Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces

TL;DR

This paper constructs uniformly bounded solutions to the divergence equation in critical spaces, extending previous theoretical results with explicit, constructive methods for various dimensions and data types.

Contribution

It provides explicit and constructive methods for solving div u=f in critical spaces, including new multi-step and hierarchical approaches for general data.

Findings

Explicit one-step solution for d=2

Multi-step construction for d>2

Hierarchical process converging to solutions

Abstract

We construct uniformly bounded solutions of the equation $div\, {\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(\Omega)$, where $\Omega$ is a domain of ${\mathbb R}^d$. This question was addressed by Bourgain & Brezis, [On the equation ${\rm div}\, Y=f$ and application to control of phases, JAMS 16(2) (2003) 393-426], who proved that although the problem has a uniformly bounded solution, it is critical in the sense that there exists no linear solution operator for general $L^d$-data. We first discuss the validity of this existence result under weaker conditions than $f\in L^d(\Omega)$, and then focus our work on constructive processes for such uniformly bounded solutions. In the $d=2$ case, we present a direct one-step explicit construction, which generalizes for $d>2$ to a $(d-1)$-step construction based on induction. An explicit construction is proposed for compactly supported data in $L^{2,\infty}(\Omega)$ in the $d=2$ case. We also present constructive approaches based on optimization of a certain loss functional adapted to the problem. This approach provides a two-step construction in the $d=2$ case. This optimization is used as the building block of a hierarchical multistep process introduced in [E. Tadmor, Hierarchical construction of bounded solutions in critical regularity spaces, CPAM 69(6) (2016) 1087-1109] that converges to a solution in more general situations.