TIFR CAM Alumni Meeting 2025

• Dates : October 28 - 29, 2025

• Venue : LH111 (TIFR CAM )

Schedule

October 28, 2025

Abstract

Title: Criticality theory for Schrödinger operators with singular potential

Prashanth K. Srinivasan  (TIFR Centre for Applicable Mathematics

Abstract: In this talk, we prove a weighted spectral gap classification for Schrodinger operators

−∆ + V to a large class of locally integrable potentials V. Harnack’s inequality will not hold in general for such potentials.  We also establish for such potentials, under some additional assumptions, versions of the Agmon-Allegretto-Piepenbrink Principle characterising the subcritical/critical operators in terms of the cone of positive distributional supersolutions.

Title: Application of Ishii-Lions method to Liouville theorems

Anup Biswas ( IISER Pune )

Abstract: The Liouville property is a frequently studied topic in the theory of partial differential equations (PDEs). Its importance stems not only from its applications in regularity theory and blow-up phenomena, but also from its intrinsic theoretical interest. In this talk, we will present a new approach to establish the Liouville property for a broader class of operators involving gradient nonlinearities. In particular, we will present a detailed calculation in the context of (p, q)-Laplacian.

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

Shyam Sundar Ghoshal (TIFR Centre for Applicable Mathematics)

Abstract: 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 (unimodal flux problem has some advantage from the regularity point of view [3]). 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 [1, 4], which is commonly used for this type of analysis. This is a joint work with John D. Towers.

References

[1] S. S. Ghoshal and J. D. Towers, A convergence rate result for front tracking approximations of conservation laws with discontinuous flux, preprint, 2025.

[2] S. S. Ghoshal, J. D. Towers and G. Vaidya, A Godunov type scheme and error estimates for multidimensional scalar conservation laws with Panov-type discontinuous flux, Numerische Mathematik, 151 (2022), no. 3, 601-625.

[3] S. S. Ghoshal, S. Junca and A. Parmar, Higher regularity for entropy solutions of conservation laws with geometrically constrained discontinuous flux, SIAM J. Math. Anal, 56 (2024) no. 5, 6121-6136.

[4] N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Computational Mathematics and Mathematical Physics, 16 (1976), 105–119.

Title: Global approximate controllability for certain nonlinear PDEs

Rajib Dutta (IISER Kolkata)

Abstract: We study the periodic Camassa–Holm equation and the FitzHugh–Nagumo (FHN) system in one spatial dimension. We first discuss the well-posedness of these models under periodic boundary conditions. We then investigate the global approximate controllability of both systems using a finite-dimensional control.

Title: Non-local conservation laws modeling traffic flow and crowd dynamics

Aekta Aggarwal ( IIM Indore )

Abstract: Nonlocal conservation laws are gaining interest due to their wide range of applications in modeling real world phenomena such as crowd dynamics and traffic flow. In this talk, the well-posedness of the initial value problems for certain class of nonlocal conservation laws, scalar as well as system, will be discussed and monotone finite volume approximations for such PDEs will be proposed. Strong compactness of the proposed numerical schemes will be presented and their convergence to the entropy solution will be proven. Some numerical results illustrating the established theory will also be presented.

Title: Strongly coupled elliptic systems: Some recent developments

Bhakti Bhusan Manna ( IIT Hyderabad )

Abstract: In this talk, we will discuss recent results related to some strongly coupled elliptic system in the entire Euclidean space. The standard variational formulation exhibits some fundamental differences compared to the scalar equation, which makes the existence result slightly hard to achieve. In the first part, we shall discuss different kinds of variational formulations used to solve these types of problems, along with specific properties of the solutions. In the second part, we will discuss the existence of sign-changing solutions for these equations.