arXiv subjects

Shyam Sundar Ghoshal

Publications and source records attributed to Shyam Sundar Ghoshal.

At least 19 recordsLinked to original sources

arXiv2026

Failure of uniqueness for scalar conservation laws

In this article, we develop the first negative results for scalar conservation laws with finite speed of propagation. While the work of Gargyants, Goritsky, and Panov establishes non-uniqueness of unbounded solutions for scalar conservation laws with spatially homogeneous flux, heterogeneity allows entropy solutions to exhibit more pathological behaviours. In particular, even for smooth and bounded initial data, the entropy solution may not be unique in the full half plane. Furthermore, the multiple entropy solutions all satisfy a finite speed of propagation property. We begin with explicit examples where bounded initial data leads to $L^{\infty}$ blow-up despite flux regularity. More strikingly, we demonstrate that Kružkov's entropy inequalities alone fail to ensure uniqueness in this regime by constructing infinitely many entropy solutions to a single Cauchy problem with bounded initial datum, each continuous in time with respect to the $L^{1}$ norm. Thus, we demonstrate that the $L^{\infty}$ assumption is essential for the doubling of variables argument, and hence for the uniqueness of entropy solutions to scalar conservation laws. On the positive side, we develop a novel theory for scalar conservation laws with spatial heterogeneity by adapting the front tracking method. We recover uniqueness by imposing a Lax-type condition in addition to the entropy inequality, motivated by the properties of our front tracking approximations. Unbounded Kružkov solutions do not necessarily satisfy the weak formulation; we show that global weak solutions may not even exist in a natural class for some Cauchy problems of this form, even when Kružkov entropy solutions exist. Finally, we detail examples demonstrating the sharpness of our assumptions and construct an explicit example of global ill-posedness with bounded initial datum.

math.AP↗

arXiv2026

$L^2$ Stability of Simple Shocks for Spatially Heterogeneous Conservation Laws

In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We identify structural assumptions under which a single shock wave connecting two constant states emerges in finite time for all $L^{\infty}$ initial data satisfying the same far-field conditions. Under a further condition on the mixed partial derivative of the flux, we establish the $L^2$-stability of these simple shock profiles: perturbations of the Riemann initial data yield solutions whose $L^2$ distance from the corresponding simple shock wave is non-increasing in time, up to a time-dependent spatial shift. We further show that these conditions are sharp: we construct explicit counterexamples demonstrating that, in particular, the existence of a contractive shift function holds only under our assumptions and fails otherwise. The main tools we use are Dafermos' generalised characteristics for the evolution analysis and the relative entropy method for stability.

math.AP↗

arXiv2025

A convergence rate result for front tracking approximations of conservation laws with discontinuous flux

We consider the initial value problem for a scalar conservation law in one space dimension with a single spatial flux discontinuity, the so-called two-flux problem. We prove that a well-known front tracking algorithm has a convergence rate of at least one-half. The fluxes are required to be smooth, but are not required to be convex or concave, monotone, or even unimodal. We require that there are no more than finitely many flux crossings, but we do not require that they satisfy the so-called crossing condition. If both fluxes are strictly increasing or strictly decreasing then our analysis yields a convergence rate of one, in agreement with a recent result. Similarly, if the fluxes are equal, i.e., there is no flux discontinuity, we obtain a convergence rate of one in this case also, in agreement with a classical result. The novelty of this paper is that the class of discontinuous-flux conservation laws for which there is a front tracking error estimate is expanded, and that the method of analysis is new; we do not use the Kuznetsov lemma which is commonly used for this type of analysis.

math.AP↗

arXiv2025

A Non-Conservative, Non-Local Approximation of the Burgers Equation

The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel inviscid, non-local regularisation in non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation and supremum norm, and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result is sharp, since we can demonstrate non-convergence when the initial data features simple discontinuities. Conservation laws with linear flux derivative, such as the Burgers equation, behave better in the presence of discontinuities. Hence, we devote special attention to the limiting behaviour of non-local solutions with respect to the Burgers equation for a simple class of discontinuous initial data.

math.AP↗

arXiv2024

On blow up of $C^1$ solutions of isentropic Euler system

In this article, we study the break-down of smooth and continuous solutions to isentropic Euler system in multi dimension. Sideris [Comm. Math. Phys. 1985] proved the blow up of smooth solutions when initial data satisfies an `integral condition'. We show that a $C^1$ solution of isentropic Euler equation breaks down if (i) gradient of initial velocity has a negative real eigenvalue at some point $x_0\in\mathbb{R}^d$ and (ii) Hessian of initial density satisfies a smallness condition in Sobolev space. Our proof also works for the data which fails to satisfy the above-mentioned `integral condition'. Furthermore, we prove the global existence of smooth solution when (i) eigenvalues of gradient of initial velocity have non-negative real-part and (ii) initial density satisfies a smallness condition. This extends the global existence result of [Grassin, Indiana Univ. Math. J. 1998]. Another goal of this article is to study the breakdown of continuous weak solutions of isentropic Euler equations. We are able to show that the `integral condition' of Sideris can cause the breakdown of continuous solutions in finite time. This improves the blow up result of Sideris from $C^1$ to continuous space.

math.AP↗

arXiv2024

Optimal regularity for all time for entropy solutions of conservation laws in $BV^s$

