My research interests are in Nonlinear Analysis and Partial Differential Equations. Most recently I have worked on the following topics that have varied applications:
- Fornberg-Whitham system in Besov spaces: The Fornberg-Whitham (FW) equation was originally derived to examine qualitative behavior of wave breaking. Thereafter it received increasing attention as study of travelling wave solutions became an important research topic in several areas of Physics and was generalized to obtain a two-component system FW system which captures several features of the Euler equations. However, it is challenging to prove local well-posedness of strong solutions for this system. It has been studied deeply in Sobolev spaces. I have conducted further research in this direction and established first results on local well-posedness of the two component FW system in Besov spaces. These spaces are important in the study of nonlinear partial differential equations as they generalize Sobolev spaces and are more effective at measuring regularity properties of functions. By showing existence and uniqueness of the solution to the FW system and continuity of the data-to-solution map when initial data belongs to Besov spaces, I have proved this problem to be well-posed in the sense of Hadamard.
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Weak diffeomorphisms for scalar conservation laws: Studies have often been conducted on families of conservation laws with the aim of developing a method for finding Lagrangian transformations. Classically, this means that the system can be described as the flow in time of a diffeomorphism of space. In our framework, once the solutions are no longer smooth, the erstwhile diffeomorphisms become invertible bi-Lipschitz mappings, referred to as weak diffeomorphisms, that also satisfy systems of conservation laws. An exciting problem is to find particle paths in the form of weak diffeomorphisms for many systems where this had not been observed before. Previous works have illustrated such a procedure for scalar conservation laws in one space variable with a convex flux. My contribution in this area has been finding a method that determines weak diffeomorphisms for any scalar conservation law with a smooth flux, not necessarily convex and in the process I have extended a prior result on existence of global large data solutions for strictly hyperbolic Temple systems.
- Metric entropy and nonlinear partial differential equations: Metric entropy has been studied extensively in a variety of literature and disciplines. It plays a central role in different areas of information theory and statistics, including nonparametric function estimation, density information, empirical processes and machine learning. 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, where it could provide a measure of the order of “resolution” and “complexity” of a numerical scheme.