arXiv subjects

Parasuram Venkatesh

Publications and source records attributed to Parasuram Venkatesh.

4 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

Front Tracking for Scalar Conservation Laws with Spatially Heterogeneous Flux

In this article, we propose a novel front tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and prove that approximations converge to the unique entropy solution. The main tools are Dafermos' generalised characteristics and Kruzkov's entropies. Crucially, our method handles fluxes where classical theory fails completely. As a concrete demonstration, we construct entropy solutions for a Cauchy problem with flux $f(x,u)=xu^2$, where bounded initial data can become unbounded in finite time, even on compact spatial domains. This finite-time blow-up violates the maximum principle, rendering all classical existence techniques--based on $L^{\infty}$ estimates and compactness--inapplicable. However, the flux $f(x,u(x,t))$ remains bounded despite $u$ blowing up, and our front tracking scheme exploits this to construct approximations that converge to an entropy solution.

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↗