Pointwise estimates for solutions of fractal Burgers equation

TL;DR

This paper derives pointwise estimates and asymptotic behaviors for solutions to a critical fractal Burgers equation in multiple dimensions, expanding understanding of its solutions under various initial conditions.

Contribution

It provides two-sided estimates and uniform asymptotics for solutions of the critical fractal Burgers equation, including cases with different initial data regularities.

Findings

Established two-sided estimates for solutions.

Derived uniform asymptotics in terms of stable semigroup operators.

Extended results to cases with different initial data regularities.

Abstract

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-\Delta^{\alpha/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $\alpha\in(1,2)$, $b\in\mathbb R^d$ for $q = (\alpha-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (\alpha-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator.