Bifurcation of periodic solutions of the Navier-Stokes equation in a thin domain (original) (raw)

On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

2019

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier-Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasiliner systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 000, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal LpL_pLp-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, wh...

Periodic Structure of 2-D Navier-Stokes Equations

Journal of Nonlinear Science, 2005

We study in this article both the structure and the structural evolution of the solutions of 2-D Navier-Stokes equations with periodic boundary conditions and the evolution of their solutions. First the structure of all eigenvectors of the corresponding Stokes problem is classified using a block structure, and is linked to the typical structure of the Taylor vortices. Then the structure of the solutions of the Navier-Stokes equations forced either by eigenmodes or by potential forcing is classified.

Periodic Solutions in {\mathbb R}^n$$ for Stationary Anisotropic Stokes and Navier-Stokes Systems

Integral Methods in Science and Engineering

and references therein. In [KMW20, KMW21a, KMW21b, KMW21c] this field has been extended to the transmission and boundary-value problems for stationary Stokes and Navier-Stokes equations of anisotropic fluids, particularly, with relaxed ellipticity condition on the viscosity tensor. In this chapter, we present some further results in this direction considering periodic solutions to the stationary Stokes and Navier-Stokes equations of anisotropic fluids, with an emphasis on solution regularity. First, the solution uniqueness and existence of a stationary, anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework are analysed on n-dimensional flat torus in a range of periodic Sobolev (Besselpotential) spaces. By employing the Leray-Schauder fixed point theorem, the linear results are employed to show existence of solution to the stationary anisotropic (nonlinear) Navier-Stokes incompressible system on torus in a periodic Sobolev space. Then the solution regularity results for stationary anisotropic Navier-Stokes system on torus are established.

On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder

Archive for Rational Mechanics and Analysis

We provide general sufficient conditions for branching out of a timeperiodic family of solutions from steady-state solutions to the twodimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem possess a simple eigenvalue that crosses the imaginary axis when the Reynolds number passes through a (suitably defined) critical value. We also show that only supercritical or subcritical bifurcation may occur. Our approach is different and, we believe, more direct than those used by previous authors in similar, but distinct, context.

Common periodic behavior in larger and larger truncations of the Navier-Stokes equations

Journal of Statistical Physics, 1988

The periodic behavior of N-mode truncations of the Navier-Stokes equations on a two-dimensional torus is studied for N= 44, 60, 80, and 98. Significant common features are found, particularly for not too high Reynolds numbers. In all models periodicity ends, giving rise, though at quite different parameter values, to quasiperiodicity.

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

Journal of Dynamics and Differential Equations, 2000

J. Dynamics and Diff. Eqns., in press, 2004. The article is devoted to the study of non-autonomous Navier-Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier-Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic,almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier-Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier-Stokes equations.