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In this paper, we consider the initial-boundary value problem to the three-dimensional primitive equations for the oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation in the domain $\Omega=M\times(-h,h)$, with $M=(0,1)\times(0,1)$. Global well-posedness of strong solutions is established, for any initial data $(v_0,T_0) \in H^1(\Omega)\cap L^\infty(\Omega)$ with $(\partial_z v_0, \nabla_H T_0) \in L^q(\Omega)$ and $v_0 \in L_z^1(B^1_{q,2}(M))$, for some $q \in (2,\infty)$, by using delicate energy estimates and maximal regularity estimate in the anisotropic setting., Comment: arXiv admin note: text overlap with arXiv:1703.02512

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization., Comment: 35 pages

We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.

We study the three-dimensional Hasegawa-Mima model of turbulent magnetized plasma with horizontal viscous terms and a weak vertical dissipative term. In particular, we establish the global existence and uniqueness of strong solutions for this model.

In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity $v_0\in H^2(\Omega)$ and the initial temperature $T_0\in H^1(\Omega)\cap L^\infty(\Omega)$ with $\nabla_HT_0\in L^q(\Omega)$, for some $q\in(2,\infty)$. Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to $H^2(\Omega)$. The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the "\,mismatching\," of regularities between the horizontal momentum and temperature equations. To handle this "mismatching" of regularities, we introduce several auxiliary functions, i.e., $\eta, \theta, \varphi,$ and $\psi$ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity $v$ or its vertical derivative $\partial_zv$. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some $L^1_t(W^{1,\infty}_\textbf{x})$-type a priori estimates on $v$, we decompose the velocity into the "temperature-independent" and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Br\'ezis-Gallouet-Wainger and Beale-Kato-Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work [12], is used, and a new logarithmic type Gronwall inequality is exploited.

In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data $(v_0, T_0)\in H^1$, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0, T_0)\in H^1\cap L^\infty$, such that $\partial_zv_0\in L^m$, for some $m\in(2,\infty)$, by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where the initial data $(v_0, T_0)$ was assumed to have $H^2$ regularity.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with $H^2$ initial data., Comment: arXiv admin note: substantial text overlap with arXiv:1312.6035

In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with $H^2$ initial data.

We establish, for smooth enough initial data, the global well-posedness (existence, uniqueness and continuous dependence on initial data) of solutions, for an inviscid three-dimensional {\it slow limiting ocean dynamics} model. This model was derived as a strong rotation limit of the rotating and stratified Boussinesg equations with periodic boundary conditions. To establish our results we utilize the tools developed for investigating the two-dimensional incompressible Euler equations and linear transport equations. Using a weaker formulation of the model we also show the global existence and uniqueness of solutions, for less regular initial data.

The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the "vortex stretching", we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely the homogeneous Besov space $\mathring{B}^1_{\infty,1}$), we are able to obtain a unique global solution under a minimal smallness assumption., Comment: 18 pages

This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-\Delta)^\beta$. We establish the global regularity for the case when $\beta>1$. This result significantly improves previous work which requires $\beta>\frac32$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when $\beta=1$., Comment: 16 pages

In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-B\'enard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations.

This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\infty}$ and $\|v\|_{L^r}$ then yields the desired global regularity., Comment: 17 pages

It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the literal boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no heat flux physical boundary condition. Consequently, the second order parabolic heat equation is over determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon's planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model.

The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions.

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 of the pressure in one-direction.

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.

In this paper we consider the three-dimensional Navier-Stokes equations in an infinite channel. We provide a sufficient condition, in terms of $\partial_z p$, where $p$ is the pressure, for the global existence of the strong solutions to the three-dimensional Navier-Stokes equations.

In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.

In this paper we study a three dimensional thermocline planetary geostrophic ``horizontal" hyper--diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper--diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models., Comment: 32 pages