17 research outputs found
On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations
We study the global regularity, for all time and all initial data in
, of a recently introduced decimated version of the incompressible 3D
Navier-Stokes (dNS) equations. The model is based on a projection of the
dynamical evolution of Navier-Stokes (NS) equations into the subspace where
helicity (the scalar product of velocity and vorticity) is sign-definite.
The presence of a second (beside energy) sign-definite inviscid conserved
quadratic quantity, which is equivalent to the Sobolev norm, allows
us to demonstrate global existence and uniqueness, of space-periodic solutions,
together with continuity with respect to the initial conditions, for this
decimated 3D model. This is achieved thanks to the establishment of two new
estimates, for this 3D model, which show that the and the time
average of the square of the norms of the velocity field remain
finite. Such two additional bounds are known, in the spirit of the work of H.
Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing
well-posedness for the 3D NS equations. Furthermore, they are directly linked
to the helicity evolution for the dNS model, and therefore with a clear
physical meaning and consequences
Infinite energy solutions to the Navier-Stokes equations in the half-space and applications
This short note serves as an introduction to the papers arXiv:1711.01651 and
arXiv:1711.04486 by Maekawa, Miura and Prange. These two works deal with the
existence of mild solutions on the one hand and local energy weak solutions on
the other hand to the Navier-Stokes equations in the half-space . We
emphasize a concentration result for (sub)critical norms near a potential
singularity. The contents of these notes were presented during the X-EDP
seminar at IH\'ES in October 2017.Comment: 18 pages. This is a review article. This note will be published in
the proceedings of the "S\'eminaire Laurent Schwartz-EDP et applications
Global Navier-Stokes flows in intermediate spaces
We construct global weak solutions of the three dimensional incompressible
Navier-Stokes equations in intermediate spaces between the space of uniformly
locally square integrable functions and Herz-type spaces which involve weighted
integrals centered at the origin. Our results bridge the existence theorems of
Lemari\'e-Rieusset and of Bradshaw, Kukavica and Tsai. An application to
eventual regularity is included which generalizes the prior work of Bradshaw,
Kukavica and Tsai as well as Bradshaw, Kukavica and Ozanski.Comment: We moved Lemma 3.1 to Lemma 2.1, and added Lemma 2.11 and several
reference