Existence and uniqueness of weak solutions to viscous primitive equations for certain class of discontinuous initial data

TL;DR

This paper proves the global existence and uniqueness of weak solutions to viscous primitive equations for initial data that can be discontinuous, extending previous results to a broader class of initial conditions.

Contribution

It establishes the existence and uniqueness of weak solutions with discontinuous initial data, generalizing prior results on continuous and z-weak solutions.

Findings

Proves global existence of weak solutions for discontinuous initial data.

Establishes conditional uniqueness of weak solutions.

Extends previous results to a broader class of initial conditions.

Abstract

We establish some conditional uniqueness of weak solutions to the viscous primitive equations, and as an application, we prove the global existence and uniqueness of weak solutions, with the initial data taken as small $L^\infty$ perturbations of functions in the space $X=\left\{v\in (L^6(\Omega))^2|\partial_zv\in (L^2(\Omega))^2\right\}$; in particular, the initial data are allowed to be discontinuous. Our result generalizes in a uniform way the result on the uniqueness of weak solutions with continuous initial data and that of the so-called $z$-weak solutions.