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

TL;DR

This paper develops a fixed-point–like relation between linear parabolic solutions and the heat equation, yielding a pointwise proportionality between $y(t)$ and $\varphi(t)$ and a shared spatial regularity. It proves that linear parabolic problems admit $L^{\infty}(0,T;L^4(\Omega))$ regularity under $L^4$-data and $H^{-1}$ right-hand sides, and uses these results to establish uniqueness for the three-dimensional Navier–Stokes equations under natural data by decomposing the flow into a smooth heat part and a linear-parabolic remainder. The approach extends to variable-coefficient diffusion operators and suggests extensions to other nonlinear parabolic problems, with potential numerical implications. Overall, the work links fixed-point ideas, maximal-regularity-type estimates, and classical Navier–Stokes theory to address regularity and uniqueness questions in parabolic PDEs.

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.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Corollary 1
• Remark 2
• Theorem 5
• Theorem 6
• Theorem 7