Stable estimates for source solution of critical fractal Burgers equation

TL;DR

This paper derives two-sided estimates for the source solution of a critical fractal Burgers equation using the density of isotropic alpha-stable processes, advancing understanding of such equations' behavior.

Contribution

It provides the first two-sided estimates for the source solution of the critical fractal Burgers equation in multiple dimensions.

Findings

Established two-sided estimates for the solution density.

Connected solution behavior to isotropic alpha-stable process densities.

Enhanced analytical tools for critical fractal PDEs.

Abstract

In this paper, we provide two-sided estimates for the source solution of $d$-dimensional critical fractal Burgers equation $u_t-\Delta^{\alpha/2}+b\cdot \nabla\left(u|u|^q\right)=0$, $q=(\alpha-1)/d$, $\alpha\in(1,2)$, $b\in\mathbb{R}^d$, by the density function of the isotropic $\alpha$-stable process.