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We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space R n \mathbb {R}^{n} . Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum.

This paper focuses on a two-dimensional tropical climate model with temperature-dependent viscosity and thermal diffusivity. We show that there is a unique global smooth solution to this model with general initial data in the Sobolev class H s for any s > 1.

In this paper we consider the Cauchy problem of a model of the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity. We show that there is a unique global smooth solution to this system for arbitrarily large initial data in Sobolev spaces. Our key argument is the De Giorgi-Nash-Moser estimates for the vorticity equation.

Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regularity issue on the 2D MHD equations with standard Laplacian magnetic diffusion., In this submitted version, we have corrected a mistake in our published version (J. Evol. Equ. 18 (2018), no. 2, 821-844). This mistake comes from the misuse of the Bochner theorem

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λ α dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α > 0 always possesses a unique global classical solution.

In this paper, we obtain a regularity criterion of the solution to the three-dimensional incompressible Hall-magnetohydrodynamic equations involving only the vorticity.

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. The global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply. To overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic-type Gronwall inequality, which may be of independent interest and have potential applications.

We investigate transcendental entire solutions of complex differential equations f ″ + A ( z ) f = H ( z ) , where the entire function A ( z ) has a growth property similar to the exponential functions, and H ( z ) is an entire function of order less than that of A ( z ) . We first prove that the lower order of the entire solution to the equation is infinity. By using our result on the lower order, we prove the entire solution does not bear any Baker wandering domains.

In this paper, we are concerned with the Cauchy problem of the multi-dimensional inhomogeneous incompressible magnetohydrodynamic equations with fractional dissipation. We prove the unique global strong solution with vacuum to the magnetohydrodynamic equations over the whole space. In addition, the large time decay rates of the corresponding strong solution are also obtained. Note that our result is proved without any smallness on the initial data. Moreover, the initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. To our best knowledge, this is the first result on the inhomogeneous fractional magnetohydrodynamic equations.

The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.

This paper establishes the global existence and regularity of solutions to a two-dimensional (2D) tropical climate model (TCM) with fractional dissipation. The inviscid counterpart of this model was derived by Frierson, Majda and Pauluis [8] as a model for tropical geophysical flows. This model reflects the interaction and coupling among the barotropic mode u, the first baroclinic mode v of the velocity and the temperature θ. The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global regularity with the least dissipation. We prove two main results: first, the global regularity of the system with ( − Δ ) β v and ( − Δ ) γ θ for β > 1 and β + γ > 3 2 ; and second, the global regularity of the system with ( − Δ ) β v for β > 3 2 . The proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal. The key tools employed here include the maximal regularity for general fractional heat operators, the Littlewood–Paley decomposition and Besov space techniques, lower bounds involving fractional Laplacian and simultaneous estimates of several coupled quantities.

The present paper is dedicated to the global-in-time existence and uniqueness issue for the three-dimensional incompressible Boussinesq equations with fractional partial dissipation.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

This paper is dedicated to establishing the global regularity for the two dimensional magnetohydrodynamic equations with fractional anisotropic dissipation when the fractional powers are restricted to some certain ranges. In addition, the global regularity results for the two dimensional magnetohydrodynamic equations with partial dissipation are also obtained. Consequently, these results bring us more closer to the resolution of the global regularity problem on the two dimensional magnetohydrodynamic equations with standard Laplacian magnetic diffusion.

In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the magnetohydrodynamics equations with the minimal dissipation. Consequently, our results significantly improve the previous works., This is the final version published in Dynamics of Partial Differential Equations

In this paper, we establish the global regularity of the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations with the Laplacian dissipation in the velocity equation and zero magnetic diffusion.

Let $$f_\omega (z)=\sum \nolimits _{j=0}^{\infty }\chi _j(\omega ) a_j z^j$$ be a transcendental random entire function, where $$\chi _j(\omega )$$ are independent and identically distributed random variables defined on a probability space $$(\Omega , \mathcal {F}, \mu )$$ . In this paper, we study a family of random entire functions, which includes Gaussian, Rademacher, and Steinhaus entire functions. Then we prove that if two random entire functions in this family share two distinct complex numbers counting multiplicities, then they are identically equal.

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Benard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity $$u_2$$ or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as $$\Delta u_2$$ , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here.

As a continuation of our previous work, we consider lower regularity global well-posedness for a model of the two-dimensional zero diffusivity Boussinesq equations with variable viscosity. More precisely, based on De Giorgi–Nash–Moser estimates and the refined logarithmic Gronwall-type inequality, we prove that it is globally well-posed, provided that the initial data belong to Hswith s > 1. Finally, we show that it is also valid for the two-dimensional zero diffusivity Boussinesq equations with variable viscosity in the non-divergence form.

In this paper, we prove the global existence and uniqueness of the strong solution to the high-dimensional inhomogeneous incompressible Boussinesq equations with zero thermal diffusion.

In this paper, we are concerned with the long-time asymptotic behavior of the two-dimensional temperature-dependent tropical climate model. More precisely, we obtain the sharp time-decay of the solution of the system with the general initial data belonging to an appropriate Sobolev space with negative indices. In addition, when such condition of the initial data is absent, it is shown that any spatial derivative of the positive integer \begin{document}$ k $\end{document}-order of the solution actually decays at least at the rate of \begin{document}$ (1+t)^{-\frac{k}{2}} $\end{document}.

