My interests are in Nonlinear Analysis and Partial Differential Equations. At present, my research is supported by NSF LEAPS-MPS grant no. DMS 2532664 .
In recent years, I have worked on the following topics:
- Weak diffeomorphisms for scalar conservation laws: Conservation laws have been studied extensively for developing a method to find Lagrangian transformations or particle paths. As their solutions are generally non-smooth, these particle paths become invertible bi-Lipschitz mappings, referred to as weak diffeomorphisms, that also satisfy systems of conservation laws. I have found a method that determines weak diffeomorphisms for any scalar conservation law with a smooth flux, not necessarily convex and in the process I extended a prior result on existence of global large data solutions for strictly hyperbolic Temple systems. Most recently, in collaboration, I have established that for scalar conservation laws, the particle paths are extremals of an action functional on the space of diffeomorphisms; that is, they are geodesics in some metric.
- Rate of convergence for Hamilton-Jacobi equations: The study of Hamilton-Jacobi (HJ) equations is motivated by several applications, traffic flow being one of note. HJ equations with state constraints admit large solutions that become the value function of an associated stochastic optimal control problem. This connection necessitates examining the rate of convergence of solutions under different assumptions. In recent times, I have worked in collaboration to determine the convergence rate in the vanishing viscosity limit for solutions to superquadratic HJ equations with state constraints.
- Fornberg-Whitham system in Besov spaces: The Fornberg-Whitham (FW) equation was originally derived to examine qualitative behavior of wave breaking. Later it became an important research topic in several areas of Physics and was generalized to obtain a two-component system FW system. I have established first results on local well-posedness in the sense of Hadamard, for the two component FW system in Besov spaces and recently showed that the data-to-solution map is not uniformly continuous for periodic initial data belonging to Besov spaces.
- Metric entropy and nonlinear partial differential equations: Metric entropy has been studied extensively in a variety of literature and disciplines. My doctoral research focused on finding sharp estimates for the metric entropy of classes of bounded total generalized variation functions and using these results to measure the set of solutions of certain nonlinear partial differential equations such as scalar conservation laws and HJ equations, where it could provide a measure of the order of “resolution” and “complexity” of numerical schemes required to provide approximate solutions to these equations.
Supported by my current NSF LEAPS-MPS grant no. DMS 2532664, in Fall 2026 I have initiated two interdisciplinary projects that utilize partial differential equations, to generate mathematical interest among undergraduates and to involve more students in my research program. Following are glimpses of the new directions:
- EEG source localization: Existing methods in this field rely on simple statistical priors that lead to convenient computations. However, they often fail to capture the complex geometry of the cerebral cortex and the wave-like propagation of neural activity within it. This poses challenges to the precision in tracking the position and time of rapidly evolving events in the brain. Combining the forward model with priors based on realistic assumptions about the spread of neural activity, our goal is to search for a PDE-based regularization strategy for the inverse problem.
- Tennis analytics: We plan to investigate whether a tennis player’s recent performance influences the outcomes of subsequent points and whether this effect changes when a game or a set ends. The goal of this study is to develop a mathematical model through discrete recurrence using parameters estimated from real tennis data and validated through simulations, and ultimately connect it to a simple transport PDE, including basic properties such as boundedness and stability.