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In this paper, we establish a new blow-up criterion for the two-component Dullin–Gottwald–Holm system. We show that strong solutions must blowup in finite time if the integral concerning initial momentum changes sign at some point $$x_0\in {\mathbb {R}}$$.

We derive various anisotropic versions of the Troisi inequality for the study of the conditional regularity of solutions to the Navier–Stokes equations. We impose additional conditions on one or two non-diagonal entries of the velocity gradient, namely or and . Formulating our results in the frame of the anisotropic Lebesgue spaces we improve two recent results from the literature. Further we present two new criteria.

In this paper, we focus on the large time behavior of compact support of the potential for a Camassa–Holm-type equation with nonlinearities of degree $$k+1$$ if the compactly supported initial potential keeps its sign. Moreover, persistence property in weighted Sobolev spaces is also investigated.

We prove, among others, the following regularity criterion for the solutions to the Navier–Stokes equations: If u is a global weak solution satisfying the energy inequality and ω = ∇ × u , then u is regular on ( 0 , T ) , T > 0 , if two components of ω belong to the space L q ( 0 , T ; B ˙ ∞ , ∞ − 3 / p ) for p ∈ ( 3 , ∞ ) and 2 / q + 3 / p = 2 . This result is an improvement of the results presented by Chae and Choe (1999) [7] or Zhang and Chen (2005) [38] . Our method of the proof uses a suitable application of the Bony decomposition and can also be used for the proofs of some other kin criteria. One such example is presented in Appendix.

In this paper we establish a new regularity criterion for the 3D incompressible MHD equations via ∂ 3 u 3 and the magnetic field b . By considering different weights in spatial variables, we show in anisotropic Lebesgue spaces if ∂ 3 u 3 and b satisfy certain space–time integrable conditions, which are almost optimal from the scaling invariant point of view, then a weak solution ( u , b ) is actually regular. This result gives new insights into the understanding of regularity theory of weak solutions.

We discuss an integrable Camassa–Holm type equation with cubic nonlinearity. Asymptotic profile has been shown in the sense that strong solutions arising from algebraic decaying initial data will keep this property at infinity in the spatial variable in its lifespan. Moreover, for a global solution, measure of potential support is presented.

We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an exterior domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.

We give asymptotic description of strong solutions in its lifespan with compactly supported initial momentum and investigate the persistence property in weighted space and blow-up phenomena for a generalized Novikov equation.

In this paper, the authors first prove an upper bound of fractal dimension of kernel sections for the discrete Zakharov equations for plasmas with a quantum correction. Then they establish the upper semicontinuity of the kernel sections when the infinite lattice system is approximated by finite ordinary differential equations.

In this work, we will establish a sufficient condition for the regularity criterion to the 3D MHD equations in terms of one directional derivative of the pressure. More precisely, we prove that if ? π ? x 3 ( x , t ) ? L 2 1 - r ( 0 , T ; M ? 2 , 3 r ( R 3 ) ) ???for? 0 < r < 1 , then the solution ( u , b ) is regular. We obtain some improvement of the related results in Berselli and Galdi (2002), Cao and Wu (2010), Jia and Zhou (2012), Lin and Du (2013) and Yamazaki (2013).

It is shown that a strong solution of the Degasperis–Procesi equation, initially decaying exponentially together with its spatial derivative, must be identically equal to zero if it also decays exponentially at a later time. The decay rate of the corresponding strong solution at infinity is also given for some kinds of initial data with exponential decay.

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields.

We discuss a new integrable two-component Camassa-Holm equation which describes the motion of fluid. This paper is concerned with the wave breaking mechanism for the Cauchy problem with periodic condition where two special classes of initial data are involved. Moreover, the estimate of momentum support is also shown.

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here, we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.

We study the regularity criteria for the incompressible Navier–Stokes equations in the whole space R 3 based on one velocity component, namely u 3 , ∇ u 3 and ∇ 2 u 3 . We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇ u 3 ∈ L β ( 0 , T ; L p ) , where 2 / β + 3 / p = 7 / 4 + 1 / ( 2 p ) and p ∈ ( 2 , ∞ ] , yields the regularity of u on ( 0 , T ] .

We consider the regularity criterion for the incompressible Navier–Stokes equations. We show that the weak solution is regular, provided [Formula: see text] for some T > 0, where Ẋris the multiplier space. This extends a result of Kukavica and Ziane [14].

