# Regularity for a parabolic problem with nonsmooth coefficients

I'm looking for references on the regularity of the (weak) solution to the parabolic problem with nonsmooth coefficients. In most literature, like Evans, the coefficients are often assumed to be smooth ($C^\infty$).

To be more specific I am interested in the equation $$\dot{u} - \nabla \cdot A \nabla u = f,$$ where $u(0) = g$ and $A = (a_{ij}(x))$ is a symmetric coefficient matrix (not dependent on time) such that $a_{ij} \in L^\infty$. I guess that we can not hope for much regularity in the spatial domain (only $H^1_0$), but what happens in the time domain? How does assumptions on $g$ and $f$ affect the regularity?

Any suggestions where I can read more about this is much appreciated. Thanks!

• Sorry, I forgot to do that. Thank you for all the reading tips! – Anna Sep 3 '15 at 6:05

## 1 Answer

You are talking about one of the great works of John Nash. The result was obtained in the elliptic setting independently by Ennio de Giorgi, and later streamlined by Jürgen Moser. This is the heart of the resolution of Hilbert's 19th problem. Basically, if $f$ is reasonable, the solution is in some Hölder space (in spacetime), which is much better than $H^1$. The regularity of $g$ does not have much effect because the equation is parabolic. Moreover, increasing the smoothness of $f$ does not help if $A$ stays rough. In order to have smoother solution one needs to increase the smoothness of both $f$ and $A$.

Possible keywords you can put in Google are Moser iteration, De Giorgi iteration, and Nash-Moser regularity theory. The theory is covered in a book by Z. Wu, J. Yin, and C. Wang called "Elliptic & parabolic equations". Also, Ladyzhenskaya and Uraltseva wrote a book on quasilinear parabolic equations, which is supposed to have a comprehensive treatment (I have not read this book though).