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In this paper, we establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the two-dimensional Euler-Boussinesq equations with temperature-dependent thermal diffusivity. More precisely, we prove the global-in-time existence and uniqueness for this system under Yudovich-type initial data. In addition, the global propagation of striated regularity is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range. [ABSTRACT FROM AUTHOR]

This paper is concerned with a model of the two-dimensional zero thermal diffusivity magnetohydrodynamics-Boussinesq equations with the temperature-dependent viscosity and electrical conductivity. The main purpose of this paper is to establish the global regularity to this system with arbitrarily large initial data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the two-dimensional incompressible generalized magnetohydrodynamics equations. We obtain some regularity criteria to the corresponding equations in terms of the current density j or the vorticity ω. Our approach is based on a key observation on the nice structure of the incompressible magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the long-time asymptotic behavior of the generalized two-dimensional quasi-geostrophic equation. More precisely, we obtain the sharp time-decay of the smooth solution of the generalized quasi-geostrophic equation with the sub-critical dissipation and the critical dissipation for the general initial data, and the sup-critical dissipation for the small initial data in the scale invariant Besov space. In addition, the sharp time-decay of any weak solution of the generalized quasi-geostrophic equation for the general initial data is also derived. Moreover, we also derive the decay estimates for the difference between the full solution and the solution to the corresponding linear part. [ABSTRACT FROM AUTHOR]

As a continuation of the previous work [48] , in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [13] . Consequently, this result significantly improves the recent works [13,45,48] . [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical three-dimensional (3D) incompressible Navier–Stokes equations. By making use of a new anisotropic Sobolev inequality, we improve a regularity criterion for the weak solution to the 3D Navier–Stokes equations involving the gradient of one velocity component. This result improves the previous work of Zhou and Pokorný (2010). [ABSTRACT FROM AUTHOR]

We prove that the Riemann zeta function and the Euler gamma function cannot satisfy a class of algebraic differential equations with functional coefficients that are connected to the zeros of the Riemann zeta function on the critical line. [ABSTRACT FROM AUTHOR]

Abstract: Under the Generalized Riemann Hypothesis for the Dedekind zeta-function , we obtain a formula for the discriminant of the algebraic number field κ in terms of an integral of on the critical line. [Copyright &y& Elsevier]

Abstract: In this paper, we define Nevanlinna functions in annuli with two independent variables and use them to prove a version of the generalized logarithmic derivative lemma in annuli. As a corollary, we can conclude the logarithmic derivative lemma in a disk and the complex plane. [Copyright &y& Elsevier]

Abstract: We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class . We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations. [Copyright &y& Elsevier]

In this study, we consider the global regularity of the high-dimensional incompressible Oldroyd-B model in the corotational case with fractional dissipation (− Δ) α u. In particular, under the assumption that α ≥ 1 2 + n 4 , we determine the global regularity of the corresponding model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Some criteria for entire coefficients A ( z ) and B ( z ) are given in terms of the lower order forcing the solutions of f ″ + A ( z ) f ′ + B ( z ) f = 0 to grow fast. In the literature similar criteria have been published in terms of the usual order. The case when the coefficient A ( z ) has an asymptotic growth T ( r , A ) ∼ α log ⁡ M ( r , A ) , α ∈ ( 0 , 1 ) , outside of an exceptional set is also discussed. Previously, Laine–Wu (2000) and Kim–Kwon (2001) have made use of this asymptotic growth in the case α = 1 . [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

The two-dimensional (2D) incompressible Boussinesq equations model geophysical fluids and play an important role in the study of the Raleigh–Bernard convection. Mathematically this 2D system retains some key features of the 3D Navier–Stokes and Euler equations such as the vortex stretching mechanism. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation or no dissipation at all. This paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases. These results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2D Boussinesq equations. [ABSTRACT FROM AUTHOR]