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


When 
Sep 21, 2022
from 02:00 PM to 03:00 PM 
Where  via zoom 
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Abstract: In the first part of the talk, we discuss uniqueness aspects of compressible Euler equations and related hyperbolic PDEs. In multidimension, the wellposedness 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) αHolder regularity with α > 1/2 and (ii) a onesided bound condition. Then we extend our discussion on uniqueness for the general hyperbolic system of conservation laws. We will also discuss a weakstrong 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 1D, 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 C^{2} flux in multiD. We exhibit a characterization of C^{2} fluxes in 1D based on BV regularizing. We will also show a global propagation of nonBV^{s} oscillation for power law type fluxes, f(u)= u^{p+1}in 1D. 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].