Related papers: The Navier-Stokes Equation and Helmholtz Decomposi…

Solutions with scaling-invariant bounds such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. In this paper, we prove that any…

It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a…

We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \begin{align} u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0(x). \label{NSa} \end{align}…

The Navier-Stokes equations are considered by the use of the method of Lagrangians with covariant derivatives (MLCD) over spaces with affine connections and metrics. It is shown that the Euler-Lagrange equations appear as sufficient…

We consider the Navier-Stokes system solution, based at parametric representation of desired function. This solution is unique and it show the velocity of a stream element as its density structure [{\rho}_S (x,y,z,t);{\rho}^\to_L (x,y,z,t)]…

We introduce a modification of the Navier-Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties…

In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $\alpha \in (0,\frac{5}{4})$ in $L^2(\mathbb{R}^3)$ and…

For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on…

In this paper we prove, if $u\in\mathcal C([0,\infty),{\bf {\mathcal X}^{-1}}(\mathbb R^3))$ is global solution of 3D Navier-Stokes equations, then $\|u(t)\|_{{\bf {\mathcal X}^{-1}}}$ decays to zero as time goes to infinity. Fourier…

In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\lim_{t\to…

This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound…

This paper is concerned with the Cauchy problem of Navier-Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in $\R^3$. For less regular data and weaker compatibility condition than those…

Landau solutions are special solutions to the stationary incompressible Navier-Stokes equations in the three dimensional space excluding the origin. They are self-similar and axisymmetric with no swirl. In fact, any self-similar smooth…

The randomly driven Navier-Stokes equation without pressure in d-dimensional space is considered as a model of strong turbulence in a compressible fluid. We derive a closed equation for the velocity-gradient probability density function. We…

We discuss the appearance of spatial asymptotic expansions of solutions of the Navier-Stokes equation on $\mathbb{R}^n$. In particular, we prove that the Navier-Stokes equation is locally well-posed in a class of weighted Sobolev and…

This paper studies local existence and the singularity formation of the solutions of the one-dimensional hyperbolic Navier-Stokes equations, in particular proving the gradient blow-up of the derivatives of the solutions. The underlying…

We show that a smooth linear unsteady velocity field $u(x,t)=A(t)x+f(t)$ solves the incompressible Navier--Stokes equation if and only if the matrix $A(t)$ has zero trace, and $\dot{{A}}(t)+A^{2}(t)$ is symmetric. In two dimensions, these…

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3<p,q<\infty$ and $u_h\in L^{\infty}(0,T; BMO^{-1}(\R^3))$…

We study the solutions of the nonstationary incompressible Navier--Stokes equations in $\R^d$, $d\ge2$, of self-similar form $u(x,t)=\frac{1}{\sqrt t}U\bigl(\frac{x}{\sqrt t}\bigr)$, obtained from small and homogeneous initial data $a(x)$.…

We consider the thin-film equation $\partial_t h + \nabla \cdot \left(h^2 \nabla \Delta h\right) = 0$ in physical space dimensions (i.e., one dimension in time $t$ and two lateral dimensions with $h$ denoting the height of the film in the…