Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

Patricio Cumsille; Takéo Takahashi

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 961-992
  • ISSN: 0011-4642

Abstract

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In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝ d , d = 2 or 3 . The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for d = 2 or 3 . Moreover, we prove that the solution is global in time for d = 2 and also for d = 3 if the data are small enough.

How to cite

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Cumsille, Patricio, and Takahashi, Takéo. "Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid." Czechoslovak Mathematical Journal 58.4 (2008): 961-992. <http://eudml.org/doc/37880>.

@article{Cumsille2008,
abstract = {In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space $\mathbb \{R\}^d$, $d=2$ or $3$. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for $d=2$ or $3$. Moreover, we prove that the solution is global in time for $d=2$ and also for $d=3$ if the data are small enough.},
author = {Cumsille, Patricio, Takahashi, Takéo},
journal = {Czechoslovak Mathematical Journal},
keywords = {Navier-Stokes equations; incompressible fluid; rigid bodies; Navier-Stokes equations; incompressible fluid; rigid bodies},
language = {eng},
number = {4},
pages = {961-992},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid},
url = {http://eudml.org/doc/37880},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Cumsille, Patricio
AU - Takahashi, Takéo
TI - Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 961
EP - 992
AB - In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space $\mathbb {R}^d$, $d=2$ or $3$. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for $d=2$ or $3$. Moreover, we prove that the solution is global in time for $d=2$ and also for $d=3$ if the data are small enough.
LA - eng
KW - Navier-Stokes equations; incompressible fluid; rigid bodies; Navier-Stokes equations; incompressible fluid; rigid bodies
UR - http://eudml.org/doc/37880
ER -

References

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