In the recent years, I have developed interest to study the transonic flows mathematically which are usually very interesting and complicated open problems in the context of mathematical fluid dynamics. The main complexity of the transonic flow is that a transonic structure consists of subsonic and supersonic parts, which are separated by either a sonic curve or transonic shock. These are usually free boundaries due to the nonlinearity of the governing system. Not only this but also the governing systems of transonic flows can change their behavior across the sonic boundary and are usually linearly degenerate on the sonic curve. Such features of transonic flow are more complicated to handle when compared to a study of purely subsonic or supersonic flow. Recently, I have been able to develop the existence and regularity of solutions of such sonic-supersonic patches and semi-hyperbolic patches arising in transonic flows over an airfoil. I will be trying to extend these solutions to the subsonic part in the near future.

Along with mixed hyperbolic-elliptic type systems, it is crucial to study purely hyperbolic system of conservation laws in multi-dimensions because they describe some of the most important physical phenomena occurring frequently in nature. One such representative example is thin film flow. Recently, we were able to develop a three-dimensional thin film flow model for anti-surfactant solutions. Under the influence of a surfactant, the surface tension of the free surface starts varying and a surface tension-driven flow (Marangoni flow) develops. We developed the evolution equations for film thickness and concentration (Bulk and surface)  and under some simplified assumptions, we were able to obtain a two-dimensional hyperbolic system of conservation laws governing the dynamics of thin film flow. We analyzed this system for different initial data and developed geometrical solutions for it by studying nonlinear wave interactions. In the near future, we want to extend this system to its multi-phase counterpart so that a better understanding of these critical phenomena can be developed. Also, the influence of gravity, temperature and mixture can be interesting to analyze.