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In this paper, we are mainly concerned with the well-posedness of the dissipative surface quasi-geostrophic equation in the framework of variable Lebesgue spaces. Based on some analytical results developed in the variable Lebesgue spaces and the $L^{p}$-$L^{q}$ decay estimates of the fractional heat kernel, we establish the local existence and regularity of solutions to the 2D dissipative surface quasi-geostrophic equation in the variable Lebesgue space., Comment: 11 pages

In this paper, we deal with some Oldroyd type models, which describe incompressible viscoelastic fluids. There are 3 parameters in these models: the viscous coefficient of fluid $\nu_{1}$, the viscous coefficient of the elastic part of the stress tensor $\nu_{2}$, and the damping coefficient of the elastic part of the stress tensor $\alpha$. In this paper, we assume at least one of the parameters is zero: $(\nu_{1}, \nu_{2},\alpha)=(+,0,+), (+,0,0), (0,+,+), (0,+,0), (0,0,+)$ and prove the global existence of unique solutions to all these 5 cases in the framework of hybrid Besov spaces. We also derive decay rates of the solutions except for the case $(\nu_{1}, \nu_{2},\alpha)=(+,0,0)$. To the best of our knowledge, decay rates in this paper are the first results in this framework, and can improve some previous works., Comment: 35 pages

In this paper, we are concerned with the three-dimensional nonhomogeneous B\'enard system with density-dependent viscosity in bounded domain. The global well-posedness of strong solution is established, provided that the initial total mass is suitably small. In particular, the initial velocity and temperature can be arbitrarily large. Moreover, the exponential decay of strong solution is also obtained. It is worth noting that the vacuum of initial density is allowed., Comment: 22 pages

Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+}\times\Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2D$ torus or a bounded convex domain in $\mathbb{R}^2$. In this paper, we show the existence of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary.

For the three dimensional Prandtl boundary layer equations, we will show that for arbitrary $M$ and sufficiently small $\epsilon$, the lifespan of the Gevrey-2 solution is at least of size $\epsilon^{-M}$ if the initial data lies in suitable Gevrey-2 spaces with size of $\epsilon$.

In this paper, we will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data, which lies in $H^1$ Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables. Proof of the main result relies on the construction of a tangentially weighted analytic energy functional, which acts on a specially designed good unknown. The constructed energy functional can find its two dimensional parallel in Ignatova-Vicol [2016ARMA] where no tangential weight is introduced and the specially good unknown is set to control the lower bound of the analytical radius, whose two dimensional similarity can be traced to Paicu-Zhang [2021ARMA]. Our result is an improvement of that in Ignatova-Vicol [2016ARMA] from the almost global existence to the global existence and an extension of that in Paicu-Zhang [2021ARMA] from the two dimensional case to the three dimensional axially symmetric case.

In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed., Comment: 20 pages, all comments are welcome

In this paper, we will show that solutions of the three-dimensional non-resistive and non-diffusive MHD-Boussinesq system are globally regular if the initial data is axisymmetric and the swirl components of the velocity and the magnetic vorticity are zero. Our main result extends previous ones on the three-dimensional non-resistive MHD system and non-diffusive Boussinesq system, and the method used here can also be applied to the magnetic Rayleigh-B\'{e}nard convection system., Comment: 17pages

In this paper, we construct a class global large solution to the 3D incompressible micropolar fluid system. Precisely speaking, by choosing a special initial data which can be arbitrarily large in $\dot{B}^{-1}_{\infty,\infty}$, the system has a unique global solution., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1904.01779

The Navier-Stokes-Maxwell-Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.

