PDE Seminar
Monday, November 26:

Kyudong Choi (UW Madison)	
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations

We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. 
We prove that k-th derivative of weak solutions is locally integrable in space-time 
for any real k such that 1 < k < 3. Up to now, only the second derivative was known 
to be locally integrable by standard parabolic regularization. We also present 
sharp estimates of those quantities in weak-L^{4/(k+1)} locally. These estimates depend 
only on the L^2 norm of the initial data and on the domain of integration. 
Moreover, they are valid even for k >= 3 as long as we have a smooth solution. 
The proof uses a standard approximation of Navier-Stokes from Leray and a 
blow-up techniques. The local study is based on De Giorgi techniques with a 
new pressure decomposition. To handle the non-locality of fractional Laplacians, 
Hardy space and Maximal functions are introduced. This is joint work with A. Vasseur.