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We construct uniformly bounded solutions of the equation div u = f for arbitrary data f in the critical spaces Ld(Ω), where Ω is a domain of Rd. This question was addressed by Bourgain & Brezis, [BB2003], 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 Ld-data. We first discuss the validity of this existence result under weaker conditions than f ∈ Ld(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 L2,∞(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 [Tad2014] that converges to a solution in more general situations.