An Existence Theorem for the Multifluid Navier-Stokes Problem

An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations

Here ν is the viscosity, p is the pressure, and f1, f2 are the components of an external forcing which may be time-dependent. As our setting is periodic, the functions u1, u2, ∇p, f1, and f2 are all periodic in x. For simplicity, we take the period to be one. The first existence and uniqueness theorems for weak solutions of (1) were proven by Leray ([Ler34]) in whole plane R. Later these result...

Absence of the Local Existence Theorem in the Critical Space for the 3D-Navier-Stokes System

We consider the 3D-Navier-Stokes system (NSS) on R3 without external forcing. After Fourier transform it becomes the system of non-linear integral equations. For one-parameter families of initial conditions A·c (0)(k) |k|2 it is known that if |A| is sufficiently small then NSS has global solution. We show that if c(0) satisfies some natural conditions at infinity then for sufficiently large A N...

Existence & Smoothness of the Navier–stokes Equation

Equation (1) is just Newton’s law f = ma for a fluid element subject to the external force f = (fi(x, t))1 i n and to the forces arising from pressure and friction. Equation (2) just says that the fluid is incompressible. For physically reasonable solutions, we want to make sure u(x, t) does now grow large as |x| → ∞. Hence, we will restrict attention to forces f and initial conditions u◦ that ...

Liouville Theorem for 2d Navier-stokes Equations

(One may modify the question by putting various other restrictions on (L); for example, one can consider only steady-state solutions, or solutions with finite rate of dissipation or belonging to various other function spaces, etc.) We have proved a positive result for dimension n = 2 which we will discuss below, but let us begin by mentioning why the basic problem is interesting. Generally spea...

Existence and bifurcation of Navier-Stokes equations