This paper examines the global (in time) regularity of classical solutions to the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-\Delta)^\beta$. We establish the global regularity for the case when $\beta>1$. This result significantly improves previous work which requires $\beta>\frac{3}{2}$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely, the case when $\beta=1$.

The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm.

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Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model

This paper studies the global well-posedness of the incompressible magnetohydrody- namic (MHD) system with a velocity damping term. We establish the global existence and unique- ness of smooth solutions when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also given.

/pdf/global-small-solution-to-the-2d-mhd-system-with-a-velocity-3h048xrg4i.pdf

Global Small Solution to the 2D MHD System with a Velocity Damping Term

This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $${\Lambda^{2\beta} b}$$
 . It is proved that when β > 1 there exists a unique global regular solution for this equations. The obtained result improves the previous known ones which require that $${\beta > \frac{3}{2}}$$
 . With help of Fourier analysis, Besov spaces and singular integral theory, some delicate estimates on the vorticity $${\omega}$$
 and the current j are established to prove our main result.

Global regularity of 2D generalized MHD equations with magnetic diffusion

Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping

http://www.ims.cuhk.edu.hk/publications/reports/2006-06.pdf

Weak and strong solutions for the incompressible Navier–Stokes equations with damping

A remark on global regularity of 2D generalized magnetohydrodynamic equations

Uniqueness of the global weak solutions to 2D compressible primitive equations

Vanishing viscous limits for the 2D lake equations with Navier boundary conditions

In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} 
\left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - \nu (-\triangle)^{\alpha} u b_t + u \cdot \nabla b & = & b \cdot \nabla u - \kappa (-\triangle)^{\beta} b 
\end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslant\alpha 3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $\alpha=0$ and $\beta>\frac 32$.

Quansen Jiu

Papers

Weak and strong solutions for the incompressible Navier–Stokes equations with damping

A remark on global regularity of 2D generalized magnetohydrodynamic equations

Uniqueness of the global weak solutions to 2D compressible primitive equations

Vanishing viscous limits for the 2D lake equations with Navier boundary conditions

A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations