Table of contents

In this paper, we study the variational properties of two special orbits: a Schubart orbit and a Broucke–Hénon orbit. We show that under an appropriate topological constraint, a minimizer must be either a Schubart orbit or a Broucke–Hénon orbit. One of the main challenges is to prove that a Schubart orbit coincides with a minimizer connecting a collinear configuration with a binary collision and an isosceles configuration. A new geometric argument is introduced to overcome this challenge.

We develop a mathematical theory for a class of compressible viscoelastic rate-type fluids with stress diffusion. Our approach is based on the concepts used in the nowadays standard theory of compressible Newtonian fluids as renormalization, effective viscous flux identity, compensated compactness. The presence of the extra stress, however, requires substantial modification of these techniques, in particular, a new version of the effective viscous flux identity is derived. With help of these tools, we show the existence of global-in-time weak solutions for any finite energy initial data.

This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one-dimensional Born–Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow-up solutions; meanwhile, Lyapunov nonlinear stability of those self-similar blow-up solutions is given inside a strictly proper subset of the backward light cone.

With the notion of the mode-j  Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff–Hopf and Perron–Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type , with x_i being an element of a cone C_i in a Banach space . We then consider a family of nonlinear integral operators f _i with positive kernel, acting on the product of spaces of continuous real-valued functions. In this setting we provide an explicit formula for the mode-j  contraction ratio, which is particularly relevant in practice, as this type of operator plays a central role in numerous models and applications.

This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz–Ladik system posed on a graph that is constituted by N semi-infinite lattices (edges) connected through some boundary conditions. We first show that analyzing this problem is equivalent to analyzing a certain matrix IBV problem; then we employ the unified transform method (UTM) to analyze this matrix IBV problem. We also compare our results with some previously known studies. In particular, we show that the inverse scattering method (ISM) for the integrable DDEs on the set of integers can be recovered from the UTM applied to our N  =  2 graph problem as a particular case, and the non-local reductions of integrable DDEs can be obtained as local reductions from our results.

In this paper, we study a two-species chemotaxis model with two chemicals in . Let be the initial mass of the two species respectively. The critical mass of the model is established as a curve of the form . That is to say the solutuons exist globally if , and the finite time blow-up of solutions may occur if .

Coarse-graining is central to reducing dimensionality in molecular dynamics, and is typically characterized by a mapping which projects the full state of the system to a smaller class of variables. While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. In this article, starting with non-reversible dynamics, we introduce and study effective dynamics which approximate the (non-closed) projected dynamics. Under fairly weak conditions on the system, we prove error bounds on the trajectorial error between the projected and the effective dynamics. In addition to extending existing results to the non-reversible setting, our error estimates also indicate that the notion of mean force motivated by these effective dynamics is a good one.

In this paper, we study the potential singular points of interior and boundary suitable weak solutions to the 3D Navier–Stokes equations. It is shown that the upper box dimensions of interior singular points and boundary singular points are bounded by 7/6 and 10/9, respectively. Both proofs rely on the recent progress of -regularity criteria at one scale.

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution . There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled tends, as , to a Maxwellian , where and T are solutions of coupled diffusion equations and estimate the error in .

When there is no independence, abnormal observations may have a tendency to appear in clusters instead of being scattered along the time frame. Identifying clusters and estimating their size are important problems arising in the statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the extremal index appears to be linked to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing limiting point processes. Smith (1988 Adv. Appl. Probab. 20 681–3), introduced a regenerative process exhibiting such disagreement. Very recently, in Abadi (2018 (arXiv:1808.02970)) the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the extremal index. Here, we consider a general regenerative process that includes Smith's model, show that it is important to consider finite time quantities instead of asymptotic ones and compare their different behaviours in relation to the cluster size. We consider other indicators, such as what we call the sojourn time, which corresponds to the size of groups of abnormal observations, when there is some uncertainty regarding where the cluster containing that group was actually initiated. We also study the decay of correlations of the non-Markovian models considered.

A map is a piecewise contraction of n intervals (n-PC) if there exist and a partition of into intervals such that for every (). An infinite word over the alphabet is a natural coding of f if there exists such that whenever . We prove that if is a natural coding of an injective n-PC, then some infinite subword of is either periodic or isomorphic to a natural coding of a topologically transitive m-interval exchange transformation (m-IET), where . Conversely, every natural coding of a topologically transitive n-IET is also a natural coding of some injective n-PC.

