Well-posedness and blowup of the geophysical boundary layer problem

TL;DR

This paper establishes local well-posedness for the geophysical boundary layer problem under analytic initial conditions and demonstrates finite-time blowup of solutions under specific initial velocity conditions.

Contribution

It provides the first proof of well-posedness in weighted Chemin-Lerner spaces and identifies conditions leading to finite-time blowup of solutions.

Findings

Local well-posedness under analytic initial conditions.

Finite-time blowup of solutions with certain initial velocities.

Use of energy methods in weighted Chemin-Lerner spaces.

Abstract

Under the assumption that the initial velocity and outflow velocity are analytic in the horizontal variable, the local well-posedness of the geophysical boundary layer problem is obtained by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outflow velocity satisfy certain condition on a transversal plane, it is proved that the $W^{1,\infty}-$norm of any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem blows up in a finite time.