Some questions about the regularity and the uniqueness of solutions of parabolic partial differential equations

TL;DR

This paper develops a fixed-point framework for analyzing solutions to linear parabolic PDEs, leading to new insights into their regularity and the uniqueness of solutions in complex cases like the Navier-Stokes problem.

Contribution

It introduces a novel fixed-point equation based on heat problem solutions, enabling regularity analysis and proving uniqueness for Navier-Stokes solutions in three dimensions.

Findings

Established a fixed-point equation for linear parabolic PDE solutions

Proved regularity properties for solutions to nonlinear parabolic problems

Demonstrated uniqueness of Navier-Stokes solutions in three dimensions

Abstract

This work obtains a fixed-point equation for the solution of linear parabolic partial differential problems based on solutions to heat problems. This is a pointwise equality, so we have required non-standard techniques that involve the study of the sign of certain solutions to linear parabolic problems. This fixed-point equation implies regularity properties of solutions to parabolic problems, not necessarily linear, and this allows us to prove the uniqueness of the solution in three dimensions for the Navier-Stokes problem.