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Limiting Behavior for an Iterated Viscosity

Ciprian Foias, Michael S. Jolly, Oscar P. Manley (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The behavior of an ordinary differential equation for the low wave number velocity mode is analyzed. This equation was derived in [5] by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It resembles the NSE in form, except that the kinematic viscosity is replaced by an iterated viscosity which is a partial sum, dependent on the low-mode velocity. The convergence of this sum as the number of iterations is taken to be arbitrarily large is explored. This leads to a limiting...

Linear flow problems in 2D exterior domains for 2D incompressible fluid flows

Paweł Konieczny (2008)

Banach Center Publications

The paper analyzes the issue of existence of solutions to linear problems in two dimensional exterior domains, linearizations of the Navier-Stokes equations. The systems are studied with a slip boundary condition. The main results prove the existence of distributional solutions for arbitrary data.

Local exact controllability for the 1 -d compressible Navier-Stokes equations

Sylvain Ervedoza (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

In this talk, I will present a recent result obtained in [6] with O. Glass, S. Guerrero and J.-P. Puel on the local exact controllability of the 1 -d compressible Navier-Stokes equations. The goal of these notes is to give an informal presentation of this article and we refer the reader to it for extensive details.

Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions

Sergio Guerrero (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we deal with the local exact controllability of the Navier-Stokes system with nonlinear Navier-slip boundary conditions and distributed controls supported in small sets. In a first step, we prove a Carleman inequality for the linearized Navier-Stokes system, which leads to null controllability of this system at any time T>0. Then, fixed point arguments lead to the deduction of a local result concerning the exact controllability to the trajectories of the Navier-Stokes system.

Local existence of solutions of the free boundary problem for the equations of compressible barotropic viscous self-gravitating fluids

G. Ströhmer, W. Zajączkowski (1999)

Applicationes Mathematicae

Local existence of solutions is proved for equations describing the motion of a viscous compressible barotropic and self-gravitating fluid in a domain bounded by a free surface. First by the Galerkin method and regularization techniques the existence of solutions of the linearized momentum equations is proved, next by the method of successive approximations local existence to the nonlinear problem is shown.

Local null controllability of a fluid-solid interaction problem in dimension 3

Muriel Boulakia, Sergio Guerrero (2013)

Journal of the European Mathematical Society

We are interested by the three-dimensional coupling between an incompressible fluid and a rigid body. The fluid is modeled by the Navier-Stokes equations, while the solid satisfies the Newton's laws. In the main result of the paper we prove that, with the help of a distributed control, we can drive the fluid and structure velocities to zero and the solid to a reference position provided that the initial velocities are small enough and the initial position of the structure is close to the reference...

Local null controllability of a two-dimensional fluid-structure interaction problem

Muriel Boulakia, Axel Osses (2008)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0 , the system can be driven at rest and the structure to its reference configuration at time T . To show this result, we first consider a linearized system....

Local null controllability of a two-dimensional fluid-structure interaction problem

Muriel Boulakia, Axel Osses (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0, the system can be driven at rest and the structure to its reference configuration at time T. To show this result, we first consider a linearized system....

Local Solutions for Stochastic Navier Stokes Equations

Alain Bensoussan, Jens Frehse (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data u 0 ∈ H s - 1 + ε , w 0 ∈ H s - 1 and b 0 ∈ H s for s > 3 2 and any 0 < ε < 1 . The initial regularity of the micro-rotational velocity w is weaker than velocity of the fluid u .

Long time estimate of solutions to 3d Navier-Stokes equations coupled with heat convection

Jolanta Socała, Wojciech M. Zajączkowski (2012)

Applicationes Mathematicae

We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in ℝ³. The domain is a cylinder along the x₃ axis. The aim of this paper is to show long time estimates without assuming smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of the L₂ norms of the x₃-derivatives of these three quantities.

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