We establish some regularity criteria for the incompressible Navier-Stokes equations in a bounded three-dimensional domain concerning the quotients of the pressure, the velocity field and the pressure gradient.

We consider the asymptotic behavior of solutions for a modified two-component Camassa–Holm (MCH2) system which arises in shallow water theory. It is proved that the corresponding solution to initial data with algebraic decay at infinity will retain this property in its lifespan.

In this paper, we establish sufficient conditions on the initial data to guarantee blow-up phenomenon for the modified two-component Camassa–Holm (MCH2) system.

In this paper, we consider the regularity problem under the critical condition to the Newton-Boussinesq equations. The Serrin type regularity criteria are established in terms of the critical Morrey-Campanato spaces and Besov spaces.

We discuss the two-component Camassa–Holm system which arises in shallow water theory. This paper is concerned with the asymptotic description of solutions in the following sense: the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan.

In this paper, we consider a modified two-component Camassa–Holm (MCH2) system which arises in shallow water theory. We analyze the wave breaking mechanism by establishing some new blow-up criteria for this system formulated either on the line or with space-periodic initial condition.

In this paper, we consider the periodic weakly dissipative Dullin–Gottwald–Holm equation. The present work is mainly concerned with blow-up phenomena for the Cauchy problem for this new kind of equation. We apply the optimal constant to give sufficient conditions via an appropriate integral form of the initial data, which guarantee the finite-time singularity formation for the corresponding solution.

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if $\mathrm{div}( \frac{u}{|u|}) $ belongs to $L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) $ with 0?r?1, then the weak solution actually is regular and unique.

We consider a two-component Camassa–Holm system which arises in shallow water theory. We analyze a wave breaking mechanism and the global existence of solutions. First, we discuss the local well posedness and a blow up mechanism, then establish some new blow up criteria for this system formulated either on the line or with space-periodic initial conditions. Finally, the existence of global solutions is analyzed.

This paper is concerned with some aspects of blow up of solutions and persistence properties. Firstly, we will try to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Then a particular class of initial data for the periodic case is also considered in this paper. Finally, we investigate the persistence properties of the strong solutions.

This work establishes a sufficient condition for the regularity criterion of the Boussinesq equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure $${\partial _{3}\pi }$$ satisfies the logarithmical Serrin-type condition $$\int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert_{L^{\lambda }}^{q}}{1+\ln (1+\left\Vert \theta \right\Vert_{L^{4}})} {d}s < \infty \quad \text{with}\quad\frac{2}{q}+\frac{3}{\lambda }=\frac{7}{4}\quad \text{and}\quad\frac{12}{7} < \lambda \leq \infty,$$ then the solution $${(u,\theta )}$$ remains smooth on $${\left[0,T\right]}$$ . Compared to the Navier–Stokes result, there is a logarithmic correction involving $${\theta}$$ in the denominator.

In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012) [6] .

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

This paper is concerned with the wave breaking phenomenon for a modified two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data to guarantee blow-up of solutions in finite time.

We consider a modified two-component rod system, which is a generalization of the two-component Camassa-Holm system to allow a dependence on the average density as well as on the pointwise density. Sufficient conditions on the initial data to guarantee finite time singularities for strong solutions are established on the real line and the periodic case.

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. We give a detailed description of the asymptotic behavior on the fluid flow in a half-space using as a starting point the theory of existence of solutions which is obtained by interpreting the coordinate perpendicular to the wall as time variable.

In this paper we establish some new logarithmically improved regularity criteria for the Navier–Stokes equations in the homogeneous Besov space B∞,∞−1.

In this paper, we consider the weakly dissipative Camassa–Holm equation. First, we try to improve other authors’ results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa–Holm equation.

In this paper, we consider the weakly dissipative Degasperis–Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis–Procesi equation are also investigated.

We will discuss a new integrable model which describes the motion of fluid. The present work is mainly concerned with global existence and blow-up phenomena which are largely due to the application of conservation laws for this integrable equations. Moreover, a new blow-up criterion for nonperiodic case is also established via the associated potential. Some interesting examples are also given to illustrate the application of our results. The precise blow-up rate is also investigated. Finally, we will emphasize the relations of classical Camassa–Holm equation and our model by analyzing the existence of global solutions.

We consider the regularity problem for 3D Navier–Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇ p / ( 1 + | u | δ 1 + | ∇ u | δ 2 ) for the Navier–Stokes equations is established.