arXiv subjects

Shyam Sundar Ghoshal

Publications and source records attributed to Shyam Sundar Ghoshal.

28 records · Page 2Linked to original sources

arXiv2020

Uniqueness of dissipative solutions to the complete Euler system

Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure-valued solutions. Motivated from [Feireisl, Ghoshal and Jana, Commun. Partial Differ. Equ., 2019], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $B^{α,\infty}_{p}$ with $α>1/2$. We prove that the Besov solution satisfying the above-mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one-sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, $B^{α,\infty}_{3}$ for $α>1/3$. Our proof relies on commutator estimates for Besov functions and the relative entropy method.

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arXiv2020

Convergence of a Godunov scheme to an Audusse-Perthame adapted entropy solution for conservation laws with BV spatial flux

In this article we consider the initial value problem for a scalar conservation law in one space dimension with a spatially discontinuous flux. There may be infinitely many flux discontinuities, and the set of discontinuities may have accumulation points. Thus the existence of traces cannot be assumed. In [6] Audusse and Perthame proved a uniqueness result that does not require the existence of traces, using adapted entropies. We generalize the Godunov-type scheme of Adimurthi, Jaffre and Gowda [2] for this problem with the following assumptions on the flux function, (i) the flux is BV in the spatial variable and (ii) the critical point of the flux is BV as a function of the space variable. We prove that the Godunov approximations converge to an adapted entropy solution, thus providing an existence result, and extending the convergence result of Adimurthi, Jaffre and Gowda.

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arXiv2019

A sufficient condition for uniqueness of weak solutions of the incompressible Euler system

We give a new sufficient criteria to prove the uniqueness of the incompressible Euler equation in dimension $N\geq2$. In their celebrated works by V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Székelyhidi Jr. [7] they have obtained the nonuniqeness of weak solutions of incompressible Euler equation. Here we obtain uniqueness criteria for the same equation under some mild regularity condition on weak solutions. Our proof is simple and can be employed to other equations like inhomogeneous incompressible Euler and Euler-Boussinesq equations. One of the key ingredients in our proof is commutator estimate [5, 11].

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arXiv2019

Optimal jump set in hyperbolic conservation laws

This paper deals with some qualitative properties of entropy solutions to hyperbolic conservation laws. In [11] the jump set of entropy solution to conservation laws has been introduced. We find an entropy solution to scalar conservation laws for which the jump set is not closed, in particular, it is dense in a space-time domain. In the later part of this article, we obtain a similar result for the hyperbolic system. We give two different approaches for scalar conservation laws and hyperbolic system to obtain the results. For the scalar case, obtained solutions are more explicitly calculated.

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arXiv2019

On uniqueness of dissipative solutions to the isentropic Euler system

The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure--valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: {\bf (i)} the weak solution enjoys certain Besov regularity; {\bf (ii)} the symmetric velocity gradient of the weak solution satisfies a one--sided Lipschitz bound.

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arXiv2019

Proof of Lions-Perthame-Tadmor conjecture and Jabin's conjecture on regularity for scalar conservation laws in higher dimension

[Note: Currently the proof is incomplete as we are using the lemma 3.2 which is not true in general]. We offer a complete resolution of a conjecture by Lions-Perthame-Tadmor mentioned in their celebrated work (1994, [34]). We prove the optimal regularizing effect to W^{s,p}_{loc} with the best exponent s, of the entropy solution for scalar conservation laws in higher dimension. In addition, we prove t (div f^\p(u))\in\mathcal{M}^1_{loc} for arbitrary flux f which was conjectured by Jabin (2010, [29]). Here we follow a new approach to understand the regularity issues in multi-dimensional scalar conservation laws.

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arXiv2018

Single shock solution for non convex scalar conservation laws

In this paper we study the finite time emergence of one shock for the solution of scalar conservation laws in one space dimension with general flux f . We give a necessary and sufficient condition to the initial data connecting to flux. The proof relies on the structure theorem for the linear degenerate flux and the finer analysis of characteristic curves.

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arXiv2016

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper deals with an optimal control problem and describes the reachable set for the scalar 1-D conservation laws with discontinuous flux. Regarding the optimal control problem we first prove the existence of a minimizer and then we prescribe an algorithm to compute it. The same method also applies to compute the initial data control. The proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics.

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arXiv2016

BV regularity near the interface for nonuniform convex discontinuous flux

In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.

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arXiv2012

Finer analysis of characteristic curves and its application to shock profile, exact and optimal controllability of a scalar conservation law with strict convex flux

Here we consider scalar conservation law in one space dimension with strictly convex flux. Goal of this paper is to study two problems. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact controllability. This was studied for Burgers equation and some partial results are obtained for large time. It was a challenging problem to know the controllability for all time and also for general convex flux. In a seminal paper, Dafermos introduces the characteristic curves and obtain some qualitative properties of a solution of a convex conservation law. In this paper, we further study the finer properties of these characteristic curves. As a bi-product we solve these two problems in complete generality. In view of the explicit formulas of Lax - Oleinik, Joseph - Gowda, target functions must satisfy some necessary conditions. In this paper we prove that it is also sufficient. Method of the proof depends highly on the characteristic methods and explicit formula given by Lax - Oleinik and the proof is constructive. This method allows to solve the optimal controllability problem in a trackable way.

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