Pressure regularity criteria of the three‐dimensional micropolar fluid flows

In the study of the regularity criteria of weak solutions of the three‐dimensional (3D) micropolar fluid flows, the regularity of solutions are examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, Multiplier space, BMO space and Besov space, respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L²‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.     

Weak solution to a one‐dimensional full compressible non‐Newtonian fluid

This paper is devoted to the existence of global‐in‐time weak solutions to a one‐dimensional full compressible non‐Newtonian fluid. A semi‐discrete finite element scheme is taken to generate approximate solutions, based on an exact projection technique. To enforce convergence of the approximate solutions, the uniform estimate is obtained using an iteration method and energy method, with the help of the weak compactness and convexity. Numerical simulations showing the existence of solutions are presented.     

Exponential stability in one‐dimensional non‐linear thermoelasticity with second sound

In this paper, we consider a one‐dimensional non‐linear system of thermoelasticity with second sound. We establish an exponential decay result for solutions with small ‘enough’ initial data. This work extends the result of Racke (Math. Methods Appl. Sci. 2002; 25 :409–441) to a more general situation. Copyright © 2004 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.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

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.     

The initial value problem for a multi‐dimensional radiation hydrodynamics model with viscosity

In this paper, we study the existence and time‐asymptotic behavior of solutions to the Cauchy problem for the equations of radiation hydrodynamics with viscosity in ?³. The global existence of the solutions is obtained by using the energy method. With more elaborate energy estimates, we also give some decay rates of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Local free boundary problem for viscous compressible magnetohydrodynamics

We consider the motion of viscous compressible magnetohydrodynamics fluid in a domain bounded by a free surface. In the external domain, there is electromagnetic field generated by some currents that keeps the magnetohydrodynamics flow in the bounded domain. Then on the free surface, transmission conditions for electromagnetic fields are imposed. In this paper, we prove the existence of local regular solutions by the method of successive approximations. The L₂ approach is used. This helps us to treat the transmission conditions.     

The 2D magneto‐micropolar equations with partial dissipation

We study the global existence and regularity of classical solutions to the 2D incompressible magneto‐micropolar equations with partial dissipation. We establish the global regularity for one partial dissipation case. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.