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On Navier-Stokes equations arising from the rotation of an obstacle in a fluid

Tahar Zamène Boulmezaoud, Nabil Kerdid, Amel Kourta

TL;DR

This work analyzes stationary Navier–Stokes-type equations in ${\mathbb R}^3$ exterior to a rotating rigid body, incorporating rotation through terms like $\hat{\omega}\times u$ and $(\hat{\omega}\times x)\cdot\nabla u$. By employing weighted Sobolev spaces, the authors characterize decay at infinity and establish an existence/uniqueness theory for weak solutions under small forcing, together with detailed regularity and asymptotic decay results. The proof combines a linear theory for the rotated-Stokes-type operator $-\Delta v + \mathscr{R}v + \nabla r$ with a fixed-point argument to handle the nonlinear convection $u\cdot\nabla u$, yielding quantitative decay rates such as $|x|^{2-\varepsilon}$ in $L^p(S^2)$ norms for the velocity and $|x|^{3-\varepsilon}$ for the pressure. The results extend prior linear/augmented analyses and provide a robust framework for understanding far-field behavior of rotating-fluid interactions with potential extensions via Mozzi–Chasles transformations.

Abstract

We consider the modified Navier-Stokes equations in R3 describing the motion of a fluid in the presence of a rotating rigid body. Weighted Sobolev spaces are used to describe the behavior of solutions at large distances. Under suitable assumptions, w e prove the existence and regularity of solutions satisfying appropriate conditions at infinity.

On Navier-Stokes equations arising from the rotation of an obstacle in a fluid

TL;DR

This work analyzes stationary Navier–Stokes-type equations in exterior to a rotating rigid body, incorporating rotation through terms like and . By employing weighted Sobolev spaces, the authors characterize decay at infinity and establish an existence/uniqueness theory for weak solutions under small forcing, together with detailed regularity and asymptotic decay results. The proof combines a linear theory for the rotated-Stokes-type operator with a fixed-point argument to handle the nonlinear convection , yielding quantitative decay rates such as in norms for the velocity and for the pressure. The results extend prior linear/augmented analyses and provide a robust framework for understanding far-field behavior of rotating-fluid interactions with potential extensions via Mozzi–Chasles transformations.

Abstract

We consider the modified Navier-Stokes equations in R3 describing the motion of a fluid in the presence of a rotating rigid body. Weighted Sobolev spaces are used to describe the behavior of solutions at large distances. Under suitable assumptions, w e prove the existence and regularity of solutions satisfying appropriate conditions at infinity.
Paper Structure (6 sections, 8 theorems, 51 equations)

This paper contains 6 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. The proof of the main result
  4. Some useful preliminary results on weighted spaces
  5. Existence, uniqueness and regularity of weak solutions
  6. Conclusion

Key Result

Theorem 2.1

Let $p > 3/2$, $p \ne 3$, and set Then, there exist two constants $\kappa > 0$ and $c > 0$ such that for any ${{{f}}} \in W^{-1, p}_k({\mathbb R}^3)^3$ satisfying and there exists a unique weak solution $({{{u}}}, \pi) \in V^{1,p}_{k}({\mathbb R}^3)^3 \times W^{0,p}_{k}({\mathbb R}^3)$ of (prob_opT) satisfying Moreover, if ${{{f}}} \in W^{0, p}_{k+1}({\mathbb R}^3)^3$, then $({{{u}}}, \pi) \

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Corollary 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • ...and 4 more