Infinite energy solutions for a 1D transport equation with nonlocal velocity

TL;DR

This paper establishes the existence of infinite energy solutions for a 1D dissipative transport equation with nonlocal velocity, using weighted Sobolev spaces and novel commutator estimates.

Contribution

It introduces new commutator estimates involving fractional operators and Muckenhoupt weights, enabling global existence results for infinite energy initial data.

Findings

Global solutions for initial data in weighted Sobolev spaces

Use of weighted Lebesgue and Sobolev spaces with specific weights

Development of new commutator estimates for fractional operators

Abstract

We study a one dimensional dissipative transport equation with nonlocal velocity and critical dissipation. We consider the Cauchy problem for initial values with infinite energy. The control we shall use involves some weighted Lebesgue or Sobolev spaces. More precisely, we consider the familly of weights given by $w_{\beta}(x)=(1+\vert x \vert^{2})^{-\beta/2}$ where $\beta$ is a real parameter in $(0,1)$ and we treat the Cauchy problem for the cases $\theta_{0} \in H^{1/2} (w_{\beta})$ and $\theta_{0} \in H^{1} (w_{\beta})$ for which we prove global existence results (under smallness assumptions on the $L^\infty$ norm of $\theta_0$). The key step in the proof of our theorems is based on the use of two new commutator estimates involving fractional differential operators and the family of Muckenhoupt weights.