Global mild solutions to three-dimensional magnetohydrodynamic system in Morrey spaces

On the local pressure expansion for the Navier-Stokes equations

We show that the pressure associated with a distributional solution of the Navier-Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on Lemarié-Rieusset's "equivalence theorem." Here, the Leray projection operator composed with a gradient is defined without using the Littlewood-Paley decomposition. Prior sufficient conditions for the local expansion assumed spatial decay or estimates on the gradient and imply the considered solution is mild. An important tool is an explicit description of t…

弱-Morrey空间上稳态Boussinesq系统解的稳定性

Stability of solutions of stationary Boussinesq systems on weak-Morrey spaces

本文在$\mathbb{R}^n$（$n \geqslant 3$）临界弱-Morrey空间笛卡尔积框架下，通过双线性估计和算子插值等技术，证明了Boussinesq系统全局小温和解的存在性及稳态解的渐近稳定性。

Existence in critical spaces for the magnetohydrodynamical system in 3D bounded Lipschitz domains

Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.

A well-posedness theory in Sobolev spaces for the stochastic magnetohydrodynamic equations in the whole space

We prove the existence of a mild solution to the three dimensional incompressible stochastic magnetohydrodynamic equations in the whole space with the initial data which belong to the Sobolev spaces.

弱 Morrey 空间中 Boussinesq 系统的周期解

Periodic solutions for Boussinesq systems in weak-Morrey spaces

本文研究临界弱 Morrey 空间中 Boussinesq 系统的周期温和解，结合热半群估计、Massera 型定理与不动点方法，证明其存在唯一性、多项式稳定性并推广至 Navier-Stokes 方程。

有界非光滑域中三维及以上维度的磁流体动力学系统

The magnetohydrodynamical system in bounded non smooth domains of dimension $n\ge3$

本文研究了在有界非光滑域中三维及以上维度磁流体动力学系统在临界空间中的温和解存在性，利用斯托克斯算子的正则性结果和微分形式的利布尼茨型公式进行证明。

关于 Navier-Stokes 方程的双线性估计与临界唯一性类

On bilinear estimates and critical uniqueness classes for Navier-Stokes equations

本文研究临界空间中 Navier-Stokes 方程温和解的双线性估计与唯一性，构建无需辅助范数的一般框架，并在 Besov、Morrey 等空间及 Besov-weak-Herz 空间中取得新成果。

Affine Variant of Fractional Sobolev Space with Application to Navier-Stokes System

It is proved that for $α\in (0,1)$, $Q_α(\rn)$, not only as an intermediate space of $W^{1,n}(\rn)$ and $BMO(\rn)$ but also as an affine variant of Sobolev space $\dot{L}^{2}_α(\rn)$ which is sharply imbedded in $L^{\frac{2n}{n-2α}}(\rn)$, is isomorphic to a quadratic Morrey space under fractional differentiation. At the same time, the dot product $\nabla\cdot\big(Q_α(\rn)\big)^n$ is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system in $\bn=(0,\infty)\times\rn$.

Wiener amalgam 空间中带阻尼的 MHD 方程与粘弹性 Navier–Stokes 方程的弱解与温和解

Weak and mild solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces

本文研究三维 MHD 方程与带阻尼粘弹性 Navier–Stokes 方程，在 Wiener amalgam 空间中证明温和解存在性并构造全局局部能量弱解，同时推广局部能量解的正则性与小–大唯一性结论。

Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $\|\f{u_r}{r}{\bf 1}_{\{u_r< -\f 1r\}}\|_{L^{3/2}(\mbR^3)}< C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\f1r$ for $\forall (r,z)\in[0,\oo)\times\mbR$, then ${\bf u}\equiv 0$. …

Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…

Espaces critiques pour le système des equations de Navier-Stokes incompressibles

In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Be…

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or…

由纯跳跃噪声驱动的随机磁流体动力学方程在 $L^p$ 空间中的研究

The stochastic MHD equations driven by pure jump noise in $L^p$ spaces

研究由纯跳跃噪声驱动的随机不可压缩磁流体动力学方程，在 $L^p$ 空间中建立了温和解的局部存在唯一性，并在二维情形下证明了全局存在性。

广义 Brezis--Van Schaftingen--Yung 公式及其在球 Banach Sobolev 空间中的应用

Generalized Brezis--Van Schaftingen--Yung Formulae and Their Applications in Ball Banach Sobolev Spaces

本文在球 Banach 函数空间中，利用外推法和 Hardy--Littlewood 极大算子有界性，恢复了齐次球 Banach Sobolev 半范数，并推导出分数阶 Sobolev 型和 Gagliardo--Nirenberg 型不等式。