Table of contents

We review recent progress in modelling the probability distribution of wave heights in the deep ocean as a function of a small number of parameters describing the local sea state. Both linear and nonlinear mechanisms of rogue wave formation are considered. First, we show that when the average wave steepness is small and nonlinear wave effects are subleading, the wave height distribution is well explained by a single 'freak index' parameter, which describes the strength of (linear) wave scattering by random currents relative to the angular spread of the incoming random sea. When the average steepness is large, the wave height distribution takes a very similar functional form, but the key variables determining the probability distribution are the steepness, and the angular and frequency spread of the incoming waves. Finally, even greater probability of extreme wave formation is predicted when linear and nonlinear effects are acting together.

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire28 283–301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.

A sufficient and necessary condition for the existence of monotone travelling waves in the nonlocal Fisher–KPP equation is established, and the uniqueness of travelling wavefronts (up to translation) is also proved.

We are interested in a problem arising, for instance, in elastoplasticity modelling, which consists in a system of partial differential equations and a constraint specifying that the solution should remain, for every time and every position, in a certain set. This constraint is generally incompatible with the invariant domains of the original model, thus this problem has to be specified in mathematical terms. Here we follow the approach proposed in Després (2007 Arch. Ration. Mech. Anal.186 275–308) that furnishes a weak formulation of the constrained problem à la Kruzhkov. More precisely, this paper deals with the study of the well-posedness of Friedrichs systems under convex constraints, in any space dimension. We prove that there exists a unique weak solution, continuous in time, square integrable in space, and with values in the constraints domain. This is done with the use of a discrete approximation scheme: we define a numerical approximate solution and prove, thanks to compactness properties, that it converges towards a solution to the constrained problem. Uniqueness is proven via energy (or entropy) estimates. Some numerical illustrations are provided.

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier–Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier–Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the nonlinearity.

We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact manifolds with variable negative sectional curvature, establishing a logarithm law for such kind of flow.

We study Birkhoff sums with at the golden mean rotation number with continued fraction p_n/q_n. The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena (Knill and Lesieutre 2010 Complex Anal. Operator Theory10.1007/s11785-010-0064-7) and because shows that g is the harmonic conjugate to the piecewise linear case studied by Hecke. We relate the boundedness of log averaged Birkhoff sums S_k/log(k) and the convergence of with the existence of an experimentally established limit function on [0, 1] which satisfies a functional equation f(α) + α²f = β with a monotone function β. The limit can be expressed in terms of f.

In Stróżyna and Żoładek (2008 Bull. Belgian Math. Soc. Simon Stevin15 927–34) a generalization of the Takens normal form for a nilpotent singularity of a vector field in was obtained. Here we present an example where the corresponding normalizing series is divergent. This indicates that the generalized Takens normal form for a general nilpotent singularity in , n ⩾ 3, is non-analytic.

We consider a model describing the behaviour of a mixture of two incompressible fluids with the same density under isothermal conditions. The model consists of three balance equations: a continuity equation, a Navier–Stokes equation for the mean velocity of the mixture and a diffusion equation (Cahn–Hilliard equation). We assume that the chemical potential depends on the velocity of the mixture in such a way that an increase in the velocity improves the miscibility of the mixture. We examine the thermodynamic consistence of the model which leads to the introduction of an additional constitutive force in the motion equation. Then, we prove the existence and uniqueness of the solution of the resulting differential problem.

A 2+1-dimensional version of a non-isothermal gas dynamic system with origins in the work of Ovsiannikov and Dyson on spinning gas clouds is shown to admit a Hamiltonian reduction which is completely integrable when the adiabatic index γ = 2. This nonlinear dynamical subsystem is obtained via an elliptic vortex ansatz which is intimately related to the construction of a Lax pair in the integrable case. The general solution of the gas dynamic system is derived in terms of Weierstrass (elliptic) functions. The latter derivation makes use of a connection with a stationary nonlinear Schrödinger equation and a Steen–Ermakov–Pinney equation, the superposition principle of which is based on the classical Lamé equation.

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner–Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to the integrability of the underlying system. Formulae for spacing distributions and level compressibility are obtained for various instances of such ensembles.

A Yukawa-field approximation of the electrostatic free energy of a molecular solvation system with an implicit or continuum solvent is constructed. It is argued through the analysis of model molecular systems with spherically symmetric geometries that such an approximation is rational. The construction extends nontrivially that of the Coulomb-field approximation which serves as a basis of the widely used generalized Born model of molecular electrostatics. The electrostatic free energy determines the dielectric boundary force that in turn influences crucially the molecular conformation, stability and dynamics. An explicit formula of such forces with the Yukawa-field approximation is obtained using local coordinates and shape differentiation.

Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation vectors are shown to exist for almost every orbit of such a dynamical system with respect to any invariant measure with compact support. The concept is then extended to flows and, as an application, it is shown how non-null rotation vectors can be used to construct a measurable semi-conjugacy between a given flow and the geodesic flow of a manifold.

We consider the system of equations governing the steady flow of a polyatomic isothermal reactive gas mixture. The model covers situations when the pressure depends on species concentration and when the diffusion coefficients of each of the species are density-dependent. It is shown that this problem admits a weak solution provided the adiabatic exponent for the mixture γ is greater than .