BV regularity of the adapted entropy solutions for conservation laws with infinitely many spatial discontinuities

Abstract

<abstract><p>In this article, we focus on the BV regularity of the adapted entropy solutions of the conservation laws whose flux function contains infinitely many discontinuities with possible accumulation points. It is well known that due to discontinuities of the flux function in the space variable, the total variation of the solution can blow up to infinity in finite time. We establish the existence of total variation bounds for certain classes of fluxes and the initial data. Furthermore, we construct two counterexamples, which exhibit $ {\rm{BV}} $ blow-up of the entropy solution. These counterexamples not only demonstrate that these assumptions are essential, but also show that the BV-regularity result of [S. S. Ghoshal, J. Differential Equations, 258 (3), 980-1014, 2015] does not hold true when the spatial discontinuities of the flux are infinite.</p></abstract>