87 research outputs found
Decay of weak solutions to the 2D dissipative quasi-geostrophic equation
We address the decay of the norm of weak solutions to the 2D dissipative
quasi-geostrophic equation. When the initial data is in only, we prove
that the norm tends to zero but with no uniform rate, that is, there are
solutions with arbitrarily slow decay. For the initial data in ,
with , we are able to obtain a uniform decay rate in . We
also prove that when the norm of the initial data
is small enough, the norms, for have uniform
decay rates. This result allows us to prove decay for the norms, for , when the initial data is in .Comment: A paragraph describing work by Carrillo and Ferreira proving results
directly related to the ones in this paper is added in the Introduction. Rest
of the article remains unchange
A study of energy concentration and drain in incompressible fluids
In this paper we examine two opposite scenarios of energy behavior for
solutions of the Euler equation. We show that if is a regular solution on a
time interval and if for some , where is the dimension of the fluid, then the energy at the
time cannot concentrate on a set of Hausdorff dimension samller than . The same holds for solutions of the three-dimensional
Navier-Stokes equation in the range . Oppositely, if the energy
vanishes on a subregion of a fluid domain, it must vanish faster than
(T-t)^{1-\d}, for any \d>0. The results are applied to find new exclusions
of locally self-similar blow-up in cases not covered previously in the
literature.Comment: an update of the previous versio
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations
We establish a connection between Optimal Transport Theory and classical
Convection Theory for geophysical flows. Our starting point is the model
designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal
Transport problems. This model can be seen as a generalization of the
Darcy-Boussinesq equations, which is a degenerate version of the
Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate
different variants of the NSB equations (in particular what we call the
generalized Hydrostatic-Boussinesq equations) to various models involving
Optimal Transport (and the related Monge-Ampere equation. This includes the 2D
semi-geostrophic equations and some fully non-linear versions of the so-called
high-field limit of the Vlasov-Poisson system and of the Keller-Segel for
Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model
can be related to the magnetic relaxation model studied by Arnold and Moffatt
to obtain stationary solutions of the Euler equations with prescribed topology
Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System
We study a complex non-newtonian fluid that models the flow of nematic liquid
crystals. The fluid is described by a system that couples a forced
Navier-Stokes system with a parabolic-type system. We prove the existence of
global weak solutions in dimensions two and three. We show the existence of a
Lyapunov functional for the smooth solutions of the coupled system and use the
cancellations that allow its existence to prove higher global regularity, in
dimension two. We also show the weak-strong uniqueness in dimension two
On formation of a locally self-similar collapse in the incompressible Euler equations
The paper addresses the question of existence of a locally self-similar
blow-up for the incompressible Euler equations. Several exclusion results are
proved based on the -condition for velocity or vorticity and for a range
of scaling exponents. In particular, in dimensions if in self-similar
variables and u \sim \frac{1}{t^{\a/(1+\a)}}, then the blow-up
does not occur provided \a >N/2 or -1<\a\leq N/p. This includes the
case natural for the Navier-Stokes equations. For \a = N/2 we exclude
profiles with an asymptotic power bounds of the form |y|^{-N-1+\d} \lesssim
|u(y)| \lesssim |y|^{1-\d}. Homogeneous near infinity solutions are eliminated
as well except when homogeneity is scaling invariant.Comment: A revised version with improved notation, proofs, etc. 19 page
Time decay for solutions to the Stokes equations with drift
In this note, we study the behavior of Lebesgue norms ||v(·, t)||pof solutions v to the Cauchy problem for the Stokes system with drift u, which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition
Time decay for solutions to the Stokes equations with drift
In this note, we study the behavior of Lebesgue norms ||v(·, t)||pof solutions v to the Cauchy problem for the Stokes system with drift u, which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition
Time decay for solutions to the Stokes equations with drift
In this note, we study the behavior of Lebesgue norms ||v(·, t)||pof solutions v to the Cauchy problem for the Stokes system with drift u, which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition
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