Regularity criteria for the three dimensional Ericksen–Leslie system in homogeneous Besov spaces

In this paper, we establish some regularity criteria involving homogeneous Besov spaces for both the simplified and the general three dimensional Ericksen–Leslie system. This improves many previous results, and can be viewed as the ultimate optimal regularity criterion in the Besov space framework.     

Global well-posedness of the incompressible fractional Navier–Stokes equations in Fourier–Besov spaces with variable exponents

We study the Cauchy problem of the fractional Navier–Stokes equations in critical variable exponent Fourier–Besov spaces $F{\stackrel{?}{B}}_{p\left(?\right),q}^{4?2\alpha ?\frac{3}{p\left(?\right)}}$. We discuss some properties of variable exponent Fourier–Besov spaces and prove a general global well-posedness result which covers some recent works about classical Navier–Stokes equations.     

Besov空间下迭代全变差正则化的图像恢复模型

在Besov空间下,提出了一种用于图像恢复领域的迭代全变差正则化模型。通过使用一个加权的参数序列,给出了一个迭代正则化的变分问题,这个变分问题实际上是一个小波软硬阈值结合的迭代程序。给出了新模型的停止标准和一些好的性质,如单调性和收敛性等。数值实验表明与传统去噪方法相比,新方法不仅能较好地恢复图像,而且收敛速度较快。     

An improved regularity criterion for the 3D Hall–MHD equations via the vorticity

In this paper, we consider the three-dimensional incompressible Hall–magnetohydrodynamic equations, and establish an improved regularity criterion of local in time classical solutions involving only the vorticity field. Consequently, this improves the previous result.     

Blow-up time estimates and simultaneous blow-up of solutions in nonlinear diffusion problems

This paper deals with some homogeneous Dirichlet problems of nonlinear diffusion equations. After demonstrating the existence and uniqueness of weak solutions, we prove the existence and non-existence of global solutions. The influence of coefficients and geometry of domain is shown clearly on the existence of global solutions. We give the complete classification of simultaneous blow-up of solutions with blow-up rates as well in one-dimensional space. Moreover, the bounds of blow-up time are studied for all dimensions of space domain.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

A numerical simulation for the blow-up of semi-linear diffusion equations

Many mathematical models have the property of developing singularities at a finite time; in particular, the solution u(x, t) of the semi-linear parabolic Equation (1) may blow up at a finite time T. In this paper, we consider the numerical solution with blow-up. We discretize the space variables with a spectral method and the discrete method used to advance in time is an exponential time differencing scheme. This numerical simulation confirms the theoretical results of Herrero and Velzquez [M.A. Herrero and J.J.L. Velzquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), pp. 131–189.] in the one-dimensional problem. Later, we use this method as an experimental approach to describe the various possible asymptotic behaviours with two-space variables.     

Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

Optimal control problem for stochastic evolution equations in Hilbert spaces

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.     

A logarithmically improved regularity criterion for the surface quasi-geostrophic equation

In this paper, we consider the two-dimensional dissipative surface quasi-geostrophic equation, and establish a logarithmically improved regularity criterion. Consequently, our result extends the regularity criterion result of Dong and Pavlovié (2009). In addition, a logarithmically improved regularity criterion to the three-dimensional Navier–Stokes equations is also derived by the same arguments. Therefore, this result extends and improves many previous works.     

On the triangle inequalities in fuzzy metric spaces

This paper presents level forms of the triangle inequalities in fuzzy metric spaces (X, d, L, R). To aid discussion, a fuzzy pre-metric condition is introduced. It is first pointed out that under the fuzzy pre-metric condition the first triangle inequality is always equivalent to its level form. The second triangle inequality is equivalent to one level form when R is right continuous, and to another level form also when further conditions are imposed on R. In a fuzzy metric space, the level form of the first triangle inequality and one of the level forms of the second triangle inequality are always valid. The other level form of the second triangle inequality holds for all but at most countable α ∈ [0, 1). Finally, a fixed point theorem for fuzzy metric spaces is derived as an application of the preceding results.     

