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Periodic solutions for Boussinesq systems in weak-Morrey spaces

Pham Truong Xuan, Nguyen Thi Van, Tran Van Thuy

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.

Periodic solutions for Boussinesq systems in weak-Morrey spaces

Abstract

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension . Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space . 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.
Paper Structure (10 sections, 14 theorems, 102 equations)

This paper contains 10 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentz and Lorentz-Morrey spaces
  4. Predual of Morrey-Lorentz spaces
  5. Boussinesq system
  6. Linear estimates and periodic solutions for linear systems
  7. Periodic solutions for Boussinesq systems and their polynomial stability
  8. Existence of periodic solutions
  9. Asymptotic stability
  10. Periodic solutions of Navier-Stokes equations revisited

Key Result

Lemma 2.1

(i) Let $1<p_{0},p_{1},r\leqslant\infty$ and $0\leqslant\beta,\lambda _{0},\lambda_{1}<n$ satisfy $\dfrac{1}{r}=\dfrac{1}{p_{0}}+\dfrac{1}{p_{1}}$ and $\dfrac{\beta}{r}=\dfrac{\lambda_{0}}{p_{0}}+\dfrac{\lambda_{1}}{p_{1}},$ and let $s\geqslant1$ be such that $\dfrac{1}{q_{0}}+\dfrac{1}{q_{1}}\geqsl where $C>0$ is a constant. (ii) Let $m\in\left\{ 0\right\} \cup\mathbb{N}$, $1<p,r\leqslant\infty

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 3.1
  • ...and 14 more