A universal variational framework for parabolic equations and systems

TL;DR

This paper introduces a universal variational method to construct fundamental solutions for parabolic equations and systems, providing new estimates and bounds without relying on traditional regularity assumptions.

Contribution

It presents a novel variational framework that constructs propagators for parabolic systems, yielding new off-diagonal estimates and Gaussian bounds under minimal regularity conditions.

Findings

Established ${L}^2$ off-diagonal estimates for fundamental solutions.

Provided Gaussian upper bounds for systems with pointwise local bounds.

Reproved Aronson's estimates using a new variational approach.

Abstract

We propose a variational approach to solve Cauchy problems for parabolic equations and systems independently of regularity theory for solutions. This produces a universal and conceptually simple construction of fundamental solution operators (also called propagators) for which we prove ${L}^2$ off-diagonal estimates, which is new under our assumptions. In the special case of systems for which pointwise local bounds hold for weak solutions, this provides Gaussian upper bound for the corresponding fundamental solution. In particular, we obtain a new proof of Aronson's estimates for real equations. The scheme is general enough to allow systems with higher order elliptic parts on full space or second order elliptic parts on Sobolev spaces with boundary conditions. Another new feature is that the control on lower order coefficients is within critical mixed time-space Lebesgue spaces or even mixed Lorentz spaces.