We consider the Neumann problem for a coupled chemotaxis–haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic 'damping' rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis–haptotaxis systems.

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem when the parameter varies. Also, we answer a question proposed in Carvalho et al (2012 Proc. Am. Math. Soc. 140 2357–73), concerning the complete description of the structure of the pullback attractor of the problem when and, more generally, for , . We construct global bounded solutions, 'non-autonomous equilibria', connections between the trivial solution and these 'non-autonomous equilibria' and characterize the -limit and -limit set of global bounded solutions. As a consequence, we show that the global attractor of the associated skew-product flow has a gradient structure. The structure of the related pullback an uniform attractors are derived from that.

We study the following nonlinear scalar field equation

Here , m  >  0 is a given constant and arises as a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity f , we show the existence of one nonradial solution for any , and obtain multiple (sometimes infinitely many) nonradial solutions when N  =  4 or . In particular, all these solutions are sign-changing.

We consider stochastic nonlinear Landau–Ginzburg equations on the half-line with Robin-type white-noise boundary conditions. We establish the existence and uniqueness of a solution to the initial boundary value problem with values in an appropriate weighted space. We are also interested in the regularity behavior of the solution, especially near the origin.

We study the dispersive blow-up phenomena for the Schrödinger–Korteweg–de Vries (S-KdV) system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. It seems that this result is the first regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties for solutions of the IVP in fractional weighted Sobolev spaces which we establish here.

We construct K-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions u of the equation such that where and for any , W_k is the Lorentz transform of the explicit standing soliton , with any speed , satisfying for , and an explicit smallness condition. The proof extends the refined method of construction of asymptotic multi-solitons from Martel and Merle (2016 Arch. Ration. Mech. Anal. 222 1113–60; Martel and Merle 2018 Inventiones Math. 214 1267–363).

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.

A measure without local dimension is a measure such that local dimension does not exist for any point in its support. In this paper, we construct such a class of Moran measures and study their lower and upper local dimensions. We show that the related 'free energy' function (L^q-spectrum) does not exist. Nevertheless, we can obtain the full Hausdroff and packing dimension spectra for level sets defined by lower and upper local dimensions. They can be viewed as a generalized multifractal formalism.

We study the shape optimization problem of variational Dirichlet and Neumann p -Laplacian eigenvalues, with area and perimeter constraints. We prove some results that characterize the optimizers and derive the formula for the Hadamard shape derivative of Neumann p -Laplacian eigenvalues. Then, we propose a numerical method based on the radial basis functions method to solve the eigenvalue problems associated to the p -Laplacian operator. Several numerical results are presented and some new conjectures are addressed.

A fluid-structure interaction model with discrete and distributed delays in the structural damping is studied. The fluid and structure dynamics are governed by the Navier–Stokes and linear elasticity equations, respectively. Due to the presence of delay, a crucial ingredient of the weak formulation is the use of hidden boundary regularity for transport equations. In two space dimension, it is shown that weak solutions are unique. For smooth and compatible data, we establish the existence of the pressure and by applying micro-local analysis, further regularity of the solutions are available. Finally, the exponential stability of the system is obtained through an appropriate Lyapunov functional.

This paper is devoted to study the asymptotic behavior, as vanishes, of a nonlinear monotone Signorini boundary value problem modelizing chemical activity in an -periodic structure of thin cylindrical absorbers, like a comb in 2D a or a brush in 3D. The novelty of this paper is the presence of a perturbed coefficient , with , in the nonlinear Signorini boundary conditions (the case was previously studied by the same authors). It is shown that the limit problem is the same as what one would get by replacing the Signorini boundary conditions with the homogeneous Dirichlet boundary condition in the original problem.

We present a study of time-independent solutions of the two-dimensional discrete Allen–Cahn equation with cubic and quintic nonlinearity. Three different types of lattices are considered, i.e. square, honeycomb, and triangular lattices. The equation admits uniform and localised states. We can obtain localised solutions by combining two different states of uniform solutions, which can develop a snaking structure in the bifurcation diagrams. We find that the complexity and width of the snaking diagrams depend on the number of 'patch interfaces' admitted by the lattice systems. We introduce an active-cell approximation to analyse the saddle-node bifurcation and stabilities of the corresponding solutions along the snaking curves. Numerical simulations show that the active-cell approximation gives good agreement for all of the lattice types when the coupling is weak. We also consider planar fronts that support our hypothesis on the relation between the complexity of a bifurcation diagram and the number of interface of its corresponding solutions.