My current research interest is in understanding various phenomena that involve shock waves and transonic flows. A sonic point is a crucial ingredient in many problems and can be interpreted either as an embedded information boundary, or an event horizon, or a critical point of the underlying ODEs. Many interesting and important mathematical issues about sonic points are still unresolved, in particular, concerning their role in unsteady flows.
Examples of past and present research areas I have been working in are:
Detonations. Detonation is a selfsustained shock wave in a chemically reactive medium that is supported by the energy released in the medium due to heating by the shock itself. Thus the shock causes the medium to burn and the burning keeps the shock going. The phenomenon is fascinating due to its intricate nonlinear dynamical features such as pulsations in one dimension and cell formation in two and three dimensions. 
Cellular detonation 
Hydraulic jumps. You have seen a hydraulic jump, if you have ever washed dishes (or have seen someone else do it). When a jet of fluid from your kitchen tap strikes a plate, the radial flow on the plate is initially very shallow, but at some distance away from the impact point, the depth suddenly increases, forming a ring  the hydraulic jump. Despite its apparent simplicity, the hydraulic jump is a source of many fascinating phenomena poorly understood to this day. For example: Why does the jump often take polygonal shapes in very viscous fluids? 

Traffic flows. The flow of vehicle traffic is a nice everyday example of a system that exhibits some basic features of a compressible flow. In particular, a traffic jam, i.e. a rapid increase in car density (the number of cars per unit length of the road) can be treated as a shock wave. Traffic shocks form as vehicles stop at a red light, but they can also form spontaneously on an open roadway if the traffic density is too high. How does the traffic jam form? What is its structure? How does it propagate, and how can one control it? These are all fundamental questions that are still awaiting for good answers. My research focuses on continuum modeling of traffic flow employing the tools of the theory of hyperbolic systems and numerical algorithms for their solution. 
Traffic jam (from Courant & Friedrichs' book) 
Numerical methods. How to simulate multidimensional shock/detonation waves so as to minimize the shockresolution errors inherent in shockcapturing methods? This is a critically important question in detonation simulations due to the extreme sensitivity of chemical reactions to such errors. My work involves development of accurate shockfitting methods for such simulations. 

Biofluid mechanics. How does a fluid flow in flexible tubes, such as arteries, interact with the surrounding deformable solid? Such interaction often leads to the formation of elastic shocks, i.e. rapid changes in the tube cross section. Why do the shocks form and what are their properties? 

Twophase and granular flows. Flow of a fluid with solid particles exhibits shocks in particle concentration in such processes as sedimentation and fluidization. The flow of granular materials is often accompanied with sharp density gradients which can also be treated as shock waves. 

Astrophysics. Accretion flows, supernovae, and many other phenomena in astrophysics involve shock waves in complex interplay with gravity, nuclear reactions, radiation, and magnetic fields. 