Geometry of Heteroclinic Cascades in Scalar Parabolic Differential Equations

We investigate the geometrical properties of the attractor for semilinear scalar parabolic PDEs on a bounded interval with Neumann boundary conditions. Using the nodal properties of the stationary solutions which are determined by an ordinary boundary value problem, we obtain crucial information about the long-time behavior for the full PDE. Especially, we prove an exact criterion for the intersection of strong-stable and strong-unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.     

$${\mathcal{A}}$$-Stability of Global Attractors of Competition Diffusion Systems

We study the structural stability of global attractors (A{\mathcal{A}}-stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally A{\mathcal{A}}-stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C ²-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.     

Topological Dynamics for Monotone Skew-Product Semiflows with Applications

Classical techniques of topological dynamics are used to prove a flow extension result for linearly stable minimal sets in monotone and differentiable skew-product semiflows. Moreover, motivated by the field of delay equations, a new version of the concept of continuous separation is introduced and studied in an abstract setting. The application of these results to the skew-product semiflows defined by almost periodic ordinary differential equations, delay equations and parabolic partial differential equations permits us to extend previous results guaranteeing the presence of almost automorphic minimal sets.     

Acoustic scattering and radiation problems, and the null-field method

The best known methods for solving the scattering and radiation problems of acoustics are integral-equation methods. However, it is also known that the simplest of these methods yield equations which are not uniquely solvable at certain discrete sets of frequencies (the irregular frequencies). In this paper, we shall analyse an alternative method (the null-field method, or T-matrix method). We prove that the infinite system of null-field equations always has precisely one solution, i.e. the unphysical irregular frequencies do not occur with this method. Moreover, we also prove that the solution of the original boundary-value problem can always be determined (at any point exterior to the scatterer) from the solution of the null-field equations. We prove these results in two dimensions, for two radiation problems (the exterior Neumann problem and the exterior Dirichlet problem) and two scattering problems (scattering by a sound-hard body and scattering by a sound-soft body); similar results can be proved in three dimensions. We also prove some subsidiary results, concerning the solvability of certain boundary integral equations and the completeness of certain sets of radiating wave-functions, and give a discussion of related numerical techniques.     

The Fredholm Alternative for Functional Differential Equations of Mixed Type

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.     

Intersection theorems concerning noncompact sets with H-convex sections and their applications

In this paper,we prove some intersection theorems concerning noncompact sets withH-convex sections which generalize the corresponding results of Ma.Fan,Tarafdar,Lassonde and Shin-Tan to H-spaces without the linear structure and to noncompact setting.An application to von Neumann type minimax theorems is given.     

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

In this paper we find topological conditions for the non existence of heteroclinic trajectories connecting saddle orbits in non singular Morse-Smale flows on S ³. We obtain the non singular Morse-Smale flows that can be decomposed as connected sum of flows and we show that these flows are those who have no heteroclinic trajectories connecting saddle orbits. Moreover, we characterize these flows in terms of links of periodic orbits.     

Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number.This work presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.     

Finite Intersection Property and Dynamical Compactness

Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800–6827, 2016). In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the \(\omega _{{\mathcal {F}}}\)-limit and the \(\omega \)-limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.     

The Longest Shortest Fence and Sharp Poincaré–Sobolev Inequalities

We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, the disc, and only the disc, maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.     

Topologically finite intersection property and minimax theorems

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Ill-posedness of the Hydrostatic Euler and Navier–Stokes Equations

We prove that the linearization of the hydrostatic Euler equations at certain parallel shear flows is ill-posed. The result also extends to the hydrostatic Navier–Stokes equations with a small viscosity.     

A note on characteristic decomposition for two-dimensional Euler system in van der Waals fluids

This paper is devoted to the generalization of a well-known result on reducible equations by Courant and Friedrichs [7] and a motivational result on compressible Euler system within the context of ideal gases by Li et al. [10]. The characteristic decomposition technique has been used to prove that any hyperbolic state, adjacent to a constant state, is simple for a pseudo-steady isentropic irrotational flow, modeled by Euler equations, in van der Waals fluids. Furthermore, this result is extended to full Euler system in self-similar coordinates provided the pseudo-flow characteristics are extending into a constant state.     

显式方法精确模拟形状记忆聚合物热力学行为

通过构建一个热耦合的多轴可压缩应变能函数,得到应力-应变、应力-温度和应变-温度之间的函数关系,建立形状记忆聚合物的本构方程.本文引入三个基于对数应变的不变量使得模型（i）可以模拟可压缩情况;（ii）适用于单轴拉伸和等双轴拉伸至少两个基准实验;（iii）多轴有效.通过显式方法（i）给出自由能和熵的具体表达,证明模型热力学定律;（ii）给出应变-应力,温度-应力以及,温度-应变的形函数具体表达.多轴模型在特定的情况下可以自动退化到各自的单轴情况. 通过调节形函数的参数,最终得到的模型结果和实验结果能够精确匹配.新方法建立的本构模型得到的结果能更加准确地指导形状记忆聚合物的工程设计。     

显式方法精确模拟形状记忆聚合物热力学行为

通过构建一个热耦合的多轴可压缩应变能函数，得到应力-应变、应力-温度和应变-温度之间的函数关系，建立形状记忆聚合物的本构方程.本文引入三个基于对数应变的不变量使得模型（i）可以模拟可压缩情况；（ii）适用于单轴拉伸和等双轴拉伸至少两个基准实验；（iii）多轴有效.通过显式方法（i）给出自由能和熵的具体表达，证明模型热力学定律；（ii）给出应变-应力，温度-应力以及，温度-应变的形函数具体表达.多轴模型在特定的情况下可以自动退化到各自的单轴情况. 通过调节形函数的参数，最终得到的模型结果和实验结果能够精确匹配.新方法建立的本构模型得到的结果能更加准确地指导形状记忆聚合物的工程设计。     

Some results on homogenization of convection-diffusion equations

We consider some models of degenerate convection-diffusion equations with oscillating coefficients. We prove that the homogenization process produces non-local and memory effects when the diffusion is longitudinal. When the diffusion is transverse we obtain a stability result. We also examine parametrized families of diffusion equations involving non-local terms.     

Symmetrization properties of parabolic equations in symmetric domains

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD R ⁿ are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.     

Pseudotrajectories generated by a discretization of a parabolic equation

A semiimplicit discretization of a parabolic equation is considered. The resulting diffepmorphism is shown to be generically Morse-Smale. Uniform bounds for the dimension of its attractor are given and numerical trajectories—including round-off errors—are shown to approximate the attractor.     

Threshold-based Quasi-static Brittle Damage Evolution

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149–189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125–152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.