A NOTE ON GLOBAL WELL-POSEDNESS OF SOLUTIONS TO BOUSSINESQ EQUATIONS WITH FRACTIONAL DISSIPATION

The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.

GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY

This paper studies the global regularity of 2D incompressible anisotropic magnetomicropolar fluid equations with partial viscosity. Ma [22](Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95–129) examined the global regularity of the 2D incompressible magnetomicropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22]and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

GLOBAL REGULARITY FOR MODIFIED CRITICAL DISSIPATIVE QUASI-GEOSTROPHIC EQUATIONS

We consider the n-dimensional modified quasi-geostrophic（SQG） equations δtθ ＋ u·△↓θ＋kΛ^αθ=0, u = Λ^α-1R^⊥θ with κ 〉 0, α∈（0, 1] and θ0∈ W^1,∞（R^n）. In this paper, we establish a different proof for the global regularity of this system. The original proof was given by Constantin, Iyer, and Wu, who employed the approach of Besov space techniques to study the global existence and regularity of strong solutions to modified critical SQG equations for two dimensional case.The proof provided in this paper is based on the nonlinear maximum principle as well as the approach in Constantin and Vicol.

On C^(1 ,α) Regularity for Monge- Ampre Equation

In this paper, we establish global C 1,1/4 estimates for the Dirichlet problem for the Monge Ampère equation, which yield the corresponding existence and regularity results. Our conditions apply to both degenerate and nondegenerate cases.

Geometric Constraints for Global Regularity of 3D Shear Thickening Fluids

We consider the relation between the direction of the vorticity and the global regularity of 3D shear thickening fluids.It is showed that a weak solution to the non-Newtonian incompressible fluid in the whole space is strong if the direction of the vorticity is (11-5p)/2-Holder continuous with respect to the space variables when 2<P<11/5.

Global strong solution of 3D tropical climate model with damping

We establish the global well-posedness of a strong solution to the 3D tropical climate model with damping.We prove that there exists the global and unique solution forα,β,γsatisfying one of the following three conditions:(1)α,β≥4;(2)7/2≤α<4,β≥(5α+7)/(2α),γ≥7/(2α−5);(3)3<α≤7/2,β,γ≥7/(2α−5).

Global Spiral Patterns in Galaxies : A Review of the Density Wave Theory from the Theoretical Perspective

The density wave theory of galactic spirals is a semi empirical theory consisting of the analysis of physically plausible mathematical models, the synthesis of extensive observational data, and an examination of the applicability of the theoretical predictions of particular models for the interpretation of observational data. In this paper, the theory is reviewed from the general perception of order and chaos in dynamical systems. Prominent among the observational data is the well established morphological classification of galaxies according to Edwin Hubble, the luminosity class of Sidney van den Bergh, and the coexistence of regular and irregular global structures in many galaxies, as was first discovered in the whirlpool galaxy M51 by Fritz Zwicky. The regular structure is prominently observed in red and in infrared, and the irregular in blue and at 21 cm radio frequency. These and other data led to the concept of the existence of an essentially standing wave pattern in the older evolved stars, which in turn acts as a backbone for the global structure of the galaxy as a whole. Dynamically, the existence of irregular structures in the interstellar medium reflects the natural state of turbulent motion in gaseous motion at high speeds. In contrast, the essentially collisionless stellar system has a substantial 'microscale' of epicyclic motion that prevents, or smoothes out, irregularities on scales. It is pointed out that the spiral structure in galaxies is but one class of interesting phenomena that crucially relates to universal gravitation. Astronomical phenomena on all scales, from planets and satellites to protostars, stars and galaxies and the whole universe present other challenges to both theorists and observers, representing a wide field of opportunities for fruitful research. At the present time, they are being successfully met in individual cases through the development of new observational techniques to cover all frequencies of electromagnetic radiation and through the development of new analytical and computational methods.

Optimal global regularity for minimal graphs over convex domains in hyperbolic space

We use the concept of the inside-(a,η,h)domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space.As an application,we prove that the Hölder exponent max{1/a,1/(n+1)}for the problem is optimal for any a∈[2,+∞].

Global Regularity for a 2D Model of Electro-Kinetic Fluid in a Bounded Domain

In this paper, we consider an initial boundary value problem for a 2D model of electro-kinetic fluid in a smooth and bounded domain. We prove that the model has a unique global-in-time smooth solution.

Global Regularity of 2D Leray-Alpha Regularized Tropical Climate Models

In this paper,we establish the global regularity of 2D leray-alpha regularized tropical climate models.The global strong solution to the system with a half Laplacian of the first baroclinic model of velocity(Av)and thermal diffusion(△θ)or with only the dissipation of the barotropic mode(-Au)are obtained.

Regularity of the Attractor for 3-D Complex Ginzburg-Landau Equation

In this paper, the existence of global attractor for 3-D complex Ginzburg Landau equation is considered. By a decomposition of solution operator, it is shown that the global attractor .Ai in Hi（Ω） is actually equal to a global attractor Aj in HJ （Ω） （i ≠j, i, j = 1, 2, .. m）.

Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities

In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is shown that the global attractor in E0 is actually equal to a global attractor in E1.