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Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on…
A main result of this paper establishes the global stability of the three-dimensional MHD equations near a background magnetic field with mixed fractional partial dissipation with $\alpha, \beta\in(\frac{1}{2}, 1]$. Namely, the velocity…
In this paper, we prove the global existence of strong solutions to the two-dimensional compressible MHD equations with density dependent viscosity coefficients (known as Kazhikhov-Vaigant model) on 2D solid balls with arbitrary large…
This study delves into a comprehensive examination of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equations in $H^{1}(\R^{3})$. The modification involves incorporating a power term in the nonlinear convection…
The purpose of this paper is to study the incompressible non-resistive MHD equations in $\mathbb{R}^3$. We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the…
We study the two-dimensional generalized magnetohydrodynamics system with generalized dissipation and diffusion in terms of fractional Laplacians. It is known that the classical magnetohydrodynamics system with full Laplacians in both…
We study the electron magnetohydrodynamics (MHD) in two dimensional geometry, which has a rich family of steady states. In an anisotropic resistivity context, we show global in time existence of small smooth solution near a shear type…
In this paper, we consider the following modified quasi-geostrophic equations $\partial_t\theta +\Lambda^\alpha\theta +u\vec\nabla\theta =0$, $u=\Lambda ^{\alpha-1}\mathcal{R}^\perp(\theta)$ where $\alpha \in ]0,1[$ is a fixed parameter.…
Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is…
We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with…
In this paper, we consider suitable weak solutions of the four dimensional incompressible magneto-hydrodynamic equations. We give two different kind $\varepsilon$-regularity criteria. One only requires the smallness of scaling $L^{p,q}$…
Whether the global existence and uniqueness of strong solutions of $n$-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding…
We solve here the so called division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that semilinear wave equations which can be written as systems involving quadratic derivative…
The nonlinear wave equation $u_{tt}-\Delta u +|u_t|^{p-1}u_t=0$ is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions $H^k_{\rm rad}({\bf R}^3)\times H^{k-1}_{\rm rad}({\bf R}^3)$ for all $p\geq 3$ and…
This work establishes two regularity criteria for the 3D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one-direction while the second one requires suitable boundedness of the derivative…
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+…
In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the…
This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in…
In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t \theta + (u \cdot \grad) \theta + \kappa \Lambda^\alpha \theta = 0 u = \Lambda^{\alpha - 1} R^{\perp}\theta. \end{gather*} with $\kappa > 0$, $\alpha…
In this paper, we consider two Approximate Deconvolution Magnetohydrodynamics models which are related to Large Eddy Simulation. We first study existence and uniqueness of solutions in the double viscous case. Then, we study existence and…