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Improved decay results for micropolar flows with nonlinear damping

Cilon F. Perusato, Franco D. Vega

Abstract

We examine the long-time behavior of solutions (and their derivatives) to the micropolar equations with nonlinear velocity damping. Additionally, we get a speed-up gain of $ t^{1/2} $ for the angular velocity, consistent with established findings for classic micropolar flows lacking nonlinear damping. Consequently, we also obtain a sharper result regarding the asymptotic stability of the micro-rotational velocity $\ww(\cdot,t)$. Related results of independent interest are also included.

Improved decay results for micropolar flows with nonlinear damping

Abstract

We examine the long-time behavior of solutions (and their derivatives) to the micropolar equations with nonlinear velocity damping. Additionally, we get a speed-up gain of for the angular velocity, consistent with established findings for classic micropolar flows lacking nonlinear damping. Consequently, we also obtain a sharper result regarding the asymptotic stability of the micro-rotational velocity . Related results of independent interest are also included.
Paper Structure (10 sections, 11 theorems, 103 equations)

This paper contains 10 sections, 11 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Main results
  4. Some mathematical preliminaries
  5. Linear operator related to \ref{['micropolar']}
  6. Proof of $L^2$ results
  7. Proof of Theorem \ref{['thm_L2']}
  8. Proof of $\dot{H}^m$ results
  9. Proof of Theorem \ref{['thm_inequalities']}
  10. Proof of Claim \ref{['Claim']}

Key Result

Theorem 1.1

Let $(\hbox{\boldmath $u$}_0,{\bf w}_0) \in {\bf L}^2_\sigma(\mathbb{R}^3) \times {\bf L}^2 (\mathbb{R}^3)$ and $(\hbox{\boldmath $u$}, {\bf w})(\cdot,t)$ any Leray solution to micropolar. Then, Moreover, for the micro-rotational field ${\bf w}(\cdot, t)$, one has

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2: Reasonability of the assumption \ref{['eqn:lambda-assumption']}
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 9 more