Fully nonlinear parabolic equations in two space variables [PDF]
arXiv, 2004 H\"older estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.
arxiv
Regularity for solutions of the two-phase Stefan problem [PDF]
arXiv, 2005 We consider local solutions of the two-phase Stefan problem with a "mushy" region. We show that if a (distributional) solution u is locally square integrable then the temperature is continuous.
arxiv
Global well-posedness for the 2 D quasi-geostrophic equation in a critical Besov space [PDF]
arXiv, 2006 We show that the the 2 D quasi-geostrophic equation has global and unique strong solution, when the (large) data belongs in the critical, scale invariant space $\dot{B}^{2-2\al}_{2, \infty}\cap L^{2/(2\al-1)}$.
arxiv
Selfsimilar Equivalence of Porous Medium and p-Laplacian Flows [PDF]
arXiv, 2007 We demonstrate the equivalence between the two popular models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. The equivalence is shown at the level of selfsimilar solutions.
arxiv
An elementary proof of the convergence of Ricci flow on compact surfaces [PDF]
arXiv, 2007 This paper has been withdrawn by the author for further modification.
arxiv
Uniqueness for solutions of the two-phase Stefan problem with signed measures as data [PDF]
arXiv, 2008 We show uniqueness of solutions to the two-phase Stefan problem which have signed measures as initial data.
arxiv
Initial-Boundary Value Problems for Parabolic Equations [PDF]
arXiv, 2008 We prove new existence and uniqueness results for weak solutions to non-homogeneous initial-boundary value problems for parabolic equations modeled on the evolution of the p-Laplacian.
arxiv
Blow up of solutions to generalized Keller--Segel model [PDF]
arXiv, 2008 The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve L\'evy diffusion operators and general potential type nonlinear terms.
arxiv
Third-order nonlinear dispersion PDEs: shocks, rarefaction, and blow-up waves [PDF]
arXiv, 2009 Various shock and rarefaction-type similarity solutions of the third-order nonlinear dispersion equation in 1D are constructed. Blow-up of some solutions are proved by different techniques.
arxiv
Metastability for Non-Linear Random Perturbations of Dynamical Systems [PDF]
arXiv, 2009 In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes.
arxiv