My research area is in mathematical physics and mathematical aspects of materials science. In addition to the theoretical aspects, I am very keen on finding real world applications. I am interested in a variety of problems that range from kinetic theory of matter to macroscopic evolution of phase boundaries to phase transitions and critical phenomena. The issues that I address in my research, I believe, will be of interest to a large academic community from different disciplines.
Currently, I am working on modeling of evaporation phenomena in droplets. In particular, I am studying the motion of colloid particles under the influence of the fluid flow in the bulk and interfacial currents on the surface of the droplet. The flows are greatly affected by external sources (such as heating) and the intrinsic properties of the system. I believe, the ideas that I develop here can be helpful in understanding the complex interfacial flows.
It is known for the Vlasov-Poisson system that the solutions may not be as nice as in the case of whole space. I am working on the stationary solutions of the Vlasov-Poisson system with boundary conditions under a variety of settings. I am also investigating the stability of the statioanary solutions.
Another area that I am working on is the applications of renormalization group and scaling methods in physics and mathematics. Such techniques can be used beyond critical phenomena for example in obtaining the time dependency of physically relevant quantities (such as characteristic length) of self similarly evolving systems or in calculating blow up and extinction exponents of certain non-linear parabolic equations.
My doctoral research was on phase transitions and free boundary problems from the diffuse interface point of view (phase field) which is also related to density functional theories for interfaces. The main goal was to understand development of spatial macroscopic patterns in anisotropic systems. This involved developing new non-local diffuse interface models that capture anisotropic interaction effects and using asymptotic analysis to get the macroscopic behavior.