Table of contents

In this survey we recall basic notions of disintegration of measures and entropy along unstable laminations. We review some roles of unstable entropy in smooth ergodic theory including the so-called invariance principle, Margulis construction of measures of maximal entropy, physical measures and rigidity. We also give some new examples and pose some open problems.

We adapt Sklyanin's K-matrix formalism to the sinh–Gordon equation, and prove that all free boundary constant mean curvature annuli in the unit ball in ${\mathbb{R}}^{3}$ are of finite type.

We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

Let G be an infinite discrete countable amenable group acting continuously on two compact metrizable spaces X, Y. Assume that φ : (Y, G) → (X, G) is a factor map. Using finite open covers, the conditional topological entropy of φ is defined. The conditional measure-theoretic entropy of φ equals the conditional measure-theoretic entropy of Y to X. With the aid of tiling system of G, the conditional variational principle of φ is studied when (X, G) is an asymptotically h-expansive system. If X = Y and φ is the identity map, the conditional topological entropy of system (X, G) is defined. In the Cartesian square (X × X, G), we define the conditional measure-theoretic entropy of (X, G) to be the defect of the upper semi-continuity of the conditional measure-theoretic entropy of X × X to the first axis. Then the conditional variational principle of (X, G) is obtained.

In this paper, we concentrate on the Lie symmetry structure of a system of multi-dimensional time-fractional partial differential equations (PDEs). Specifically, we first give an explicit prolongation formula involving Riemann–Liouville time-fractional derivative for the Lie infinitesimal generator in multi-dimensional case, and then show that the infinitesimal generator has an elegant structure. Furthermore, we present two simple conditions to determine the infinitesimal generators where one is a system of linear time-fractional PDEs, the other is a system of integer-order PDEs and plays the dominant role in finding the infinitesimal generators. We study three time-fractional PDEs to illustrate the efficiencies of the results.

We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem in Sobolev spaces from two different points of view. In the first one, we study the pure Cauchy problem and establish local well-posedness in ${H}^{s}\left({\mathbb{R}}^{2}\right)$, s > 1/2. In the second one, we study the Cauchy problem on the background of a Korteweg–de Vries solitary traveling wave in a less regular space. To obtain our results we make use of the smoothing properties of solutions for the linear problem corresponding to the Zakharov–Kuznetsov equation for the latter problem. For the former problem we use bilinear estimates in Fourier restriction spaces introduced in [24].

Let A, B be matrices in ${\mathrm{S}\mathrm{L}}_{2}\mathbb{R}$ having trace greater than or equal to 2. Assume the pair A, B is coherently oriented, that is, can be conjugated to a pair having nonnegative entries. Assume also that either A, B⁻¹ is coherently oriented as well, or A, B have integer entries. Then the Lagarias–Wang finiteness conjecture holds for the set {A, B}, with optimal product in {A, B, AB, A²B, AB²}. In particular, it holds for every pair of 2 × 2 matrices with nonnegative integer entries and determinant 1.

In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov–Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in ${H}^{2}({\mathbb{R}}^{2})$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in Dohnal and Dörfler, [2013 Multiscale Model. Simul. 11 162–191].

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing with the qualitative analysis of this problem with unbalanced growth. Using the frozen variable method and eventually a fixed point theorem, the main result of this paper establishes that the problem has a positive smooth solution.

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically linked star subnetworks of identical oscillators with shear and Kuramoto–Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order ɛ, the chimera states persist on time scales at least of order 1/ɛ in general, and on time-scales at least of order 1/ɛ² if the intra-star coupling is of Kuramoto–Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a Möbius symmetry group and techniques from averaging theory and normal hyperbolicity.

The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on ${\mathbb{R}}^{d}$ (d ⩾ 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.

In this paper we address two questions about the synchronisation of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronisation for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm.

In the present paper, we are interested in the nonlocal dispersal logistic equations in heterogeneous environment. This leads to study the spectrum theory and asymptotic behavior of nonlocal dispersal problems. We establish the existence and uniqueness of positive stationary solutions. By employing continuous lemma and nonlocal estimates, we obtain the limiting behavior of positive solutions when the dispersal rate is small or large. We also analyze the effect of spatial heterogeneity on the long time behavior of evolution equations.

This paper focuses on a system of three-dimensional (3D) Boussinesq equations modeling anisotropic buoyancy-driven fluids. The goal here is to solve the stability and large-time behavior problem on perturbations near the hydrostatic balance, a prominent equilibrium in fluid dynamics, atmospherics and astrophysics. Due to the lack of the vertical kinematic dissipation and the horizontal thermal diffusion, this stability problem is difficult. When the spatial domain is ${\Omega}={\mathbb{R}}^{2}{\times}\mathbb{T}$ with $\mathbb{T}=[-1/2,1/2]$ being a 1D periodic box, this paper establishes the desired stability for fluids with certain symmetries. The approach here is to distinguish the vertical averages of the velocity and temperature from their corresponding oscillation parts. In addition, the oscillation parts are shown to decay exponentially to zero in time.

In this paper, we study the existence of solutions for a diagonal hyperbolic system, that is not necessarily strictly hyperbolic, in one space dimension, considering discontinuous BV initial data without any restrictions on the size of its norm. This system appears naturally in various physical domains, particularly in isentropic gas dynamics and dislocation dynamics in materials. In the case of strictly hyperbolic systems, an existence and uniqueness of a discontinuous solution result is available for BV initial data with small norm, whereas several existence and uniqueness results have been presented for non-decreasing continuous solutions. In the present paper, we show the global in time existence of discontinuous viscosity solutions to a diagonal hyperbolic system for every initial data of bounded total variation, without the assumption that the system is strictly hyperbolic. Up to our knowledge, this is the first global existence result of large discontinuous solutions to this system.

