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On the Asymptotic Behavior in Time of the Kinetic Energy in a Rigid Body-Liquid Problem

Giovanni P. Galdi, Paolo Maremonti

Abstract

We give sufficient conditions on the initial data for the decay in time of the kinetic energy, $E$, of solutions to the system of equations describing the motion of a rigid body in a Navier-Stokes liquid. More precisely, assuming the initial data ``small" in appropriate norm, we show that if, in addition, the initial velocity field of the liquid, $v_0$, is in $L^q$, $q\in(1,2)$, then $E(t)$ vanishes as $t\to\infty$ with a specific order of decay. The order remains, however, unspecified if $v_0\in L^2$.

On the Asymptotic Behavior in Time of the Kinetic Energy in a Rigid Body-Liquid Problem

Abstract

We give sufficient conditions on the initial data for the decay in time of the kinetic energy, , of solutions to the system of equations describing the motion of a rigid body in a Navier-Stokes liquid. More precisely, assuming the initial data ``small" in appropriate norm, we show that if, in addition, the initial velocity field of the liquid, , is in , , then vanishes as with a specific order of decay. The order remains, however, unspecified if .
Paper Structure (4 sections, 83 equations)

This paper contains 4 sections, 83 equations.

Table of Contents

  1. Governing Equations and Preliminary Results
  2. Main Results
  3. Initial datum $v_0\in L^q(\Omega)$, $q\in(1,2)$
  4. Initial datum $v_0\in L^2(\Omega)$

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['ABLTN']}