This paper studies the Cauchy problem to the two-dimensional (2D) anisotropic magnetic Benard system with partial dissipation. The system considered in this paper has only vertical dissipation. We establish here the global regularity of classical solutions to this system. In proving the global regularity of classical solutions, the main difficulties are the absence of the horizontal dissipation and the presence of the buoyancy force term θ e 2 as well as the Rayleigh–Benard convection term u 2 . To overcome them, we establish two new Gronwall type inequalities, which may be of independent interest. In addition, the global regularity of the 2D magnetic Benard system with vertical dissipation, vertical magnetic diffusion and horizontal thermal diffusivity also holds true. This settles the global regularity issue unsolved in the previous works.

This paper examines the global regularity problem for a 2D tropical climate model with fractional dissipation. The inviscid version of this model was derived by Frierson, Majda and Pauluis for large-scale dynamics of precipitation fronts in the tropical atmosphere. The model considered here has some very special features. This nonlinear system involves interactions between a divergence-free vector field and a non-divergence-free vector field. In addition, the fractional dissipation not only models long-range interactions but also allows simultaneous investigations of a family of system. Our study leads to the global regularity of solutions when the indices of the fractional Laplacian are in two very broad ranges. In order to establish the global-in-time bounds, we introduce an efficient way to control the gradient of the non-divergence-free vector field and make sharp estimates by controlling the regularity of related quantities simultaneously.

In this note we establish the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional incompressible micropolar equations with a nonlinear damping term.

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

This paper examines the global regularity problem of the two-dimensional Oldroyd-B-type model. When the initial $$L^2$$ -energy is suitably small or the initial stress tensor is nonnegative definite, we show that the corresponding system admits a unique global regular solution.

This paper examines the global regularity problem on the two-dimensional (2D) incompressible Boussinesq equations with fractional horizontal dissipation and thermal diffusion. The goal is to establish the global existence and regularity for the Boussinesq equations with minimal dissipation and thermal diffusion. By working with this general 1D fractional Laplacian dissipation, we are no longer constrained to the standard partial dissipation and this study will help understand the issue on how much dissipation is necessary for the global regularity. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques such as integration by parts no longer apply and new tools including several anisotropic embedding and interpolation inequalities involving fractional derivatives are derived. These tools allow us to obtain very sharp upper bounds for the nonlinearities.

In this paper, we consider the Euler-Boussinesq equations in dimension two. Some new regularity criteria via the temperature are established by making use of the Littlewood-Paley technique and the new logarithmic Gronwall inequality.

Micropolar equations, modeling micropolar fluid flows, consist of coupled equations obeyed by the evolution of the velocity \begin{document}$ u$\end{document} and that of the microrotation \begin{document}$ w$\end{document} . This paper focuses on the two-dimensional micropolar equations with the fractional dissipation \begin{document}$ (-Δ)^{α} u$\end{document} and \begin{document}$ (-Δ)^{β}w$\end{document} , where \begin{document}$ 0 . The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent efforts have resolved the two borderline cases \begin{document}$ α = 1$, $β = 0$\end{document} and \begin{document}$ α = 0$\end{document} , \begin{document}$ β = 1$\end{document} . However, the situation for the general critical case \begin{document}$ α+β = 1$\end{document} with \begin{document}$ 0 is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases and different ranges of \begin{document}$ α$\end{document} and \begin{document}$ β$\end{document} require different estimates and tools. We aim at the subcritical case \begin{document}$\alpha+\beta>1$\end{document} and divide \begin{document}$\alpha\in (0,1)$\end{document} into five sub-intervals to seek the best estimates so that we can impose the minimal requirements on \begin{document}$\alpha$\end{document} and \begin{document}$\beta$\end{document} . The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators.

This paper examines the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity in the whole space. Whether or not its classical solutions global in time existenc...

In this paper, we investigate the Cauchy problem for the two-dimensional (2D) incompressible Benard equations. On the one hand, we prove global-in-time existence of smooth solutions to the 2D Benard equations with critical dissipation. On the other hand, we establish several regularity criteria involving temperature for 2D Benard equations with supercritical dissipation.

In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution.

In this paper, we consider the 2D incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. On the basis of the previous works and some new observations, we show that the condition 1−α

In this paper, we establish the global regularity of a modified version of the tropical climate model originally derived by Frierson–Majda–Pauluis (2004) [6] which couples the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature. In proving the global regularity, to bypass the difficulty caused by the very weak dissipation (any small power α > 0 ) of the barotropic mode, we utilize the “weakly nonlinear” energy estimate approach and the Maximal L t q L x p regularity for the heat kernel.

This paper addresses the Cauchy problem of the three-dimensional inhomogeneous incompressible micropolar equations. We prove the global existence and exponential decay-in-time of strong solution with vacuum over the whole space \begin{document}$ \mathbb{R}^{3} $\end{document} provided that the initial data are sufficiently small. The initial vacuum is allowed.