Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System

Masmoudi (2010) obtained global well-posedness for 2D Maxwell-Navier-Stokes system. In this paper, we reprove global existence of regular solutions to the 2D system by using energy estimates and Brezis-Gallouet inequality. Also we obtain a blow-up criterion for solutions to 3D Maxwell-Navier-Stokes system

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A

Normal surfaces in knot complements

We extend normal surface Q-theory developed in compact triangulated 3-manifolds to some non-compact 3-manifolds and apply the Q-theory to knot complements. We also give an algorithm to find a normal surface representing a minimal Seifert surface of a non-fibered knot in the knot complement. The figure-eight knot is presented as a fibered knot which does not have any either normal or almost normal representation of a minimal Seifert surface of the knot in its complement in S 3.

Regularizing Model for the 2D MHD Equations with Zero Viscosity

We consider the regularity of two dimensional incompressible magneto-hydrodynamics equations with zero viscosity. We provide an approximating system to the equations and prove global-in-time existence of classical solution to this approximating system. By using approximating system, a priori estimates for the equations can be justified