Global strong solutions for 3D viscous incompressible heat conducting Navier‐Stokes flows with the general external force

In this paper, we consider the initial boundary value problem for the nonhomogeneous heat–conducting fluids with non‐negative density and the general external force. We prove that there exists a unique global strong solution to the 3D viscous nonhomogeneous heat–conducting Navier‐Stokes flows if is suitably small.     

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices keeping the subspace invariant, we compute miniversal deformations of a given pair when it is observable, and the subspace is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs, and study the effect of each deformation parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

Expression and behavior of the solutions of some rational recursive sequences

We obtain in this paper the expression of the solutions of the following recursive sequences: where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solution of these equations. Copyright © 2016 John Wiley & Sons, Ltd.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]?multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b . Copyright © 2015 John Wiley & Sons, Ltd.     

Global stability for the fractional Navier‐Stokes equations in the Fourier‐Herz space

We consider global stability for the fractional incompressible Navier‐Stokes equations in a 3‐D critical Fourier‐Herz space. By introducing a weighted norm space and using Fourier localization technique, the stability of mild solutions with small initial perturbation is established. With the Friedrichs method, the stability of weak solutions is proved under arbitrary large initial perturbation.     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 John Wiley & Sons, Ltd.     

Semistability of two systems of difference equations using centre manifold theory

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: and where a, b, c and d are positive constants and the initial conditions x₀ and y₀ are positive numbers. We study the stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. Copyright © 2016 John Wiley & Sons, Ltd.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear wave equations of carrier type on manifolds

In this paper, we investigate the existence and uniqueness of a solution for differential equations of the carrier type on lateral boundary Σ of the cylinder Q, cf. (1). The main point is to transform this initial value problem into a differential operator equation of the type , cf. (6). The operator is defined in Section 2, and it acts in Sobolev spaces on Γ, boundary of Ω. The initial value problem (6) is investigated in Section 3 by the method of Faedo–Galerkin. Thus, we obtain the existence of a weak solution for (6), and in Section 4, we prove its uniqueness. Copyright © 2012 John Wiley & Sons, Ltd.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces

In recent paper, we prove the well‐posedness for the heat flow of harmonic maps with initial data and for the hydrodynamic flow of nematic liquid crystals with initial data . Copyright © 2011 John Wiley & Sons, Ltd.     

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

Forced oscillation of sublinear impulsive differential equations via nonprincipal solution

In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form where γ∈(0,1), under the assumption that associated homogenous linear equation is nonoscillatory.     

Nonstationary Stokes system in Besov spaces

We examine the solvability in Besov spaces of an initial–boundary value problem for the nonstationary Stokes system with the slip boundary conditions. We prove the existence and uniqueness of solutions to the problem in a bounded domain . The existence is shown by localizing the system to interior and boundary subdomains of Ω. The localized Stokes system is transformed by the Helmholtz–Weyl decomposition to the heat and the Poisson equations, which are solved in the Besov spaces. Next, by the properties of the partition of unity and a perturbation argument, the existence is proved in domain Ω. Copyright © 2013 John Wiley & Sons, Ltd.     

Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient

The low Mach number limit for classical solutions of the compressible magnetohydrodynamic equations without thermal conductivity is, here, studied. A uniform existence result for the Cauchy problem in is proved under the assumption that the initial data are uniformly bounded with respect to the Mach number in and are well‐prepared in . Copyright © 2011 John Wiley & Sons, Ltd.     

Exponential stability for a plate equation with p‐Laplacian and memory terms

This paper is concerned with the energy decay for a class of plate equations with memory and lower order perturbation of p‐Laplacian type, with simply supported boundary condition, where Ω is a bounded domain of , g > 0 is a memory kernel that decays exponentially and f(u) is a nonlinear perturbation. This kind of problem without the memory term models elastoplastic flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

In this paper, we numerically study the water wave model with a nonlocal viscous term where is the Riemann‐Liouville half‐order derivative in time. We propose and compare different numerical schemes based on the diffusive realizations of fractional operators.