425 research outputs found
Large, global solutions to the Navier-Stokes equations, slowly varying in one direction
In to previous papers by the 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 aim of this article is to provide new examples of
arbitrarily large initial data giving rise to global solutions, in the whole
space. Contrary to the previous examples, the initial data has no particular
oscillatory properties, but varies slowly in one direction. The proof uses the
special structure of the nonlinear term of the equation.Comment: References adde
On the global wellposedness of the 3-D Navier-Stokes equations with large initial data
We give a condition for the periodic, three dimensional, incompressible
Navier-Stokes equations to be globally wellposed. This condition is not a
smallness condition on the initial data, as the data is allowed to be
arbitrarily large in the scale invariant space ,
which contains all the known spaces in which there is a global solution for
small data. The smallness condition is rather a nonlinear type condition on the
initial data; an explicit example of such initial data is constructed, which is
arbitrarily large and yet gives rise to a global, smooth solution
Sums of large global solutions to the incompressible Navier-Stokes equations
Let G be the (open) set of~ divergence free vector fields
generating a global smooth solution to the three dimensional incompressible
Navier-Stokes equations. We prove that any element of G can be perturbed by an
arbitrarily large, smooth divergence free vector field which varies slowly in
one direction, and the resulting vector field (which remains arbitrarily large)
is an element of G if the variation is slow enough. This result implies that
through any point in G passes an uncountable number of arbitrarily long
segments included in G.Comment: Accepted for publication in Journal f\"ur die reine und angewandte
Mathemati
Tempered distributions and Fourier transform on the Heisenberg group
The final goal of the present work is to extend the Fourier transform on the
Heisenberg group \H^d, to tempered distributions. As in the Euclidean
setting, the strategy is to first show that the Fourier transform is an
isomorphism on the Schwartz space, then to define the extension by duality. The
difficulty that is here encountered is that the Fourier transform of an
integrable function on \H^dis no longer a function on \H^d : according to
the standard definition, it is a family of bounded operators on
Following our new approach in\ccite{bcdFHspace}, we here define the Fourier
transform of an integrable functionto be a mapping on the
set~\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}endowed with a suitable
distance \wh d.This viewpoint turns out to provide a user friendly
description of the range of the Schwartz space on \H^d by the Fourier
transform, which makes the extension to the whole set of tempered distributions
straightforward. As a first application, we give an explicit formula for the
Fourier transform of smooth functions on \H^d that are independent of the
vertical variable. We also provide other examples
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
Representing three-dimensional cross fields using 4th order tensors
This paper presents a new way of describing cross fields based on fourth
order tensors. We prove that the new formulation is forming a linear space in
. The algebraic structure of the tensors and their projections on
\mbox{SO}(3) are presented. The relationship of the new formulation with
spherical harmonics is exposed. This paper is quite theoretical. Due to pages
limitation, few practical aspects related to the computations of cross fields
are exposed. Nevetheless, a global smoothing algorithm is briefly presented and
computation of cross fields are finally depicted
- …