A mixed finite element method for the Navier–Stokes equations is introduced in which thestress is a primary variable. The variational formulation retains the mathematicalstructure of the Navier–Stokes equations and the classical theory extends naturally tothis setting. Finite element spaces satisfying the associated inf–sup conditions aredeveloped.

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.

Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

We study a form of optimal transportation surplus functions which arise in hedonicpricing models. We derive a formula for the Ma–Trudinger–Wang curvature of thesefunctions, yielding necessary and sufficient conditions for them to satisfy(A3w). We use this to give explicit new examples of surplus functionssatisfying (A3w), of the formb(x,y) = H(x + y)where H is a convex function on ℝn. We alsoshow that the distribution of equilibrium contracts in this hedonic pricing model isabsolutely continuous with respect to Lebesgue measure, implying that buyers are fullyseparated by the contracts they sign, a result of potential economic interest.

We consider a multi-polaron model obtained by coupling the many-body Schrödinger equationfor N interacting electrons with the energy functional of a mean-fieldcrystal with a localized defect, obtaining a highly non linear many-body problem. Thephysical picture is that the electrons constitute a charge defect in an otherwise perfectperiodic crystal. A remarkable feature of such a system is the possibility to form a boundstate of electrons via their interaction with the polarizable background. We prove firstthat a single polaron always binds, i.e. the energy functional has aminimizer for N = 1. Then we discuss the case of multi-polaronscontaining N ≥ 2 electrons. We show that their existence is guaranteedwhen certain quantized binding inequalities of HVZ type are satisfied.

In this paper, starting from classical non-convex and nonlocal3D-variational model of the electric polarization in a ferroelectricmaterial, via an asymptotic process we obtain a rigorous2D-variational model for a thin film. Depending on the initial boundaryconditions, the limit problem can be either nonlocal or local.

A nation-wide vaccination campaign began in New Zealand in 2004 with the aim of stopping the epidemic of meningococcal B disease. Approximately 80% of those under 20 years of age when the campaign was launched were vaccinated with three doses of a tailor-made vaccine. We propose a framework for a mathematical model based on the susceptible–carrier–infectious–removed (SCIR) structure. We show how the model could be used to calculate the predicted yearly incidence of infection in the absence of vaccination, and compare this to the effect that vaccination had on the course of the epidemic. Our model shows that vaccination led to a considerable decrease in the incidence of infection compared to what would have been seen otherwise. We then use our model to explore the potential effect of alternative vaccination schemes, and show that the one that was implemented was the best of all the possibilities we consider.

A susceptible–exposed–infectious theoretical model describing Tasmanian devil population and disease dynamics is presented and mathematically analysed using a dynamical systems approach to determine its behaviour under a range of scenarios. The steady states of the system are calculated and their stability analysed. Closed forms for the bifurcation points between these steady states are found using the rate of removal of infected individuals as a bifurcation parameter. A small-amplitude Hopf region, in which the populations oscillate in time, is shown to be present and subjected to numerical analysis. The model is then studied in detail in relation to an unfolding parameter which describes the disease latent period. The model’s behaviour is found to be biologically reasonable for Tasmanian devils and potentially applicable to other species.

We prove pointwise gradient bounds for entire solutions of pde’s of the form

     ℒu(x) = ψ(x, u(x), ∇u(x)),

where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

A lower semicontinuity result in BV is obtained for quasiconvexintegrals with subquadratic growth. The key steps in this proof involve obtainingboundedness properties for an extension operator, and a precise blow-up technique thatuses fine properties of Sobolev maps. A similar result is obtained by Kristensen in[Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], wherethere are weaker asssumptions on convergence but the integral needs to satisfy a strongergrowth condition.

We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness.