Fractional Burgers equation with singular initial condition

TL;DR

This paper proves the existence of solutions to a fractional Burgers equation with singular initial conditions, including self-similar solutions with detailed properties, expanding understanding of such equations with non-standard initial data.

Contribution

It establishes existence results for fractional Burgers equations with initial data outside traditional function spaces and constructs self-similar solutions with detailed analysis.

Findings

Existence of solutions for initial data not in any L^p space.

Construction of self-similar solutions with explicit properties.

Derivation of smoothness, estimates, and asymptotics for solutions.

Abstract

We consider the fractional Burgers equation $ \Delta^{\alpha/2} u + b\cdot \nabla (u|u|^{(\alpha-1)/\beta})$ on ${\mathbf R}^d$, $d\geq2$, with {$\alpha \in (1,2)$ and} $\beta>1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf R}^d)$, $1\leq p\leq\infty$. Next, we apply the general results to the initial condition $u_0(x)=M|x|^{-\beta}$, $1<\beta<d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.