Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces
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2024
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| author | Cohen, Albert DeVore, Ronald Tadmor, Eitan |
| author_facet | Cohen, Albert DeVore, Ronald Tadmor, Eitan |
| contents | We construct uniformly bounded solutions of the equation $div\, {\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(Ω)$, where $Ω$ 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 $f\in L^d(Ω)$, 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 $L^{2,\infty}(Ω)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces Cohen, Albert DeVore, Ronald Tadmor, Eitan Analysis of PDEs We construct uniformly bounded solutions of the equation $div\, {\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(Ω)$, where $Ω$ 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 $f\in L^d(Ω)$, 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 $L^{2,\infty}(Ω)$ 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. |
| title | Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.12703 |