Electron. J. Differential Equations, Vol. 2016 (2016), No. 325, pp. 1-18.

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

Martin Lazar, Darko Mitrovic

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.

Submitted May 18, 2016. Published December 22, 2016.
Math Subject Classifications: 35L65.
Key Words: Scalar conservation law; discontinuous coefficient; existence; velocity averaging; H-measure.

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Martin Lazar
University of Dubrovnik, Croatia
email: mlazar@unidu.hr
Darko Mitrović
Faculty of Mathematics
University of Montenegro
Montenegro
email: darkom@ac.me

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