Periodic solutions for Boussinesq systems in weak-Morrey spaces

TL;DR

This paper establishes the existence, uniqueness, and polynomial stability of periodic solutions for Boussinesq systems in weak-Morrey spaces, extending to Navier-Stokes equations, using advanced functional analysis techniques.

Contribution

It introduces new methods for proving periodic solutions in weak-Morrey spaces for Boussinesq systems, including dispersive estimates and a Massera-type theorem.

Findings

Existence of bounded mild solutions for linear Boussinesq systems

Uniqueness and polynomial stability of periodic solutions

Application to Navier-Stokes equations

Abstract

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension $n\geqslant3$. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space $\mathbb{R}^{n}$. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear {systems} corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear {systems} on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear {systems} to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.