This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional $BV$ spaces. We show that optimality of the regularizing effect for the initial value problem from $L^\infty$ to fractional Sobolev space and fractional $BV$ spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover , we prove sharp smoothing in $BV^s$ for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

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↗

arXiv2023

Existence of BV solution for the Euler-Poisson system in one dimension with large initial data

This paper deals with the existence of BV solution for the Euler-Poisson system endowed with a $γ$ pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when $γ=1+2ε$ satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differential Equations, 1995]. Existence of BV solution of 1-D isentropic Euler equation for large data and $γ=1+2ε$ is proved in [Nishida and Smoller, Comm. Pure Appl. Math, 1973]. Due to the presence of electric field, the difficulty arises while controlling the Glimm functional for the Euler-Poisson system. It requires a subtle study of wave interaction. In the later part of this article, we discuss the initial-boundary value problem for the Euler-Poisson system. We prove the existence of $BV$ solution for the initial-boundary value problem with large initial and boundary data. By an explicit example, we also show ill-posedness of initial-boundary value problem for the isentropic Euler equation.

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↗

arXiv2021

Weak-Strong Uniqueness for the Isentropic Euler Equations with Possible Vacuum

We establish a weak-strong uniqueness result for the isentropic compressible Euler equations, that is: As long as a sufficiently regular solution exists, all energy-admissible weak solutions with the same initial data coincide with it. The main novelty in this contribution, compared to previous literature, is that we allow for possible vacuum in the strong solution.

math.AP↗

arXiv2021

A Godunov type scheme and error estimates for multidimensional scalar conservation laws with Panov-type discontinuous flux

This article concerns a scalar multidimensional conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and prove that entropy solutions are unique. We propose a Godunov-type finite volume scheme and prove that the Godunov approximations converge to an entropy solution, thus establishing existence of entropy solutions. We also show that our numerical scheme converges at an optimal rate of $\mathcal{O}(\sqrt{\D t}).$ To the best of our knowledge, convergence of the Godunov type methods in multi-dimension and error estimates of the numerical scheme in one as well as in several dimensions are the first of it's kind for conservation laws with discontinuous flux. We present numerical examples that illustrate the theory.

math.NA↗

arXiv2020

Convergence of a Godunov scheme for degenerate conservation laws with BV spatial flux and a study of Panov type fluxes

In this article we prove convergence of the Godunov scheme of [16] 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 contrast to the study appearing in [16], we do not restrict the flux to be unimodal. We allow for the case where the flux has degeneracies, i.e., the flux may vanish on some interval of state space. Since the flux is allowed to be degenerate, the corresponding singular map may not be invertible, and thus the convergence proof appearing in [16] does not pertain. We prove that the Godunov approximations nevertheless do converge in the presence of flux degeneracy, using an alternative method of proof. We additionally consider the case where the flux has the form described in [21]. For this case we prove convergence via yet another method. This method of proof provides a spatial variation bound on the solutions, which is of independent interest. We present numerical examples that illustrate the theory.

math.AP↗

arXiv2020

Energy-balance for the incompressible Euler equations with stochastic forcing

We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder regularity $C^α$, $α>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having Hölder regularity $C^α$, $α>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.

math.AP↗

arXiv2020

Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity $C^α,α>1/2$ in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager's conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying Hölder regularity $C^α,α>1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.

math.AP↗

arXiv2020

Well-posedness for conservation laws with spatial heterogeneities and a study of BV regularity

In this article, we consider scalar conservation laws with fluxes having spatial discontinuities and possible flat regions and study the following three aspects: (i) existence, (ii) uniqueness and (iii) BV regularity of solutions. We propose a uniqueness condition and prove existence of a weak solution via the method of wave front tracking. In the later part of the article, a BV bound of the solution is achieved under a suitable condition on the initial data and flux. We construct two counterexamples showing BV blow-up of the solution which proves the optimality on the assumptions

math.AP↗

arXiv2020

Non existence of the BV regularizing effect for scalar conservation laws in several space dimension

This article deals with the regularity aspects of entropy solutions to scalar conservation laws. We show that for each C2 flux in multi-D, there exists an entropy solution which does not belong to BV locally for all time. For this purpose, we construct a non-BVloc solution in 1-D for a special class of C2 fluxes whose second derivative has a zero. It covers all the C2 functions for which Lax-Oleinik's BV regularizing result is not applicable and provides a classification of one dimensional C2 fluxes based on L\infty-BVloc regularizing of entropy solution. In the later part of this article, we extend our result to fractional Sobolev spaces for a class of non-degenerate fluxes.

math.AP↗

arXiv2020

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.

math.AP↗

arXiv2020

On the uniqueness of solutions to hyperbolic systems of conservation laws

For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{α,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative solutions. The exponent $α>1/2$ is universal independently of the nature of the nonlinearity and the Besov regularity need only be imposed in space when the system is expressed in appropriate variables. The proof utilises a commutator estimate which allows for an extension of the relative entropy method to the required regularity setting. The systems of elasticity, shallow water magnetohydrodynamics, and isentropic Euler are investigated, recovering recent results for the latter. Moreover, the article explores a triangular system motivated by studies in chromatography and constructs an explicit solution which fails to be Lipschitz, yet satisfies the conditions of the presented uniqueness result.

math.AP↗