Regularity of solutions for the critical $N$-dimensional Burgers' equation

TL;DR

This paper proves the existence of smooth solutions for the critical fractional Burgers' equation in N dimensions using a parabolic De-Giorgi method, extending understanding of regularity in nonlinear PDEs.

Contribution

It introduces a novel application of the De-Giorgi method to the critical fractional Burgers' equation, establishing regularity results for solutions with initial data in L^2.

Findings

Existence of smooth solutions for the critical fractional Burgers' equation.

Application of De-Giorgi's method to a new class of nonlinear PDEs.

Extension of regularity theory to N-dimensional fractional equations.

Abstract

We consider the fractional Burgers' equation on $\R^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur \cite{Driftdiffusion} and show existence of smooth solutions given any initial datum in $L^2(\R^N)$.