Related papers: Global Regularity of the 2D HVBK equations

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…

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space…

This is a remark that by using an adaptation of the technique invented by A. Kiselev, F. Nazarov, and A. Voldberg, with a modified scaling argument, we can prove global regularity of the critical 2-D dissipative quasi-geostrophic equation…

This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data,…

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial…

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term $-\partial_x^2u$. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in…

In this brief note we study the $n$-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.

We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…

The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct…

We prove the global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical (that is, $2<\gamma<3$) Hartree equation with low regularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a unique and global…

In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small…

This article studies the global regularity problem of the two-dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by $(-\Delta)^{\alpha}u$ in the barotropic mode equation and by $(-\Delta)^{\beta}v$…

Answering a question left open in \cite{MZ2}, we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski…

Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians is studied. First, global well-posedness of small-energy solutions with general initial data in $H^s$, $s>\frac{5}{2}$, is proved.…

In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for…

We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid-structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on…

The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport…

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation…