Primitive Equations with Linearly Growing Initial Data

TL;DR

This paper studies the primitive equations with linearly growing initial data in a 3D infinite layer, establishing semigroup properties and local existence of solutions using advanced mathematical techniques.

Contribution

It introduces a novel analysis of primitive equations with linearly growing initial data, employing semigroup theory and Sobolev space methods for the first time.

Findings

Established semigroup smoothing properties.

Proved local existence and uniqueness of mild solutions.

Analyzed boundary condition effects on solutions.

Abstract

The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann or mixed boundary conditions are imposed. The Ornstein-Uhlenbeck type operator appears in the linear parts, so the semigroup theory is established by Trotter's arguments due to decomposition of infinitesimal generators. To obtain smoothing properties of the semigroup, derivatives of the associated kernel are calculated. For proving time-local existence and uniqueness of mild solutions, the adapted Fujita-Kato scheme is used in certain Sobolev spaces.