arXiv subjects

Akash Parmar

Publications and source records attributed to Akash Parmar.

3 recordsLinked to original sources

arXiv2026

Existence and stability estimates for weak solutions of $p$-systems by front tracking scheme

We prove the existence of global weak entropy solutions to the $p$-system with the piecewise affine flux function. The construction is based on a wave-front tracking scheme for which we identify a necessary and sufficient condition for the occurrence of infinitely many fronts. For smooth fluxes, the theory is well established. However, the lower Lipschitz regularity of the flux requires a new method to obtain the interaction estimates. We also prove the stability of the solution in the sense of \cite{bia-col-02}, showing that solutions are Lipschitz continuous with respect to the $L^{\infty}$ norm of the difference of the flux derivatives. We obtain the convergence rate of the solutions from the piecewise affine flux to smooth flux by using the stability estimate.

math.AP↗

arXiv2023

Higher regularity for entropy solutions of conservation laws with geometrically constrained discontinuous flux

For the Burgers equation, the entropy solution becomes instantly BV with only $L^\infty$ initial data. For conservation laws with genuinely nonlinear discontinuous flux, it is well known that the BV regularity of entropy solutions is lost. Recently, this regularity has been proved to be fractional with s = 1/2. Moreover, for less nonlinear flux the solution has still a fractional regularity 0 < s \leq 1/2. The resulting general rule is the regularity of entropy solutions for a discontinuous flux is less than for a smooth flux. In this paper, an optimal geometric condition on the discontinuous flux is used to recover the same regularity as for the smooth flux with the same kind of nonlinearity.

math.AP↗

arXiv2022

Fractional regularity for conservation laws with discontinuous flux

This article deals with the regularity of the entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such equation does not admit BV regularity in general, even when the initial data belongs to BV. Due to this phenomenon fractional BVs spaces wider than BV are required, where the exponent 0<s\leq 1 and BV = BV1. It is a long standing open question to find the optimal regularizing effect for the discontinuous flux with L^\infty initial data. The optimal regularizing effect in BVs is proven on an important case using control theory. The fractional exponent s is at most 1/2 even when the fluxes are uniformly convex.

math.AP↗