The primitive equations on curved surfaces

M. Hieber; Y. Shao; G. Simonett; M. Wilke

analysis of partial differential equations

The primitive equations on curved surfaces

arXiv:2607.27724

summary

The paper investigates the primitive equations defined near a smooth closed surface in three-dimensional space, proving existence, uniqueness, and smoothness of strong solutions globally without smallness assumptions on the initial data.

Abstract

In this paper, we study the primitive equations in a collar neighborhood of a smooth closed hypersurface in $R^{3}$ . Using the framework of maximal $L_{q}$ -regularity and the hydrostatic Helmholtz projection, we establish existence, uniqueness, and regularity results for strong solutions. Building on these local well-posedness results, we derive suitable a priori estimates and prove the global existence of strong solutions without imposing any smallness assumptions for the initial data. Moreover, we show that the solutions are $C^{\infty}$ jointly in time and space.

52 pages

Topics & keywords

#primitive equations #curved surfaces #maximal regularity #global existence #hydrostatic helmholtz projectionmaximal L_q-regularityhydrostatic Helmholtz projectionstrong solutionsglobal well-posednesssmooth closed hypersurface