Most of my research falls in the realm of analyzing singularity formation (or the lack thereof) in nonlinear partial differential equations arising as models in mathematical fluid and plasma dynamics.

Current research includes :

-full space-time localization of various geometric scenarios preventing finite time blow-up in the 3D NSE

-exploring the interplay between the geometric structure of the regions of intense vorticity/magnetic field and the smoothness in the 3D magnetohydrodynamics (MHD) flows

-detailed study of spatial analyticity properties of nonlinear dispersive wave equations and systems -- in particular, Kdv-KdV-type Boussinesq systems -- deriving both the lower and the upper bounds on the radii of spatial analyticity and hence approximating dynamics of complex singularities in time (jointly with J.L. Bona, U. of Illinois at Chicago and H. Kalisch, U. of Bergen, Norway) 

-further study of the finite-energy global-in-time solutions to a nonlinear fluid-structure interaction model (jointly with  I. Lasiecka and A. Tuffaha)





An article in "UVa Top News Daily."






[Leonardo's drawings of coherent /self-similar structures in turbulent flows]

 PUBLICATIONS

1. Space analyticity for the Navier-Stokes and related equations with initial data in L^p (with I. Kukavica), J. Funct. Anal. 152 (1998), 447-466.
2. The role of spatial analyticity in the local alignment of vorticity directions in 3D viscous fluids Nonlinearity, 12 (1999), 1239-124.
3. Space analyticity for the nonlinear heat equation in a bounded domain (with I. Kukavica), J. Diff. Eq.152 (1999), 42-54.
4. On the smallness of the (possible) singular set in space for 3D Navier-Stokes equations, Electronic J. Diff. Eq.1999 (1999), 1-8.
5. Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eq. 12 (2000), 217-227.
6. Dynamics of complex singularities in 1D nonlinear parabolic PDE's , Studia Math. 147 (2001), 183-195.
7. The geometric structure of the super-level sets and regularity for 3D Navier-Stokes equations, Indiana Univ. Math. J. 50 (3) (2001), 1309-1317.
8. Local well-posedness of the generalized KdV equation in spaces of analytic functions (with H. Kalisch), Diff. Integral Eq. 15 (2002), 1325-1334.
9. A remark on time-analyticity for the Kuramoto-Sivashinsky equation (with I. Kukavica), Nonlinear Anal. 52 (1) (2003), 69-78.
10. Spatial analyticity properties of nonlinear waves (with J.L. Bona), Math. Models & Methods in Appl. Sci. 13 (3) (2003), 345-360.
11. The derivative nonlinear Schrodinger equation in analytic classes (with H. Kalisch), J. Nonlinear Math. Physics 10 (2003), 1-10.
12. Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE (with A. Ruzmaikina) , Indiana Univ. Math. J. 53 (4) (2004), 1073-1080.
    
13. On depletion of the vortex-stretching term in the 3D Navier-Stokes equations (with A. Ruzmaikina), Comm. Math. Physics 247 (2004), 601-611.
    
14. Algebraic lower bounds on the uniform radius of spatial analyticity for the generalized KdV equation (with J.L. Bona and H. Kalisch), Ann. Inst. Henri Poincare, Anal. Non Lineaire 22 (2005), 783-797.
    
15. Regularity of forward-in-time self-similar solutions to the 3D NSE, Discrete Cont. Dynamical Systems 14 (2006), 837-843.
    
16. Global solutions of the derivative Schrodinger equation in a class of functions analytic in a strip (with J.L. Bona and H. Kalisch), J. Diff. Eq.  229 (2006), 186-203.
    
17. Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE (with Qi Zhang), Comm. Math. Physics 262 (2006), 555-564.
18. Boundary control model of fluid-structure interactions  (with V. Barbu, I. Lasiecka and A. Tuffaha), Proceedings of the 12th IEEE International Conference on Methods and Models in Automation and Robotics, 29th-31st August 2006, Miedzyzdroje, Poland.
    
19. Existence of the energy-level weak solutions to a nonlinear fluid-structure interaction model (with V. Barbu, I. Lasiecka and A. Tuffaha), Contemporary Mathematics 440, Fluids and waves, Recent trends in applied analysis, AMS (2007), 55-82.
20. Smoothness of  solutions to a nonlinear fluid-structure interaction model  (with V. Barbu, I. Lasiecka and A. Tuffaha), Indiana Univ. Math. J. 57 (3) (2008), 1173-1207.
21. Analytic solutions of a class of nonlinear dispersive wave equations  (with H. Kalisch) (submitted).
22. A bound on oscillations in an unsteady undular bore  (with H. Kalisch) (submitted).
23. A Kdv-Kdv-type Boussinesq system in a scale of Bourgain-type spaces: from the energy level to analytic spaces (with J.L. Bona and H. Kalisch) (submitted).
24. On localization and geometric depletion of vortex-stretching in the 3D NSE  (submitted).