In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations with periodic boundary conditions using the same strategy. In this paper, we mo...

A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical meth...

We study how the number of numerically determining modes in the Navier–Stokes equations depends on the Grashof number. Consider the two-dimensional incompressible Navier–Stokes equations in a periodic domain with a fixed time-independent forcing function. We increase the Grashof number by rescaling the forcing and observe through numerical computation that the number of numerically determining ...

We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in R 2. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enst...

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear FokkerPlanck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]) Key wordsNonlinear Fokker-Planck equations, Navier-Stokes equations, Smoluchowski equation, micro-macro interactions. AMS subject classification 35Q30, 82C31, 76A05.

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct a representation of the strong solution to the Navier–Stokes equations in terms of diffusion processes.

This paper presents asymptotically stable schemes for patching of nonoverlapping subdomains when approximating the compressible Navier–Stokes equations given on conservation form. The scheme is a natural extension of a previously proposed scheme for enforcing open boundary conditions and as a result the patching of subdomains is local in space. The scheme is studied in detail for Burgers’s equa...

This article describes a new numerical solver for the Navier-Stokes equations. The proposed solver is written in Python which is a newly developed language. The Python packages are built to solve the Navier-Stokes equations with existing libraries. We have created discretized coefficient matrices from systems of the Navier-Stokes equations by the finite difference method. In addition we focus o...

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair (Xh,Mh) which satisfies some approximate assumpt...

We investigate the self-similar solutions to the isothermal compressible Navier–Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions with finite total energy. This generalizes the results for the incompressible case in Nečas, J., Ru̇žička, M. & Šverák, V. (1996, On Leray’s self-similar solutions of the Navier-Stokes equations. Ac...