Hierarchical construction of bounded solutions of div U = F in critical regularity spaces

Abstract

We implement the hierarchical decomposition introduced in (Tadmor in Hierarchical construction of bounded solutions in critical regularity spaces, arXiv:1003.1525v2), to construct uniformly bounded solutions of the problem divU = F, where the two-dimensional data is in the critical regularity space, F ε L2#(T2). Criticality in this context, manifests itself by the lack of linear mapping, F ε L2#(T2) → U ε L∞(T2), (Bourgain and Brezis in J. Am. Math. Soc. 16(2): 393-426, 2003). Thus, the intriguing aspect here is that although the problem is linear, the construction of its uniformly bounded solutions is not. © Springer-Verlag Berlin Heidelberg 2012.