Solutions Of 3D Navier-Stokes Equations Research Articles

Enhanced dissipation for the third component of 3D anisotropic Navier-Stokes equations

In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations.

Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $\rho_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2\nu}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.

Helical symmetric solution of 3d Navier-Stokes equations arising from geometric shape of the boundary

In this article, we investigate three-dimensional solution with helical symmetry in a gap between two concentric rotating cylinders, inside is a helicoidal surface (screw propeller) while outside is a cylindrical surface. Establish the partial differential equations and its variational formulation satisfied by a helical solution in a helical coordinate system using tensor analysis method, we provide a computational method for the power and propulsion of the screw. The existence and uniqueness of weak helical solutions are proved.

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.

Computational aeroacoustics of human phonation

The current paper presents a CFD model of flow past vibrating vocal folds coupled to an acoustic solver, which calculates the sound sources from the flow field in a hybrid approach. The CFD model is based on the numerical solution of 3D Navier-Stokes equations on a time-dependent domain, solved by cell-centered finite volume method. To capture the fine turbulent scales important for the acoustic source calculations, the equations are discretized and solved on large computational meshes up to 3.2M elements. The CFD simulations were run in parallel using domain decomposition method and OpenMPI implementationof the MPI standard.

On the regularity criterion for the solutions of 3D Navier-Stokes equations in weak multiplier spaces

In this work, we improved the regularity criterion on the Cauchy problem for the Navier–Stokes equations in multiplier space in terms of the two partial derivatives of velocity fields, ∂1u1 and ∂2u2. Copyright © 2011 John Wiley & Sons, Ltd.

Rooted trees for 3D Navier-Stokes equation

We establish a representation of a class of solutions of 3d Navier-Stokes equations in $\R^3$ using sums over rooted trees. We study the convergence properties of this series recovering in a simplified manner some results obtained recently by Sinai and other known results for solutions in spaces of pseudo-measures introduced initially by Le Jan and Sznitman. The series representation make sense also in the critical case where there exists global solutions for small initial data and it allows the study of their long-time or small-distance behavior.

The geometric structure of the super-level sets and regularity for 3D Navier-Stokes equations

We utilize L°°-estimates on the complexified solutions of 3D Navier-Stokes equations (NSE) via plurisubharmonic measure to show that, under certain geometric assumptions on the structure of the super-level sets, blow-up in a finite time can not occur.

On regularity of solutions of 3D navier-stokes equations

We consider fluid in a rotating frame under periodic boundary conditions. In our previous work we show that for typical aspect ratios of periods the solutions are regular if the rotation is fast enough and the initial data and forcing terms are very smooth. Here we use a kind of closing argument to extend the results by continuity to less smooth data. The required smoothness of the data follows from natural requirement of continuous dependence of solutions of the equations on the data in appropriate spaces.