**Research Interests**

I am interested in the qualitative analysis of Partial Differential Equations (PDE) from fluid dynamics.

Basically, imagine a fluid with an initial configuration of particles that is

One of the most important system of PDEs in fluid dynamics are the Euler equations, derived by Leonhard Euler in 1757 as a model for ideal fluids (i.e. incompressible and inviscid). The importance of the Euler equations resides, partially, in the fact that when viscous effects are incorporated, one obtains the Navier-Stokes equations, used to design/model aircraft, automobiles, pipe flow, ocean currents and exploding supernova. The global regularity problem for both the Euler and NS equations is currently open; in fact, the global regularity issue for the later is a

In my research, I have studied particular classes of solutions to the

Similarly, I worked with Professor J. Wu from OSU on special solutions of the 2D Boussinesq equations, a system describing the motion of a fluid under gravitational forces and which also serves as a mathematical analogue for the 3D axisymmetric Euler equations. It is worth noting that the regularity problem for the 2D Boussinesq equations is open. In this project, we derived conditions on the initial data leading to finite-time blowup. On a slightly different note, with Professor S.C. Preston from Brooklyn College we considered the 3D axisymmetric Euler equations with swirl, which is obtained from the 3D Euler equations by passing to cylindrical coordinates and assuming the flow to be independent of the angular variable (e.g. a fluid in a rotating cylinder). In this project, (anti)symmetries in the initial data that are preserved by the equations of motion were exploited to explore regularity properties of the velocity field and the scalar pressure.

In addition to the Euler and Boussinesq equations, I have also worked on:

Lastly, as a graduate student I had the opportunity to do some work on solid mechanics. Professors D. Wei and M. Elgindi, and I, studied critical buckling loads of plastic axial columns using Hollomon's power-law, which is an extension of the Euler critical buckling loads of perfect elastic columns. We calculated the critical buckling loads in terms of Euler-type analytic formulas and the associated deformed shapes were presented in terms of generalized trigonometric functions.

Basically, imagine a fluid with an initial configuration of particles that is

*smooth*(i.e. described by an infinitely differentiable function). As time evolves, I study whether such a smooth initial configuration ceases to be smooth at some finite time or stays smooth for all time. To figure this out, one generally studies properties of the solution of a PDE which initially coincides with the aforementioned (smooth) initial configuration. These are commonly referred to as*global**regularity problems.*One of the most important system of PDEs in fluid dynamics are the Euler equations, derived by Leonhard Euler in 1757 as a model for ideal fluids (i.e. incompressible and inviscid). The importance of the Euler equations resides, partially, in the fact that when viscous effects are incorporated, one obtains the Navier-Stokes equations, used to design/model aircraft, automobiles, pipe flow, ocean currents and exploding supernova. The global regularity problem for both the Euler and NS equations is currently open; in fact, the global regularity issue for the later is a

*Millennium Prize Problem.*In my research, I have studied particular classes of solutions to the

*n*-dimensional Euler equations. These solutions are referred to as*stagnation-point*form solutions: they depend linearly on one of the coordinate variables (thus have, at best, finite local kinetic energy) and reduce the study of the*n*-dimensional Euler equations to either a single one-dimensional PDE or an n-1 dimensional equation. Briefly, such solutions can be written down in closed form when considered along their associated Lagrangian trajectories, which allowed us to explore their regularity properties in great detail. The upshot of all this is that by considering this special class of solutions one hopes to gain some intuition about what might be going on in the more general problem.Similarly, I worked with Professor J. Wu from OSU on special solutions of the 2D Boussinesq equations, a system describing the motion of a fluid under gravitational forces and which also serves as a mathematical analogue for the 3D axisymmetric Euler equations. It is worth noting that the regularity problem for the 2D Boussinesq equations is open. In this project, we derived conditions on the initial data leading to finite-time blowup. On a slightly different note, with Professor S.C. Preston from Brooklyn College we considered the 3D axisymmetric Euler equations with swirl, which is obtained from the 3D Euler equations by passing to cylindrical coordinates and assuming the flow to be independent of the angular variable (e.g. a fluid in a rotating cylinder). In this project, (anti)symmetries in the initial data that are preserved by the equations of motion were exploited to explore regularity properties of the velocity field and the scalar pressure.

In addition to the Euler and Boussinesq equations, I have also worked on:

- N-dimensional Euler and magnetohydrodynamic flow with damping.
- 1D models for waves in shallow water.
- Special solutions of higher-dimensional analogue of such 1D models.
- A generalized PDE (and system of PDEs) with applications to the study of 1D fluid convection and stretching, and gas dynamics.

Lastly, as a graduate student I had the opportunity to do some work on solid mechanics. Professors D. Wei and M. Elgindi, and I, studied critical buckling loads of plastic axial columns using Hollomon's power-law, which is an extension of the Euler critical buckling loads of perfect elastic columns. We calculated the critical buckling loads in terms of Euler-type analytic formulas and the associated deformed shapes were presented in terms of generalized trigonometric functions.

*Students who are interested in finding out more and/or working on projects should feel free to email me.*