Table of contents

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension n. Furthermore, in the one-dimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous.

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation (ODE) with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ODE with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two-parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ODE with a parameter. This paper provides an exponential-asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study.

If the stable, centre and unstable foliations of a partially hyperbolic system are quasi-isometric, the system has Global Product Structure. This result also applies to Anosov systems and to other invariant splittings.

If a partially hyperbolic system on a manifold with an abelian fundamental group has quasi-isometric stable and unstable foliations, the centre foliation is without holonomy. If, further, the system has Global Product Structure, then all centre leaves are homeomorphic.

In this paper we study hyperelliptic limit cycles of the Liénard systems where, respectively, f_m(x) and g_n(x) are polynomials of degree m and n, g_n(0) = 0. We prove that, if m ⩾ 5 and m + 1 < n < 2m, then there always exist Liénard systems of the above form such that they have a hyperelliptic limit cycle. This gives a positive answer to the open problem posed in the paper by Yu and Zhang (2011 J. Math. Anal. Appl.376 535–9). By combining all the results obtained up to now, we in fact give a complete classification of the hyperelliptic limit cycles of the Liénard systems: Liénard systems of the above form have hyperelliptic limit cycles only in the following cases: (i) m = 2, 3 and m + 3 ⩽ n; (ii) 4 ⩽ m and m + 2 ⩽ n.

We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an N-dimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is one-dimensional, is carried out along a staircase within this multidimensional lattice. The solutions obtained are dependent on all N lattice variables and parameters. We further show that the soliton solutions derived from the Cauchy matrix approach are exactly the solutions obtained from reflectionless potentials, and we give a short discussion on solutions of some previously known lattice equations, such as the lattice KdV equation.

We undertake the mathematical analysis of a model describing equilibrium binary electrolytes surrounded by charged solid walls. The problem is formulated in terms of the electrostatic potential and the ionic concentrations which have prescribed spatial mean values. The free energy of the system is decomposed as the difference of the internal energy and entropy functionals. The entropy functional is the sum of an ideal entropy and an excess entropy, the latter taking into account non-ideality due to electrostatic correlations at low ionic concentrations and steric exclusion effects at high ionic concentrations. We derive sufficient conditions to achieve convexity of the entropy functional, yielding a convex–concave free energy functional. Our main result is the existence and uniqueness of the saddle point of the free energy functional and its characterization as a solution of the original model problem. The proof hinges on positive uniform lower bounds for the ionic concentrations and uniform upper bounds for the ionic concentrations and the electrostatic potential. Some numerical experiments are presented in the case where the excess entropy is evaluated using the mean spherical approximation.

We study the cyclicity of period annuli of some reversible non-Hamiltonian quadratic systems under quadratic perturbations. In general, the first integral, which we study, is non-algebraic. Via estimating the number of zeros of the associated Abelian integral (first order Melnikov function), we show that the cyclicity is 2.

We consider the problem of strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier–Stokes equations, under the Navier slip-type boundary condition (1.4), to the solution of the Euler equations under the no-penetration condition. In two dimensions, the above strong convergence holds in any smooth domain. Furthermore, in three dimensions, arbitrarily strong convergence results hold in the half-space case. In spite of the above results, recently we presented an explicit family of smooth initial data in the 3D sphere, for which the result fails. The result was proved as a by-product of the lack of time persistency for the above boundary condition under the Euler flow. Our aim here is to show a more general, and simpler proof, displayed in arbitrary, smooth domains.

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back) operator . Harmonic analysis yields a natural decomposition , where j indexes irreducible representation spaces. Using semi-classical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centred on 0 in the complex plane.

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the point (x₀, t₀). We show that it is sufficient to impose conditions on the Serrin-type integrability of v and the associated pressure p in a parabolic neighbourhood of (x₀, t₀), intersected with the exterior of a certain space–time paraboloid with the vertex at point (x₀, t₀). We make no special assumptions on v or p in the interior of the paraboloid.

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, the existence of a symmetry no longer implies the existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincaré section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.

The compressible magnetohydrodynamic equations can be derived from the equations describing electromagnetic dynamics as the dielectric constant tends to zero. Under the assumption that the initial data are well prepared, we justify this singular limit rigorously for smooth solutions to the electromagnetic fluid system in three dimensions by employing an elaborate nonlinear energy method.

We investigate the dimension of intersections of the Sierpiński gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrand's theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension.

In this paper, we first obtain a formula of averaged Lyapunov exponents for ergodic Szegő cocycles via the Herman–Avila–Bochi formula. Then using acceleration, we construct a class of analytic quasiperiodic Szegő cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in Young (1997 Ergod. Theory Dyn. Syst.25 483–504) allows us to estimate the Lebesgue measure of support of the measure associated with certain class of C¹ quasiperiodic 2-sided Verblunsky coefficients. Using the same method, we also recover the Sorets and Spencer (1991 Commun. Math. Phys.142 543–66) results for Schrödinger cocycles with nonconstant real analytic potentials and obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno frequency and for certain C¹ potentials.

The Novikov–Veselov (NV) equation is a (2 + 1)-dimensional nonlinear evolution equation generalizing the (1 + 1)-dimensional Korteweg–deVries equation. The inverse scattering method (ISM) is applied for numerical solution of the NV equation. It is the first time the ISM is used as a computational tool for computing evolutions of a (2 + 1)-dimensional integrable system. In addition, a semi-implicit method is given for the numerical solution of the NV equation using finite differences in the spatial variables, Crank–Nicolson in time, and fast Fourier transforms for the auxiliary equation. Evolutions of initial data satisfying the hypotheses of part I of this paper are computed by the two methods and are observed to coincide with significant accuracy.

We present a proof of the existence of a renormalization fixed point for Lorenz maps of the simplest non-unimodal combinatorial type ({0, 1}, {1, 0, 0}) and with a critical point of arbitrary order ρ > 1.

We investigate existence and asymptotic completeness of the wave operators for nonlinear Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger–Moser type inequality. We prove that if the Hamiltonian is below the critical value, then the solution approaches a free Schrödinger solution at the time infinity.

We obtain a Schwarzian variable associated with discrete equations of Korteweg–de Vries-type (KdV-type). In the generic case, including the primary model Q4, the new variable satisfies the lattice Schwarzian Kadomtsev–Petviashvili equation in three dimensions. For the degenerate sub-cases of Q4 the construction reveals an invertible transformation to the lattice Schwarzian KdV equation, as well as a new auto-transformation of the Schwarzian equation itself.

In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing–sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing–sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing–sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist.

Dynamical systems that are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this paper, we study stability properties of a class of structurally stable heteroclinic cycles in which we call heteroclinic cycles of type Z. It is well known that a heteroclinic cycle that is not asymptotically stable can nevertheless attract a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any δ > 0 the measure of its local basin of attraction is positive. A local basin of attraction is the set of such points that trajectories starting there remain in the δ-neighbourhood of X for all t > 0, and are attracted by X as t → ∞. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearizations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, we discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.

An Abelian integral is the integral over the level curves of a Hamiltonian H of an algebraic form ω. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the degrees H and ω. Petrov and Khovanskii have shown that this number grows at most linearly with the degree of ω, but gave a purely existential bound. Binyamini, Novikov and Yakovenko have given an explicit bound growing doubly exponentially with the degree.

We combine the techniques used in the proofs of these two results, to obtain an explicit bound on the number of zeros of Abelian integrals growing linearly with deg ω.