794 research outputs found
Global well-posedness of -D anisotropic Navier-Stokes system with small unidirectional derivative
In \cite{LZ4}, the authors proved that as long as the one-directional
derivative of the initial velocity is sufficiently small in some scaling
invariant spaces, then the classical Navier-Stokes system has a global unique
solution. The goal of this paper is to extend this type of result to the 3-D
anisotropic Navier-Stokes system with only horizontal dissipation. More
precisely, given initial data u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12}, has
a unique global solution provided that |D_\h|^{-1}\pa_3u_0 is sufficiently
small in the scaling invariant space $\cB^{0,\f12}.
On the existence and structures of almost axisymmetric solutions to 3-D Navier-Stokes equations
In this paper, we consider 3-D Navier-Stokes equations with almost
axisymmetric initial data, which means that by writing in the cylindrical coordinates, then
and are small in some sense (recall axisymmetric means these three
quantities vanish). Then with additional smallness assumption on ,
we prove the global existence of a unique strong solution , and this
solution keeps close to some axisymmetric vector field. We also establish some
refined estimates for the integral average in variable for .
Moreover, as and here depend on , it is
natural to expand them into Fourier series in variable. And we shall
consider one special form of , with some small parameter to
measure its swirl part and oscillating part. We study the asymptotic expansion
of the corresponding solution, and the influences between different profiles in
the asymptotic expansion. In particular, we give some special symmetric
structures that will persist for all time. These phenomena reflect some
features of the nonlinear terms in Navier-Stokes equations
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