Global attractor of the three dimensional primitive equations of large-scale ocean and atmosphere dynamics in an unbounded domain

TL;DR

This paper investigates the long-term behavior of solutions to the 3D primitive equations modeling large-scale ocean and atmosphere dynamics in an unbounded domain, establishing the existence of a global attractor despite compactness challenges.

Contribution

It proves the existence of a global attractor for the primitive equations in an unbounded domain by overcoming the lack of Sobolev compactness using tail-estimates and energy methods.

Findings

Existence of a global attractor for the primitive equations.

Overcoming Sobolev embedding limitations in unbounded domains.

Establishment of asymptotic compactness in a weak phase space.

Abstract

This paper is concerned with the long-time behavior of solutions for the three dimensional primitive equations of large-scale ocean and atmosphere dynamics in an unbounded domain. Since the Sobolev embedding is no longer compact in an unbounded domain, we cannot obtain the asymptotical compactness of the semigroup generated by problem (2.4)-(2.6) by the Sobolev compactness embedding theorem even if we obtain the existence of an absorbing set in more regular phase space. To overcome this difficulty, we first prove the asymptotical compactness in a weak phase space of the semigroup generated by problem (2.4)-(2.6) by combining the tail-estimates and the energy methods. Thanks to the existence of an absorbing set in more regular phase space, we establish the existence of a global attractor of problem (2.4)-(2.6) by virtue of Sobolev interpolation inequality.