Parabolic problems with rough coefficients
Using form methods, one can solve linear parabolic PDE in divergence form with $L^2$ data in appropriate energy spaces, even when the coefficients are merely bounded measurable in time and space, and no maximum principle is available. This goes back, at least, to the work of Lions and his school in the 1950s. When dealing with $L^p$ data, it is not so clear which $L^p$ like solution space one should use as a replacement of Lions energy space. Depending on the choice, one can solve, for instance, time dependent single equations (Aronson 1968), time independent systems for a range of values of $p$ (Auscher 2005), or stochastic problems with some spatial regularity (starting with Krylov 1994).
In this talk, I explain that, by choosing tent spaces as solution spaces (inspired by their role in elliptic boundary value problems on rough domains), one gets fairly general results, including both deterministic and stochastic problems, all values of $p$, systems as well as single equations, and rough coefficients in both time and space.
This summarises joint works with Pascal Auscher, Sylvie Monniaux, Jan van Neerven, and Mark Veraar.