Related papers: Integral formulation of 3-D Navier-Stokes and long…
Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms \cite{TT10,TT11}. There…
We consider the Navier-Stokes initial value problem, $$v_t - \nabla v = -\mathcal{P} [ v \cdot \nabla v \right ] + f, v(x, 0) = v_0 (x), x \in \mathbb{R}^3 $$ where $\mathcal{P}$ is the Hodge-Projection to divergence free vector fields in…
In this paper, we prove that there exists a unique global solution of $3D$ Navier-Stokes equation if $\exp(a|D|^{1/\sigma})u^0\in{\mathcal{X}}^{-1}(\mathbb R^3)$ and $\|u^0\|_{{\mathcal{X}}^{-1}}<\nu$. Moreover, we will show that…
Given initial data $u_0=(u_0^\h,u_0^3)\in H^{-\d,0}\cap H^{\f12}(\R^3)$ with both~$\uh_0$ and~$\nabla_{\rm h}\uh_0$ belonging to ~$L^2(\R^3)\cap L^\infty(\R_\v;L^2(\R^2_\h))$ and $u_0^\h\in L^\infty(\R_\v, H^{-\d}(\R^2_\h))$ for some…
In this paper we develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces $E^s_{p,q}$ with exponentially decaying weights $(s<0, \…
We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous.…
In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = \nu\Delta u + \mathbb{P} [v\nabla u]$ we derive the corresponding equations for the time…
In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier--Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates…
A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions. In this paper we shall study this problem. The Navier-Stokes equations are given by:…
The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$ there exists a global smooth solution u. In this note we observe (using a simple…
In this paper, we prove that if $u\in C([0,\infty), \dot{H}^{1/2}_{a,1}(\mathbb{R}^3))$ is a global solution of 3D incompressible Navier-Stokes equations, then $\|u\|_{\dot{H}^{1/2}_{a,1}}$ decays to zero as time approaches infinity.…
We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this…
The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from the Navier-Stokes equations. The solution $v$ to NS problem is defined in this paper as the solution to an integral equation. The kernel $G$ of this equation…
In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: $A(r,z)\cos N\theta +B(r,z)\sin…
In this paper, we simplify and extend the results of \cite{GZ} to include the case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb…
In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian…
The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a…
It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces…
Consider the equations of Navier-Stokes in $\R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided the initial data is small with respect to the norm…
In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier-Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier-Stokes equation, then…