28 research outputs found
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
Decay Results for Solutions to the Magneto-Hydrodynamics equations(Mathematical Analysis of Phenomena in fluid and Plasma Dynamics)
A priori estimates of higher order derivatives of solutions to the FitzHugh-Nagumo equations
Global solvability of the Cauchy problem for the Navier–Stokes equation in for some class of initial data
Analyticity and Decay Estimates of the Navier-Stokes Equations in Critical Besov Spaces
In this paper, we establish analyticity of the Navier-Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151-180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < a. We extend these results for the intricate limiting case p = a in a suitably designed E (a) space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spacesclose8