Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation

TL;DR

This paper introduces a universal variational method to establish the existence, uniqueness, and representation of fundamental solutions for a broad class of parabolic equations and systems with minimal assumptions.

Contribution

It presents a new systematic variational approach that avoids traditional regularity assumptions, proving fundamental solutions for general parabolic problems.

Findings

Existence and uniqueness of fundamental solutions in broad settings

Representation formulas for all weak solutions

Applications to elliptic, integro-differential, and degenerate elliptic equations

Abstract

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational definition. Our classes of weak solutions are taken with minimal assumptions. We prove the existence and uniqueness of a fundamental solution which seems new in this generality: it is shown to always coincide with the associated evolution family for the initial value problem with zero source and it yields representation of all weak solutions. Our strategy is a variational approach avoiding density arguments, a priori regularity of weak solutions or regularization by smooth operators. One of our main tools are embedding results which yield time continuity of our weak solutions going beyond the celebrated Lions regularity theorem and that is addressing a variety of source terms. We illustrate our results with three concrete applications : second order uniformly elliptic part with Dirichlet boundary condition on domains, integro-differential elliptic part, and second order degenerate elliptic part.