Research

My research interest lies in the field of partial differential equations. In particular, I am now working in the regularity theory of incompressible Navier-Stokes equations, Navier-Stokes equation on the Riemannian manifold setting, regularity theory for fractional Burgers' and other non-local equations.

Publications

1.On possible isolated blow-up phenomena of the 3D-Navier-Stokes equation and a regularity criterion in terms of supercritical function space condition and smoothness condition along the streamlines, (with Tsuyoshi Yoneda). Submitted. pdf

2.Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting, (with Magdalena Czubak). Submitted. pdf

3.Regularity theory for nonlinear integral operators, (with Luis Caffarelli and Alexis Vasseur). To appear in the Journal of AMS. pdf

4.Eventual regularization of the slightly supercritical fractional Burgers equation, (with Magdalena Czubak and Luis Silvestre). Discrete and Continuous Dynamical Systems, Volume: 27, Number: 2, June 2010. A special issue Trends and Developments in DE/Dynamics Part I. pdf

5.Regularity of solutions for the critical N-dimensional Burgers' equation, (with Magdalena Czubak). Annales de l'Institut Henri Poincare (C) Non Linear Analysis Volume 27, Issue 2, March-April 2010, Pages 471-501. pdf

6.Smoothness criteria for Navier-Stokes equations in terms of regularity along the stream lines. Methods Appl. Anal. 17 (2010), no. 1, 81–103 pdf

7. Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, (with Alexis Vasseur). Methods Appl. Anal. 14 (2007), no. 2, 197-212. pdf

8.The De Giorgi's Method as Applied to The Regularity Theory for Incompressible Navier Stokes Equations. Ph.D. Thesis