[Paper Review] Regularity of solutions for the critical $N$-dimensional Burgers' equation

Chi Hin Chan, Magdalena Czubak|ArXiv.org|Oct 17, 2008

Advanced Mathematical Physics Problems10 references4 citations

TL;DR

This paper establishes the global existence and smoothness of solutions to the critical $N$-dimensional Burgers' equation with initial data in $L^2(bR^N)$, using a modified parabolic De Giorgi method inspired by Caffarelli and Vasseur. It proves that weak solutions are locally Hölder continuous and thus smooth, extending regularity results from the 1D periodic case to the general $N$-dimensional setting.

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 $L^2(\R^N)$.

Motivation & Objective

• To establish the existence of globally regular solutions to the critical $N$-dimensional Burgers' equation for initial data in $L^2(bR^N)$.
• To extend the parabolic De Giorgi method of Caffarelli and Vasseur to the non-quasi-geostrophic, nonlinear setting of the Burgers' equation.
• To prove that weak solutions are locally Hölder continuous and hence smooth, even without periodicity or Besov space assumptions.
• To address the regularity problem in higher dimensions where prior results were limited to 1D or periodic settings.
• To demonstrate that the method of Caffarelli and Vasseur can be adapted to equations with nonlinear advection and critical fractional dissipation.

Proposed method

• Adapts the parabolic De Giorgi method from Caffarelli and Vasseur [2] to the critical $N$-dimensional Burgers' equation with $ heta_0 otin L^p$ for $p>2$.
• Employs a local energy inequality for the positive part of the solution, derived from the structure of the fractional Laplacian $(- riangle)^{1/2}$.
• Uses harmonic extension techniques to control the nonlocal term $(- riangle)^{1/2} heta$ via boundary estimates in the upper half-space.
• Applies a bootstrap argument based on Hölder continuity estimates, showing that $C^eta$ regularity improves iteratively to $C^ ho$ for all $ ho < 1$.
• Controls singular integral operators arising from the Riesz transforms via pointwise estimates and decay properties of the Poisson kernel.
• Establishes $L^ ho$ bounds on the solution and its derivatives through iterative decay and scaling arguments in parabolic cylinders.

Experimental results

Research questions

• RQ1 Can the parabolic De Giorgi method be extended from the critical quasi-geostrophic equation to the critical Burgers' equation in $N$ dimensions?
• RQ2 Does every $L^2(bR^N)$ initial datum generate a globally smooth solution to the critical $N$-D Burgers' equation?
• RQ3 Is local Hölder continuity of weak solutions sufficient to imply smoothness in the critical $N$-dimensional setting?
• RQ4 Can the method of Caffarelli and Vasseur be adapted to handle the nonlinearity $ heta abla heta$ in the absence of divergence-free velocity fields?
• RQ5 What is the role of the critical dissipation $(- riangle)^{1/2}$ in ensuring regularity for general $L^2$ initial data in higher dimensions?

Key findings

• For any initial datum $ heta_0 otin L^p$ for $p>2$, a global weak solution $ heta otin L^ ho$ for $ ho>2$ exists in $L^ ho_{ ext{loc}}((0, ho) imesbR^N)$ for all $ ho< ho_0$.
• The solution $ heta$ belongs to $L^ ho_{ ext{loc}}((0, ho) imesbR^N)$ for all $ ho< ho_0$, with $ ho_0 = rac{2N}{N-2}$ if $N>2$, and $ ho_0 = ho$ for $N=2$.
• The solution is locally Hölder continuous in space and time, with Hölder exponent $eta$ satisfying $2eta < 1$.
• The solution $ heta$ is smooth (i.e., $C^ ho$ for all $ ho < 1$) due to a bootstrap argument based on the regularity of the nonlocal term.
• The method ensures that the solution remains bounded in $L^ ho$ for all $ ho < ho_0$, with $ ho_0 = rac{2N}{N-2}$ for $N>2$.
• The proof establishes that $C^eta$ regularity implies $C^{eta'}$ regularity for all $eta' < 1$, leading to $C^ ho$ regularity for all $ ho < 1$.