Chi Hin Chan

Regularity theory for parabolic nonlinear integral operators

Received by the editors March 8, 2010 and, in revised form, August 2, 2010, October 26, 2010, and December 17, 2010. 2010 Mathematics Subject Classification. Primary 35B65, 45G05, 47G10.

The formulation of the Navier–Stokes equations on Riemannian manifolds

Abstract We consider the generalization of the Navier–Stokes equation from R n to the Riemannian manifolds. There are inequivalent formulations of the Navier–Stokes equation on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier–Stokes to the Riemannian manifolds.

De Giorgi Techniques Applied to the Hölder Regularity of Solutions to Hamilton–Jacobi Equations

This article is dedicated to the proof of C^{\alpha} regularization effects of Hamilton- Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent on the regularity of the Hamiltonian.