Related papers: A regularity criterion of the 3D MHD equations inv…

We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free…

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions…

It has been well known that if $\Omega$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $\Omega$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(\Omega )$ of the Poisson…

We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of…

In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak…

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small $L^2$-norm which may display codimension-one…

We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm…

We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations…

In this article, we initiate the study of the Cauchy problem for the two-dimensional relativistic Euler equations in a low-regularity setting. By introducing good variables--a rescaled velocity, logarithmic enthalpy, and an appropriately…

We address a threshold problem of the Couette flow $(y,0)$ in a uniform magnetic field $(\beta,0)$ for the 2D MHD equation on $\mathbb{T}\times\mathbb{R}$ with fluid viscosity $\nu$ and magnetic resistivity $\mu$. The nonlinear enhanced…

In this paper, we study the global regularity of strong solution to the Cauchy problem of 3D incompressible Navier-Stokes equations with large data and non-zero force. We prove that the strong solution exists globally for $\nabla u\in…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity $v$ determined by $v=\mathcal{R}^{\perp}\Lambda^{\gamma-1}\theta$. It is shown that the $L^p$ type energy norm of weak solutions is…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

The micropolar fluid system is a model based on the Navier-Stokes equations which considers two coupled variables: the velocity field $\vec u$ and the microrotation field $\vec\omega$. Assuming an additional condition over the variable…

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3<p,q<\infty$ and $u_h\in L^{\infty}(0,T; BMO^{-1}(\R^3))$…

We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^\alpha$ with $\alpha=1/200$, for which the total energy and the cross helicity (i.e., the so-called Els\"asser energies) are not conserved. The solutions do…

We prove that given any $\beta<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v \in…

This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…