Optimal regularity for conservation laws with discontinuous flux  and existence for the isentropic Euler equations with large BV data

PhD Synopsis by Mr Akash Parmar ( TIFR CAM )

Abstract

In this talk, first we discuss the regularity aspects of the scalar conservation laws with discontinuous flux. The optimal regularity of the entropy solution was an open question since 2011 [Adimurthi et. al, Comm. Pure Appl. Math., 2011], as they established that the entropy solution of scalar conservation laws with discontinuous flux does not belong to BV space even when the initial data is in BV. We prove the  optimal regularity of the entropy solution  for bounded initial data in the "fractional BV" space,  (i.e.,  we prove the Lax-Oleinik type regularizing effect). Furthermore, we demonstrate the higher regularity of the entropy solution in  the "fractional BV"  space for the geometrically restricted fluxes. In the second part of the talk, we discuss the global existence of a weak solution for the isentropic Euler equation in one space dimension with large initial data in BV. We use the Glimm's scheme to get the approximate weak solution and then use the Riemann invariant to get the uniform bounds of the total variation. We also provide a criterion under which the total variation of Riemann invariants decays.