda) \in \mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ where $\mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ is the space of Radon measures. Moreover, the function s is measurable and both f and s satisfy certain growth rate assumptions with respect to $\lambda$. The result is obtained by means of the H-measures.

Keywords

• Scalar conservation law
• discontinuous coefficient
• existence
• velocity averaging
• H-measure