Ph.D. Thesis Viva: Mr. Animesh Jana: Uniqueness and some regularity aspects of hyperbolic conservation laws

Abstract: In the first part of the talk, we discuss uniqueness aspects of compressible Euler equations and related hyperbolic PDEs. In multi-dimension, the well-posedness of entropy solution is an open question for the hyperbolic system of conservation laws. The existence of infinitely many entropy solutions for Riemann type planar data has been shown [Chiodaroli, De Lellis and Kreml, Comm. Pure Appl. Math. 2015] for isentropic Euler equations. We prove the uniqueness of Holder continuous solution for compressible Euler equations in the set of weak solutions satisfying entropy conditions. To prove the uniqueness we impose (i) α-Holder regularity with α > 1/2 and (ii) a one-sided bound condition. Then we extend our discussion on uniqueness for the general hyperbolic system of conservation laws. We will also discuss a weak-strong uniqueness result for the is entropic Euler system  when the strong solution may contain a vacuum region. In the second part of this talk, we focus on regularity aspects of hyperbolic conservation laws. Even for the scalar case, it is still an open question to obtain the optimal regularity for entropy solutions arising from bounded initial data. For uniformly convex flux in 1-D, Lax and Oleinik showed that entropy solutions belong to BV_loc in any positive time even for L^∞ data. By explicit examples, we  show that BV regularizing is not true for any C² flux in multi-D. We exhibit a characterization of C² fluxes in 1-D based on BV -regularizing. We will also show a global propagation of non-BV^s oscillation for power law type fluxes, f(u)= |u|^p+1in 1-D. Lastly, we discuss jump sets of entropy solutions for hyperbolic conservation laws. With explicit construction of entropy solution, we   prove that jump sets for hyperbolic conservation laws may not be closed in general, in fact, they are dense. This solves a conjecture proposed by [Silvestre, Comm. Pure Appl. Math. 2019].