Logarithmic improvement of regularity criteria for the Navier-Stokes equations in terms of pressure

XY is partially supported by a grant from NSERC.In this article we prove a logarithmic improvement of regularity criteria in the multiplier spaces for the Cauchy problem of the incompressible Navier-Stokes equations in terms of pressure. This improves the main result in [S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009) 1438–1443].PostprintPeer reviewe

On the existence and smoothness problem of the magnetohydrodynamics system

Fluid mechanics plays a pivotal role in engineering application to daily lives. The prominently famous fluid dynamics partial differential equations (PDE) due to its remarkable utility is the Navier-Stokes equations of which its mathematical and physical significance is so highly regarded that it has become one of the seven Millennium Prize problems declared by the Clay Research Institute. We study closely related systems of partial differential equations with focus on the magnetohydrodynamics system, of which its special case is the Navier-Stokes equations. Other systems of PDEs of our concern include the surface quasi-geostrophic equations, incompressible porous media equation governed by Darcy's law, Boussinesq system, Leray, Lans-alpha models, micropolar and magneto-micropolar fluid models. We discuss the properties of solutions to these systems such as the global regularity issue with fractional Laplacians, logarithmic supercriticality, component reduction results of regularity criteria

On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations

We improve and extend some known regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony's para-product decomposition.Comment: 13page

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

X.Y. is partially supported by the Discovery Grant No. RES0020476 from NSERC.In this article we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical Lr(0,T;Ls) regularity criteria for both velocity and pressure by factors of certain nagative powers of the scaling invariant norms ||u||L3 and ||u||H1/2.PostprintPeer reviewe

Stationary field-aligned MHD flows at astropauses and in astrotails. Principles of a counterflow configuration between a stellar wind and its interstellar medium wind

A stellar wind passing through the reverse shock is deflected into the astrospheric tail and leaves the stellar system either as a sub-Alfvenic or as a super-Alfvenic tail flow. An example is our own heliosphere and its heliotail. We present an analytical method of calculating stationary, incompressible, and field-aligned plasma flows in the astrotail of a star. We present a recipe for constructing an astrosphere with the help of only a few parameters, like the inner Alfven Mach number and the outer Alfven Mach number, the magnetic field strength within and outside the stellar wind cavity, and the distribution of singular points of the magnetic field within these flows. Within the framework of a one-fluid approximation, it is possible to obtain solutions of the MHD equations for stationary flows from corresponding static MHD equilibria, by using noncanonical mappings of the canonical variables. The canonical variables are the Euler potentials of the magnetic field of magnetohydrostatic equilibria. Thus we start from static equilibria determined by the distribution of magnetic neutral points, and assume that the Alfven Mach number for the corresponding stationary equilibria is finite. The topological structure determines the geometrical structure of the interstellar gas - stellar wind interface. Additional boundary conditions like the outer magnetic field and the jump of the magnetic field across the astropause allow determination of the noncanonical transformations. This delivers the strength of the magnetic field at every point in the astrotail region beyond the reverse shock. The mathematical technique for describing such a scenario is applied to astrospheres in general, but is also relevant for the heliosphere. It shows the restrictions of the outer and the inner magnetic field strength in comparison with the corresponding Alfven Mach numbers in the case of subalfvenic flows.Comment: 19 pages, 17 figures, accepted for publication in A&