Stationary Solutions Of Navier-Stokes Equations Research Articles

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

(−1)-homogeneous axisymmetric no-swirl solutions of three dimensional incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles have been classified. In this paper we study the vanishing viscosity limit of sequences of these solutions. As the viscosity tends to zero, some sequences of solutions Clocm converge to solutions of Euler equations on the sphere minus the poles, while for other sequences of solutions, transition layer behaviors occur. For every latitude circle, there are sequences which Clocm converge respectively to different solutions of the Euler equations on the spherical caps above and below the latitude circle. We give detailed analysis of these convergence and transition layer behaviors.

Converge rates towards stationary solutions for the outflow problem of planar magnetohydrodynamics on a half line

AbstractIn this paper, convergence rates of solutions towards stationary solutions for the outflow problem of planar magnetohydrodynamics (MHD) are investigated. Inspired by the relationship between MHD and Navier-Stokes, we prove that the global solutions of the planar MHD converge to the corresponding stationary solutions of Navier-Stokes equations. We obtain the corresponding convergence rates based on the weighted energy method when the initial perturbation belongs to some weighted Sobolev space.

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities

All $ (-1) $-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $ (-1) $-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.

Stability of stationary solutions for inflow problem on the planar magnetohydrodynamics

In this paper, stability of stationary solutions for the inflow problem on the planar magnetohydrodynamics is being investigated. Motivated by the relationship between magnetohydrodynamics and Navier-Stokes, we can prove that the global solutions of the planar magnetohydrodynamics asymptotically converge to the corresponding stationary solutions of Navier-Stokes equations when time t → ∞ in both the subsonic and transonic cases. The proof is based on the basic energy method.

Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970's, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution is a particular type of a statistical solution in the sense of Foias and Prodi which is constructed in a way akin to the definition given by Vishik and Fursikov, in such a way that it possesses a number of useful analytical properties.

Multiscales and cascading in isotropic turbulence

The explicit mapping method is used to analyze the nonlinear dynamical behavior for cascade in isotropic turbulence. This deductive scale analysis is shown to provide the first visual evidence of the celebrated Richardson-Kolmogorv cascade, and reveals in particular its multiscale character based on the statistical solutions of Navier-Stokes equations. The results also indicate that the energy cascading process has remarkable similarities with the deterministic construction rules of the logistic map. Cascade of period-doubling bifurcations have been seen in this isotropic turbulent system that exhibit chaotic behavior. The “cascade” appears as an infinite sequence of period-doubling bifurcations.

Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations

We are interested in space-time spatially homogeneous statistical solutions of Navier-Stokes equations in space dimension three. We first review the construction of such solutions, and introduce convenient tools to study the pressure gradient. Then we show that given a spatially homogeneous initial measure with finite energy density, one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality.

Nearly singular two-dimensional Kolmogorov flows for large Reynolds numbers

We study the convergence of two-dimensional stationary Kolmogorov flows as the Reynolds number increases to infinity. Since the flows considered are stationary solutions of Navier-Stokes equations, they are smooth whatever the Reynolds number may be. However, in the limit of an infinite Reynolds number, they can, at least theoretically, converge to a nonsmooth function. Through numerical experiments, we show that, under a certain condition, some smooth solutions of the Navier-Stokes equations converge to a nonsmooth solution of the Euler equations and develop internal layers. Therefore the Navier-Stokes flows are “nearly singular” for large Reynolds numbers. In view of this nearly singular solution, we propose a possible scenario of turbulence, which is of an intermediate nature between Leray's and Ruelle-Taken's scenarios.

Résolution des équations de Navier-Stokes et Détection des bifurcations stationnaires par une Méthode Asymptotique-Numérique

Perturbation methods (asymptotic expansions) are usually considered as powerful methods for solving many kinds of non-linear problems. However, these methods are very often applied in a purely analytic framework, and the calculation is limited to the first few terms of the series. Since a few years, we have shown that the combination of perturbation techniques and finite element method can lead to a robust numerical method for some categories of non-linear problems. In this paper, we apply these techniques to compute branches of stationary solutions of Navier-Stokes equations and to detect stationary bifurcations.

STATISTICAL SOLUTIONS OF NAVIER-STOKES EQUATIONS BY NONSTANDARD DENSITIES

We present a method of constructing nonstandard densities of measures living on infinite-dimensional spaces. This gives the existence of statistical solutions of Navier-Stokes equations as standard measures corresponding to the solution of a first order PDE for the nonstandard densities. For linear equations the method gives a simple proof of uniqueness of statistical solutions.

Crystallographic groups and homogeneous statistical solutions of navier-stokes equations

In this paper, we investigate the connection between crystallographic groups and homogeneous statistical solutions of Navier-Stokes equations. Several results of Foias and Temam are extended. Fluid flows invariant under crystallographic groups are studied. This idea may be of interest to the understanding of bifurcation and turbulence.

Approaches to the solution of stationary Navier-Stokes equations

Approaches to the solution of stationary Navier-Stokes equations