On the collective sort problem for distributed tuple spaces

In systems coordinated with a distributed set of tuple spaces, it is crucial to assist agents in retrieving the tuples they are interested in. This can be achieved by sorting techniques that group similar tuples together in the same tuple space, so that the position of a tuple can be inferred by similarity. Accordingly, we formulate the collective sort problem for distributed tuple spaces, where a set of agents is in charge of moving tuples up to a complete sort has been reached, namely, each of the N tuple spaces aggregate tuples belonging to one of the N kinds available. After pointing out the requirements for effectively tackling this problem, we propose a self-organizing solution resembling brood sorting performed by ants. This is based on simple agents that perform partial observations and accordingly take decisions on tuple movement. Convergence is addressed by a fully adaptive method for simulated annealing, based on noise tuples inserted and removed by agents on a need basis so as to avoid sub-optimal sorting. Emergence of sorting properties and scalability are evaluated through stochastic simulations.     

Blind Wavelet Compression of the Solution of a Nonlinear PDE with Singular Forcing Term Within Optimal Order Cost: Stability of Restricted Approximation to Small Errors

Level lifting of the wavelet expansion is related to an interpolation result for Sobolev H ^s spaces; the nonlinear N-term approximation is linked with a nonlinear interpolation result for Sobolev spaces W ^s,p noncompactly included into L ². Cohen et al. (2000, Constr. Approx. 16 (1), 85–113) introduced the intermediate notion of restricted approximation. Based on this, we construct an optimal order resolution algorithm extending beyond the linear elliptic case of Cohen et al. (2001, Math. Comput. 70 (233), 27–75), as we illustrate numerically. We undeline that optimal order adaptivity implies the blind compression of the unknown of the PDE. We illustrate on a univariate version of the bivariate PDE $\Delta u + e^{cu} = 0Level lifting of the wavelet expansion is related to an interpolation result for Sobolev H ^s spaces; the nonlinear N-term approximation is linked with a nonlinear interpolation result for Sobolev spaces W ^s,p noncompactly included into L ². Cohen et al. (2000, Constr. Approx. 16 (1), 85–113) introduced the intermediate notion of restricted approximation. Based on this, we construct an optimal order resolution algorithm extending beyond the linear elliptic case of Cohen et al. (2001, Math. Comput. 70 (233), 27–75), as we illustrate numerically. We undeline that optimal order adaptivity implies the blind compression of the unknown of the PDE. We illustrate on a univariate version of the bivariate PDE , c > 0, used to benchmark three nonadaptive multilevel methods Hackbusch (1992, Z. Angew. Math. Mech. 72 (2), 148–151). The adaptiveness of our algorithm is highlighted by the addition in this illustration of a singular forcing term. This term is an element of H ⁻¹ but does not belong to H ^−5/6: more precisely, it is the second derivative of . This illustration passed the numerical implementation test flops). The algorithm’s convergence and cost where ɛ is the final error in H ¹ norm) in both univariate (d=1) and bivariate (d=2) general cases is shown to have optimal order, with s less than three. ? John Wiley and Sons, Inc.     

On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces

The objective of this paper is to establish the existence of solutions of nonlinear impulsive fractional integrodifferential equations of Sobolev type with nonlocal condition. The results are obtained by using fractional calculus and fixed point techniques.     

On null controllability of linear systems in Banach spaces

Our aim here is to give characterizations for the null controllability of linear systems in general Banach spaces. The starting point of this paper is the work of Pengnian Chen and Huashu Qin (Systems Control Lett. 45 (2002) 155), which solves the problem of exact controllability. In fact, the results of (Systems Control Lett. 45 (2002) 155) say that, in case of setting in general Banach spaces, there are the same characterizations of exact controllability as in the case of reflexive Banach spaces. An open problem raised in (Systems Control Lett. 45 (2002) 155) is the validity of a similar result for null controllability. We give here a positive answer to this problem. However, it seems to us that the technique of (Systems Control Lett. 45 (2002) 155) does not work for null controllability, and, consequently, our approach is completely different.     

Stochastic differential equations driven by a Wiener process and fractional Brownian motion: Convergence in Besov space with respect to a parameter

A stochastic differential equation involving both a Wiener process and fractional Brownian motion, with nonhomogeneous coefficients and random initial condition, is considered. The coefficients and initial condition depend on a parameter. The assumptions on the coefficients and the initial condition supplying continuous dependence of the solution on a parameter, with respect to the Besov space norm, are established.     

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.