Non-equilibrium Ensembles for the three-dimensional Navier-Stokes Equations

On a time-discrete approach to solving Navier-Stokes systems

We present a new scheme for solving Navier-Stokes systems. This is inspired by material differentiation and combined with discrete Morse semi-flow. The solution has the first energy inequality, we set some assumption though. This result crucially does not provide any new result, but I believe that by making my scheme is more sophisticated, we can develop the study of Navier-Stokes systems.

Symmetry reduction and exact solutions of the Navier-Stokes equations

Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

A rigorous justification of the Euler and Navier-Stokes equations with geometric effects

We derive the 1D isentropic Euler and Navier-Stokes equations describing the motion of a gas through a nozzle of variable cross section as the asymptotic limit of the 3D isentropic Navier-Stokes system in a cylinder, the diameter of which tends to zero. Our method is based on the relative energy inequality satisfied by any weak solution of the 3D Navier-Stokes system and a variant of Korn-Poincare's inequality on thin channels that may be of independent interest.

Comparing Poiseuille with 1D Navier-Stokes Flow in Rigid and Distensible Tubes and Networks

A comparison is made between the Hagen-Poiseuille flow in rigid tubes and networks on one side and the time-independent one-dimensional Navier-Stokes flow in elastic tubes and networks on the other. Analytical relations, a Poiseuille network flow model and two finite element Navier-Stokes one-dimensional flow models have been developed and used in this investigation. The comparison highlights the differences between Poiseuille and one-dimensional Navier-Stokes flow models which may have been unjustifiably treated as equivalent in some studies.

Quotients of Navier--Stokes equation on space curves

A Navier--Stokes system on a curve is discussed. The quotient equation for this system is found. The quotient is used to find some solutions of Navier--Stokes system. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations.

Navier-Stokes方程的Reynolds平均解

Reynolds Averaged Solutions of the Navier-Stokes Equation

本文证明一般初始条件下Navier-Stokes方程Young测度解的均值属于Leray-Hopf类PDE解。

Multi-dimensional Jordan chain and Navier-Stokes Equation

Multi-dimensional Jordan chain is presented. It is shown that it admits the finite-component reductions to the Navier-Stokes equation and other important equations of continuous media theory.

On the Dynamics of Navier-Stokes and Euler Equations

This is a rather comprehensive study on the dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a). zero viscosity limit of the spectra of linear Navier-Stokes operator, (b). heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Besides Navier-Stokes and Euler equations, we also study two models of them.

Some preliminary observations on a defect Navier-Stokes system

Some implications of the simplest accounting of defects of compatibility in the velocity field on the structure of the classical Navier-Stokes equations are explored, leading to connections between classical elasticity, the elastic theory of defects, plasticity theory, and classical fluid mechanics.

Geometric Regularity Criteria for Incompressible Navier--Stokes Equations with Navier Boundary Conditions

We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on regular domains with a class of oblique derivative boundary conditions, providing that the vorticity of the fluid is coherently aligned. In particular, we establish the boundary regularity on round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.