# Research Interests

I work on the analysis of Partial Differential Equations and its applications to fluid dynamics, plasma physics, and other scientific disciplines. Specifically, my research has been concerned with the following (selected) areas:

- The theory of Prandtl boundary layers in fluid dynamics.
- Dynamics of coherent structures in oscillatory media.
- Kinetic theory of plasmas confined in a bounded domain.
- Inflow/outflow boundary layers in compressible flows.
- Stability theory of regularized shock waves (e.g., in viscous gas dynamics and radiative hydrodynamics).

Below are more details on what I have worked on in each research area. Below is NOT up to date!

## Snapshots in Differential Equations (working group)

A weekly group meeting to study new ideas / techniques in PDEs.

+ Continue reading## Boundary layers in Fluid Dynamics

In fluid dynamics, one of the most classical and challenging issues is to completely understand the dynamics of fluid flows past solid bodies (e.g., aircrafts, ships, etc...), especially when the viscosity or the inverse of the physical Reynolds number of the fluid is small. The theory of boundary layers was introduced and developed to simplify the dynamics of a viscous fluid near the boundary by dividing it into two regions: one near the boundary (or so-called the boundary layer region), where viscosity is significant, and a second one away from the boundary where the fluid is essentially inviscid. One of the great achievements was then the discovery of the boundary layer equations, which significantly simplify the Navier-Stokes equations near the boundary (for example, the pressure is no longer an unknown quantity inside the boundary layer, and is completely determined from the outer Euler flow via the well-known Bernoulli's law). Since then, study of boundary layer solutions has become a physical and practical problem that greatly interests many physicists, and especially, aerodynamicists.

+ Continue reading## Coherent Structures

Patterns are ubiquitous in nature, and many interesting patterns can be found as spatially periodic traveling waves or wave trains of certain partial differential equations (e.g., reaction-diffusion systems). A coherent structure or defect is formed by two co-existing patterns which are separated in an organized fashion by an interface or so-called core of the defect. Defects can be found in many biological, chemical, and physical experiments. Mathematically, they can be described as special solutions of the PDEs that are time-periodic in an appropriate moving frame and spatially asymptotic at infinities to (generally different) wave trains. Motivated by the mentioned practical applications, I am interested in investigating the formation of these defects, their stability and nonlinear dynamics, and especially, how stable they can be under small disturbances in nature.

+ Continue reading## Kinetic Theory

In theory of a confined plasma, most mathematical studies are based on macroscopic magnetohydrodynamic or other approximate fluids-like models. Many plasma instability phenomena however have an essentially microscopic nature, reflecting the collective behavior of the particles. It is therefore of great physical interest to investigate the stability properties of a plasma in kinetic Vlasov-like models. Yet, there are not many mathematical works in literature that study plasmas governed by kinetic models, especially in a domain with boundaries due to the complication of solid-gas interactions and the development of singularity.

+ Continue reading## Shock Waves

Hyperbolic conservation laws are systems of PDEs that include many of the most fundamental physical principles such as conservations of mass, momentum, and energy. In such an ideal (hyperbolic) model, shock waves are known to exist, and determination of their physical admissibility is the central issue in theory of conservation laws. Relating to the so-called entropy admissible condition, hyperbolic-parabolic (or viscous) conservation laws are introduced as an approximation of the hyperbolic system with small dissipation or regularization (such as viscosity and heat dissipation in context of gas dynamics or magnetohydrodynamics). There are natural traveling waves in the latter system that are associated with inviscid shock waves. These are called viscous shock profiles, and their stability and dynamics play a crucial role in studies of validity of the viscous approximations, convergence in the inviscid limit, or convergence of numerical approximation schemes.

+ Continue reading## Elliptic/parabolic regularity theory

While the regularity theory for scalar elliptic or parabolic equations is classical (via the well-known De Giorgi-Nash-Moser theorem, proved back in the 50s), its analog for the system case is challenging and left open in general. During my time at UT-San Antonio, I worked on the existence and regularity theory for general elliptic and parabolic 2x2 systems with a strong couple in diffusion matrices.

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