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The transition problem between time-independent motions of a body in a viscous liquid

Giovanni P. Galdi, Toshiaki Hishida

TL;DR

The paper addresses the attainment problem for transitions between steady exterior Navier–Stokes flows around a moving body, from an initial translational state with velocity $\\gamma_1$ to a final rigid-motion state with translation $\\gamma_2$ and rotation $\\omega_0$. The authors reformulate the transition as an integral equation in a weak-$L^3$ setting using the Oseen-type evolution operator $T(t,s)$ and a lifting construction, then apply a contraction mapping principle to obtain a unique global solution that decays to the final steady state. Under small data, they derive explicit decay rates in $L^{3,\\infty}$ and various Lorentz spaces for the velocity and its gradient, along with continuity and uniqueness properties. The results extend Finn’s starting problem to general rigid motions in an exterior domain, providing rigorous attainability and detailed asymptotic behavior for transitional flows in 3D Navier–Stokes.

Abstract

A body $\mathscr B$ moves in an unbounded Navier-Stokes liquid by time-independent translatory motion. Suppose that at time $t=0$, $\mathscr B$ smoothly changes its motion to an arbitrary rigid motion, reached at time $t=1$. We then show that the associated Navier-Stokes problem has a unique solution connecting the two steady-states generated by the motion of $\mathscr B$, provided all the involved velocities of $\mathscr B$ are sufficiently small.

The transition problem between time-independent motions of a body in a viscous liquid

TL;DR

The paper addresses the attainment problem for transitions between steady exterior Navier–Stokes flows around a moving body, from an initial translational state with velocity to a final rigid-motion state with translation and rotation . The authors reformulate the transition as an integral equation in a weak- setting using the Oseen-type evolution operator and a lifting construction, then apply a contraction mapping principle to obtain a unique global solution that decays to the final steady state. Under small data, they derive explicit decay rates in and various Lorentz spaces for the velocity and its gradient, along with continuity and uniqueness properties. The results extend Finn’s starting problem to general rigid motions in an exterior domain, providing rigorous attainability and detailed asymptotic behavior for transitional flows in 3D Navier–Stokes.

Abstract

A body moves in an unbounded Navier-Stokes liquid by time-independent translatory motion. Suppose that at time , smoothly changes its motion to an arbitrary rigid motion, reached at time . We then show that the associated Navier-Stokes problem has a unique solution connecting the two steady-states generated by the motion of , provided all the involved velocities of are sufficiently small.

Paper Structure

This paper contains 5 sections, 7 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. Statement of the main result
  4. Preparatory results
  5. Proof of Theorem \ref{['main']}

Key Result

Theorem 3.1

Let $\gamma_1, \gamma_2, \omega_0\in \mathbb R^3$, and set For every $\varepsilon\in (0,\frac{1}{4})$, there is a constant $\delta=\delta(\varepsilon,\Omega)\in (0,1]$ independent of $|\psi^\prime|_0$ such that if where $|\psi^\prime|_0$ is given by size-psi, see also trans, then problem IE admits a global solution $w(t)$ of class for all $r\in (3,\infty)$ as well as sol-cl, which enjoys for a

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 14 more