In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n) there exists a unique linearly stable soliton, which we call the n-kink. We give numerical evidence that the n-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree n. When the radius of the wormhole throat a is large enough, the convergence to the n-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the one-kink using the Soffer–Weinstein weakly nonlinear perturbation theory.

We consider traveling front solutions connecting an invading state to an unstable ground state in a Ginzburg–Landau equation with an additional conservation law. This system appears as the generic amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the Bénard–Marangoni problem. We prove the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. The main challenges are the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.

We study a rock–paper–scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transition from stability to instability with the continuation scheme developed by Rademacher et al (2007 Physica D 229 166–83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves.

When a liquid crystal elastomer layer is bonded to an elastic layer, it creates a bilayer with interesting properties that can be activated by applying traction at the boundaries or by optothermal stimulation. Here, we examine wrinkling responses in three-dimensional nonlinear systems containing a monodomain liquid crystal elastomer layer and a homogeneous isotropic incompressible hyperelastic layer, such that one layer is thin compared to the other. The wrinkling is caused by a combination of mechanical forces and external stimuli. To illustrate the general theory, which is valid for a range of bilayer systems and deformations, we assume that the nematic director is uniformly aligned parallel to the interface between the two layers, and that biaxial forces act either parallel or perpendicular to the director. We then perform a linear stability analysis and determine the critical wave number and stretch ratio for the onset of wrinkling. In addition, we demonstrate that a plate model for the thin layer is also applicable when this is much stiffer than the substrate.

Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart et al (2003 SIAM J. Appl. Dyn. Syst.2 609–646), Golubitsky et al (2005 SIAM J. Appl. Dyn. Syst.4 78–100) and Field (2004 Dyn. Syst.19 217–243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields—these networks are said to be ordinary differential equation (ODE)-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n − 1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n − 1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

This paper focuses on a two-dimensional tropical climate model with temperature-dependent viscosity and thermal diffusivity. We show that there is a unique global smooth solution to this model with general initial data in the Sobolev class H^s for any s > 1.

We find radial and nonradial solutions to the following nonlocal problem $-{\Delta}u+\omega u=\left({I}_{\alpha }\;{\ast}\;F(u)\right)f(u)-\left({I}_{\beta }\;{\ast}\;G(u)\right)g(u)\enspace \text{in}\enspace {\mathbb{R}}^{N}$ under general assumptions, in the spirit of Berestycki and Lions, imposed on f and g, where N ⩾ 3, 0 ⩽ β ⩽ α < N, ω ⩾ 0, $f,g:\mathbb{R}\to \mathbb{R}$ are continuous functions with corresponding primitives F, G, and I_α, I_β are the Riesz potentials. If β > 0, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.

In this paper, we introduce a three-component Schnakenberg model, whose key feature is that it has a solution consisting of N spikes that undergoes Hopf bifurcations with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values beyond the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of dimension 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive the long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady state whose amplitude can be computed explicitly. For two spikes, the long-time behavior can be either in-phase or out-of-phase oscillations. Both in-phase and out-of-phase oscillations are stable, coexist for the same parameter values, and the fate of motion depends solely on the initial conditions. Further away from the Hopf bifurcation points, we offer numerical experiments indicating the existence of highly complex oscillations.

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos). While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.

We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP-convergence, i.e. convergence in the sense of the energy-dissipation principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

In this paper we consider the problem

$\begin{equation*}\begin{cases}-{\Delta}u{\pm}\phi u+{W}^{\prime }(x,u)=0\;\text{in}\;{\mathbb{R}}^{2},\quad \\ {\Delta}\phi ={u}^{2}\;\text{in}\;{\mathbb{R}}^{2},\quad \end{cases}\end{equation*}$

where W is assumed nonnegative. In dimension three, the problem with the sign + (we call it $({\mathcal{P}}_{+})$) was considered and solved in [22], whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign − (say it $({\mathcal{P}}_{-})$). We provide a general existence result for $({\mathcal{P}}_{+})$ and two examples falling in the case $({\mathcal{P}}_{-})$ for which there exists at least a nontrivial solution.

Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier–Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier–Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.

We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations ${\partial }_{t}u-a\left(\int \vert \nabla u{\vert }^{2}\right){\Delta}u+\alpha (x)u=f(x)\hspace{2pt}\text{in}\enspace {\Omega}{\times}(0,\infty ),$ on bounded domains of ${\mathbb{R}}^{N}$, N ⩾ 2, with non-homogeneous flux boundary conditions $a\left(\int \vert \nabla u{\vert }^{2}\right)\frac{\partial u}{\partial \nu }+\beta (x)u=g(x)\hspace{2pt}\text{on}\enspace \partial {\Omega}{\times}(0,\infty )$ of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) ∈ L²(Ω) × L²(∂Ω) × H¹(Ω). Approximated solutions are constructed using time rescaling and a complete set in H¹(Ω) relating the equation and the boundary condition. Uniform global estimates are derived and used to prove existence, uniqueness, continuous dependence on data, a priori estimates and higher regularity for the parabolic problem. Existence and uniqueness of stationary solutions are shown, as well as a description about their role on the asymptotic behavior regarding to the evolutionary equation. Furthermore, a sufficient condition for the existence of isolated local energy minimizers is provided. They are shown to be asymptotically stable stationary solutions for the parabolic equation.

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka–Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.