Blow‐up criterion for two‐dimensional magneto‐micropolar fluid equations with partial viscosity

In this paper, we derive a blow‐up criterion of smooth solutions to the incompressible magneto‐micropolar fluid equations with partial viscosity in two space dimensions. Our proof is based on careful Hölder estimates of heat and transport equations and the standard Littlewood–Paley theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (?_yyu₁,?_xxu₂) and the partial magnetic diffusion (?_yyb₁,?_xxb₂). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.     

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.     

A uniform error estimate in time for spectral Galerkin approximations of the magneto‐micropolar fluid equations

We consider Galerkin approximations for the equations modeling the motion of an incompressible magneto‐micropolar fluid in a bounded domain. We derive an optimal uniform in time error bound in the H¹ and L² ‐norms for the velocity. This is done without explicit assumption of exponential stability for a class of solutions corresponding to decaying external force fields. Our study is done for no‐slip boundary conditions, but the results obtained are easily extended to the case of periodic boundary conditions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 689–706, 2012     

On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation

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.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation

The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by $?{(?\mathrm{\Delta })}^{\alpha }u$ and $?{(?\mathrm{\Delta })}^{\beta }b$, respectively, the resulting equations with $\alpha \ge \frac{5}{4}$ and $\alpha +\beta \ge \frac{5}{2}$ then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion $?{(?{\mathrm{\Delta }}_h)}^{\frac{5}{4}}b$, where ${\mathrm{\Delta }}_h={?}_1^2+{?}_2^2$.     

Global regularity for the 2D magneto-micropolar system with partial and fractional dissipation

This paper focuses on the 2D incompressible magneto-micropolar sysytem with the kinematic dissipation given by the fractional operator (−Δ)^α, the magnetic diffusion by the fractional operator (−Δ)^β and the spin dissipation by the fractional operator (−Δ)^γ. α,β, and γ are nonnegative constants. We proved that this system with any α+β=2,1 ≤ α ≤ 2,γ=0, and α+γ ≥ 1,β=1 always possesses a unique global smooth solution $(\mathbf{u},\mathbf{b},\mathrm{w})\in H^s({\mathbb{R}}^2)(s>2)$ if the initial data is sufficiently smooth. In addition, we also obtained the global regularity results for several partial dissipation cases.     

Global smooth solution to the 2D Boussinesq equations with fractional dissipation

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 with suffices in order for the solution pair of velocity and temperature to remain smooth for all time. Copyright © 2017 John Wiley & Sons, Ltd.     

The D incompressible Navier–Stokes equations with partial hyperdissipation

The three‐dimensional incompressible Navier–Stokes equations with the hyperdissipation always possess global smooth solutions when . Tao [6] and Barbato, Morandin and Romito [1] made logarithmic reductions in the dissipation and still obtained the global regularity. This paper makes a different type of reduction in the dissipation and proves the global existence and uniqueness in the H¹‐functional setting.     

Global regularity of the 2D magnetohydrodynamic equations with fractional anisotropic dissipation

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.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

In this paper, we prove the existence and uniqueness of a global solution for 2‐D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor. Copyright © 2006 John Wiley & Sons, Ltd.     

On the global well‐posedness for the two‐dimensional Boussinesq equations with horizontal dissipation

In this paper, we study the global well‐posedness for the two‐dimensional nonlinear Boussinesq equations with horizontal dissipation. The method we adopted is the smoothing effect in horizontal direction and the low‐high decomposition technique.     

On the global regularity of the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the two‐dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely with . Therefore, this result significantly improves the previous work 31 which obtained the global regularity result for with , where is an explicit function.     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations

In this paper, we consider the 2D micropolar fluid equations in the whole space . We prove the global wellposedness of the system with rough initial data and show the vanishing microrotation viscosity limit in the case of zero kinematic viscosity or zero angular viscosity. Copyright © 2011 John Wiley & Sons, Ltd.     

Ladder estimates for micropolar fluid equations and regularity of global attractor

This paper is devoted to obtain ladder inequalities for 2D micropolar fluid equations on a periodic domain Q=(0, L)². The ladder inequalities are differential inequalities that connect the evolution of L² norms of derivatives of order N with the evolution of the L² norms of derivatives of other (usually lower) order. Moreover, we find (with slight assumption on external fields) long‐time upper bounds on the L² norms of derivatives of every order, which implies that a global attractor is made up from C^∞ functions. Copyright © 2010 John Wiley & Sons, Ltd.