On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.     

On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier-Stokes equations (n?3), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping

In this paper, we study a quasilinear hyperbolic equation with strong damping. Firstly, by use of the successive approximation method and a series of classical estimates, we prove the local existence and uniqueness of a weak solution. Secondly, via some inequalities, the potential method and the concave method, we derive the asymptotic and blow-up behavior of the weak solution with different conditions.     

Gevrey class regularity for quasi-linear sub-elliptic equations of second order

In this work we study the Gevrey regularity of solutions to a general class of second order quasi-linear equations. Under some kind of sub-ellipticity conditions, we obtain the Gevrey regularity of weak solutions to these equations.     

Regularity criteria for the 3D generalized MHD equations in terms of vorticity

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

Remarks on regularity criteria for the weak solutions of liquid crystals

In this paper, we consider the regularity criteria for weak solutions of liquid crystals. It is proved that the solution is in fact smooth if the velocity or the velocity gradient belongs to some critical multiplier spaces or Tribel–Lizorkin spaces. As a corollary, we obtain the Beal–Kato–Majda criteria for liquid crystals.     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nehari manifold and by exploring some properties of the best Hardy-Sobolev constant together with an approach developed by Brezis and Nirenberg.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces

In this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz–Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity.     

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

An Extended Beam Theory for Smart Materials Applications Part II: Static Formation Problems

In this paper we further develop the theory of the extended Timoshenko beam model, as first introduced in Part I [5] of this work, with particular emphasis on applications of the model in formation theory [11], [12]. We begin with formal development of the equilibrium equations of static formation theory in the context of the extended Timoshenko model, giving a rigorous discussion of existence, uniqueness, and regularity of weak solutions with appropriate assumptions on the coefficients. We continue to obtain the fundamental duality relationship in the context of weak solutions and indicate its usefulness in investigations of approximate formability. Optimal formation problems and corresponding necessary conditions for optimality are discussed. We conclude with a discussion of a particular problem of joint optimization of controls and actuator densities in the context of a prismatic extended Timoshenko beam and we present the results of some computational studies. Accepted 14 November 1996