Pre-Publicación 2026-31
Gabriel N. Gatica:
A note on sufficiently small solenoidal liftings of Dirichlet data
Abstract:
In this paper we establish the existence of solenoidal liftings that are arbitrarily small in suitable Lebesgue norms for all Dirichlet boundary data satisfying the zero-flux condition. This result is needed to derive, by means of a translated velocity unknown, existence of solutions to mixed variational formulations of the Navier--Stokes equations without imposing any smallness assumption on the Dirichlet boundary data for the velocity. Our approach relies first on a density property of the divergence-free vector test functions, whose proof makes use, among other tools, of the classical Hahn--Banach theorem and the distributional de Rham lemma, and then on the vector-valued version of the trace theorem and a standard right inverse of the divergence operator. Finally, the necessity of the zero-flux condition is also shown.
