[Paper Review] Blow up and regularity for fractal Burgers equation

Alexander Kiselev, Fëdor Nazarov+1 more|ArXiv.org|Apr 22, 2008

Navier-Stokes equation solutions9 references19 citations

TL;DR

This paper establishes the critical threshold for global existence and regularity in the fractal Burgers equation with fractional dissipation. It proves finite-time blow-up for α < 1/2 and global existence with analyticity for α ≥ 1/2, using a novel nonlocal maximum principle and weighted Sobolev space estimates to handle rough initial data and extend results to the critical case α = 1/2.

ABSTRACT

The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $α&lt; 1/2,$ and global existence as well as analyticity of solution for $α\geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \in L^p,$ $1 &lt; p &lt; \infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.

Motivation & Objective

• To determine the threshold value of α for which solutions to the fractal Burgers equation remain globally regular or blow up in finite time.
• To establish existence and uniqueness of solutions for rough initial data in L^p spaces with 1 < p < ∞.
• To prove space analyticity of solutions for α ≥ 1/2, extending regularity results to the critical case α = 1/2.
• To extend the analysis to a broader class of equations, including the surface quasi-geostrophic equation, via shared analytical techniques.
• To develop a nonlocal maximum principle tailored to the critical case α = 1/2, enabling global existence and analyticity results.

Proposed method

• Use of weighted Sobolev spaces H^{s,φ} with slowly growing weight functions φ to control growth in high-frequency modes.
• Application of a nonlocal maximum principle to handle the critical case α = 1/2, which avoids standard energy estimates and enables analyticity proofs.
• Employment of Schur test estimates on dyadic frequency shells to bound nonlinear terms in the energy inequality.
• Derivation of a modified energy inequality involving ∥u∥_{H^{s,φ}}^2 and ∥u∥_{H^{q,φ}} with q = 3/2 - 2α, enabling control in the critical regime.
• Use of weak continuity in L^2 and duality arguments to define and characterize weak solutions in the L^2 framework.
• Extension of results to s = 3/2 - 2α via careful limiting arguments and unbounded weight functions φ, ensuring persistence of regularity.

Experimental results

Research questions

• RQ1 For which values of α does the fractal Burgers equation admit global classical solutions with smooth initial data?
• RQ2 What is the sharp threshold for finite-time blow-up in the supercritical regime α < 1/2?
• RQ3 Can solutions be shown to be real analytic in space for t > 0 when α ≥ 1/2, and what techniques enable this?
• RQ4 How does the regularity of solutions depend on the initial data in L^p or H^s spaces for p ∈ (1, ∞) and s ≥ 3/2 - 2α?
• RQ5 Can the methods developed for the Burgers equation be extended to other equations such as the surface quasi-geostrophic equation?

Key findings

• For α > 1/2, global existence and real analyticity in x for t > 0 are established for initial data in H^s with s > 3/2 - 2α.
• For α = 1/2, global existence and space analyticity are proven for initial data in H^s with s > 1/2, using a new nonlocal maximum principle.
• For α < 1/2, finite-time blow-up occurs in H^s for s ≥ 3/2 - 2α, even for smooth periodic initial data.
• Solutions exist globally in time for initial data in H^s with s ≥ 3/2 - 2α, and are C^∞ for t > 0 in the interval (0, T), with T depending on α and the initial data norm.
• The results extend to s = 3/2 - 2α by using weighted norms H^{s,φ} with unbounded φ, and T depends on φ and ‖u₀‖_{H^{s,φ}}.
• The solution is unique in the class C([0,T], L^2) ∩ L^{3/(2δ)}([0,T], H^δ) for some δ ∈ (1/2, 1], ensuring uniqueness beyond classical solutions.