Regularity aspects of hyperbolic conservation laws

par Dr Shyam Sundar Ghoshal (TIFR-CAM, Bangalore, India)

jeudi 17 mars 2022, 15:30 → 16:30 Europe/Paris

E2290 (Tours)

For scalar conservation laws, it is an open question to obtain the optimal regularity for entropy
solutions arising from bounded initial data. For uniformly convex flux in 1-D, solution belongs
to BVloc in any positive time even for L

∞ data. By explicit examples, we show [1] that BV

regularizing is not true for any C
2
flux in multi-D. We are also able to characterize the 1-D C
2

fluxes based on BV-regularizing.
In the second part of the talk, we will discuss about jump sets of entropy solutions for hyperbolic
conservation laws. With explicit construction of entropy solution, we prove [2] that jump sets of
scalar conservation laws may not be closed in general, in fact they are dense. This solves a question
proposed by [Silvestre, Comm. Pure Appl. Math. 2019]. Then we extend the result for hyperbolic
system of conservation laws via a different construction.

 

References
[1] S. S. Ghoshal and A. Jana, Non existence of the BV regularizing effect for scalar conservation
laws in several space dimension for C
2
fluxes. SIAM J. Math. Anal. 53, 2, 1908–1943, 2021.
[2] S. S. Ghoshal and A. Jana, Optimal jump set in hyperbolic conservation laws. J. Hyperbolic
Differ. Equ. 17, 4, 765–784, 2020.