This paper establishes the global existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyperresistivity. More precisely, the equation of $b_1$ (the horizontal component of the magnetic field) involves only vertical hyperdiffusion (given by $\Lambda_2^{2\beta} b_1$) while the equation of $b_2$ (the vertical component) has only horizontal hyperdiffusion (given by $\Lambda_1^{2\beta} b_2$), where $\Lambda_1$ and $\Lambda_2$ are directional Fourier multiplier operators with the symbols being $|\xi_1|$ and $|\xi_2|$, respectively. We prove that, for $\beta>1$, this system always possesses a unique global-in-time classical solution when the initial data is sufficiently smooth. The model concerned here is rooted in the MHD equations with only magnetic diffusion, which play a significant role in the study of magnetic reconnection and magnetic turbulence. In certain physical regimes and under suitable scaling, the magnetic diffusion becomes partial (given by part of the Laplacian operator). There have been considerable recent developments on the fundamental issue of whether classical solutions of these equations remain smooth for all time. The papers of Cao-Wu-Yuan \cite{CaoWuYuan} and of Jiu-Zhao \cite{JiuZhao2} obtained the global regularity when the magnetic diffusion is given by the full fractional Laplacian $(-\Delta)^\beta$ with $\beta>1$. The main result presented in this paper requires only directional fractional diffusion and yet we prove the regularization in all directions. The proof makes use of a key observation on the structure of the nonlinearity in the MHD equations and technical tools on Fourier multiplier operators such as the H\"{o}rmander-Mikhlin multiplier theorem. The result presented here appears to be the sharpest for the 2D MHD equations with partial magnetic diffusion., Comment: Accepted for publication in International Mathematics Research Notices

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh-Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving \emph{a priori} estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field., Comment: 33 pages

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. A local (in time) existence of weak solution is established, in particular, the density is positive and the solution is regular away from the free boundary., Comment: Acta Mathematica Scientia 2012

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t \theta + \mathcal{R}^\perp \theta \cdot \nabla \theta + \Lambda^\gamma \theta = 0, \qquad \theta(\cdot,0) =\theta_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $\gamma \in (0,1)$. The coefficient in front of the dissipative term $\Lambda^\gamma = (-\Delta)^{\gamma/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|\theta_0\|_{L^2}^{\gamma/2} \|\theta_0\|_{\dot{H}^2}^{1-\gamma/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $\gamma_1 = \gamma_1(R) \in (0,1)$ such that for $\gamma \geq \gamma_1$, the solution of the supercritical SQG equation with dissipation $\Lambda^\gamma$ does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle., Comment: 12 pages

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain $\Omega\subset \mathbb{R}^N$, where N=2 or 3. By exploiting a maximum principle, Nirenberg's interpolation inequality and a smallness condition imposed on the $N$-th component of initial direction field $\mf{d}_0$ to overcome the difficulties induced by the supercritical nonlinearity $|\nabla{\mathbf d}|^2{\mathbf d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity., Comment: arXiv admin note: substantial text overlap with arXiv:1210.3565

We present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three. We prove that suitable weak solutions are H\"older continuous near boundary provided that either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\le 2$, $2

We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy's Law. At the triple point where air and liquid meet the solid substrate, the liquid assumes a constant, non-zero contact angle ({\it partial wetting}). We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $\eps$., Comment: 38 pages, 6 figures

This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $\theta$ through $\mathcal{R} \Lambda^{-1} P(\Lambda) \theta$ where $\mathcal{R}$ denotes a Riesz transform, $\Lambda=(-\Delta)^{1/2}$ and $P(\Lambda)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(\Lambda)=I$ while the surface quasi-geostrophic (SQG) equation to $P(\Lambda) =\Lambda$. We obtain the global regularity for a class of equations for which $P(\Lambda)$ and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with $P(\Lambda)= (\log(I-\Delta))^\gamma$ for any $\gamma>0$ are globally regular.

We study steady flow of a second grade fluid past an obstacle in three space dimensions. We prove existence of solution in weighted Lebesgue spaces with anisotropic weights and thus existence of the wake region behind the obstacle. We use properties of the fundamental Oseen tensor together with results achieved in \cite{Koch} and properties of solutions to steady transport equation to get up to arbitrarily small $\ep$ the same decay as the Oseen fundamental solution.

Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $\theta$ through $u_j =\mathcal{R} \Lambda^{-1} P(\Lambda) \theta$ where $\mathcal{R}$ is a Riesz transform and $\Lambda=(-\Delta)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(\Lambda)=I$ while the SQG equation to the case $P(\Lambda) =\Lambda$. We develop tools to bound $\|\nabla u||_{L^\infty}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(\Lambda)= (\log(I+\log(I-\Delta)))^\gamma$ with $0\le \gamma\le 1$. In addition, a regularity criterion for the model corresponding to $P(\Lambda)=\Lambda^\beta$ with $0\le \beta\le 1$ is also obtained.

The compressible Navier-Stokes-Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions., Comment: 28 pages