80 research outputs found
On the striated regularity for the 2D anisotropic Boussinesq system
In this paper, we investigate the global existence and uniqueness of strong
solutions to 2D Boussinesq system with anisotropic thermal diffusion or
anisotropic viscosity and with striated initial data. Using the key idea of
Chemin to solve 2-D vortex patch of ideal fluid, namely the striated regularity
can help to bound the gradient of the velocity, we can prove the global
well-posedness of the Boussinesq system with anisotropic thermal diffusion with
initial vorticity being discontinuous across some smooth interface. In the case
of an anisotropic horizontal viscosity we can study the propagation of the
striated regularity for the smooth temperature patches problem.Comment: 36 page
Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density
In this paper, we prove the global existence and uniqueness of solution to
d-dimensional (for ) incompressible inhomogeneous Navier-Stokes
equations with initial density being bounded from above and below by some
positive constants, and with initial velocity for in
2-D, or satisfying |u_0|_{L^2}|\na u_0|_{L^2} being
sufficiently small in 3-D. This in particular improves the most recent
well-posedness result in [10], which requires the initial velocity for the local well-posedness result, and a smallness condition on
the fluctuation of the initial density for the global well-posedness result
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}.
Global regularity for some classes of large solutions to the Navier-Stokes equations
In three previous papers by the two first authors, classes of initial data to
the three dimensional, incompressible Navier-Stokes equations were presented,
generating a global smooth solution although the norm of the initial data may
be chosen arbitrarily large. The main feature of the initial data considered in
the last paper is that it varies slowly in one direction, though in some sense
it is ``well prepared'' (its norm is large but does not depend on the slow
parameter). The aim of this article is to generalize the setting of that last
paper to an ``ill prepared'' situation (the norm blows up as the small
parameter goes to zero).The proof uses the special structure of the nonlinear
term of the equation
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