Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH DATA

2015 ◽  
Vol 16 (5) ◽  
pp. 947-985 ◽  
Author(s):  
Pascal Auscher ◽  
Dorothee Frey
Keyword(s):  
Parabolic Equations ◽  
Stokes System ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Tent Spaces ◽  
Quadratic Nonlinearities ◽  
Kernel Bounds

We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

A note on the regularity of weak solutions to the coupled 2D Allen–Cahn–Navier–Stokes system

2019 ◽  
Vol 25 (1) ◽  
pp. 111-117 ◽  
Author(s):  
T. Tachim Medjo
Keyword(s):  
Weak Solutions ◽  
Stokes System ◽  
Stokes Equations ◽  
Regularity Result ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Weak Formulation ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Dimensional Domain

Abstract In this article, we study a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We present an equivalent weak formulation for the model, and we prove a new regularity result for the weak solutions.

Sign in / Sign up

Export Citation Format

Share Document