In this thesis we study well-posedness problems for certain reformulations and models of the Euler equations and the Navier{Stokes equations. We also prove several global well-posedness results for the diffusive Burgers equations. We discuss the Eulerian-Lagrangian formulation of the incompressible Euler equations considered by Constantin (2000). Using this formulation we give a new proof that the Euler equations are locally well-posed in Hs (Td ) for s > d/2 + 1. Constantin proved a local well-posedness result for this system in the Hӧlder spaces C1; for μ> 0, but an analysis in Sobolev spaces is perhaps more natural. After suggesting a possible Eulerian-Lagrangian formulation for the incompressible Navier{Stokes equations in which the back-to-labels map is not di used, we obtain the formulation written in terms of the so-called magnetization variables, as studied by Montgomery-Smith and Pokornẏ (2001). We give a rigorous analysis of the equivalence between this formulation and the classical one, in the context of weak solutions. Noting certain similarities between this formulation and the diffusive Burgers equations we begin a study of the latter. We prove that the diffusive Burgers equations are globally well-posed in Lp ∩ L2 (Ω ) for certain domains Rd , p > d, and d = 2 or d = 3. Moreover, we prove a global well-posedness result in H1= 2 (T3 ). Lastly, we consider a new model of the Navier{Stokes equations, obtained by modifying one of the nonlinear terms in the magnetization variables formulation. This new system admits a maximum principle and we prove a global well-posedness result in H1=2 (T3 ) following our analysis of the Burgers equations.

In this thesis I present the results I have been developing during my PhD studies at City University London. The original results are based on D Bianchini et al, D Bianchini, O Castro-Alvaredo and B Doyon, D Bianchini and F Ravanini, D Bianchini et al and D Bianchini and O Castro-Alvaredo. In all but one publications, we compute the entanglement of various systems. Using the celebrated “replica trick” we compute the entanglement entropy of non unitary systems using integrable tools in continuum and discrete models. In particular, in the first article we generalise the method described in the seventh article in order to take into account non unitary conformal systems. In the second article we use a form factor expansion to probe a non unitary system outside the critical point. In the fourth article we derive the explicit expressions of one dimensional quantum Hamiltonians which provide a lattice realisation of off critical non unitary minimal models. Using a Corner Transfer Matrix approach we compute the scaling of the entanglement of such spin chains. In the fifth article we study the scaling of various twist field correlation functions in order to compute the entanglement entropy and the logarithmic negativity in free boson massive theories.

Thin liquid films are found everywhere in nature. Their flows play a fundamental role in a wide range of applications and processes. They are central to a number of biological, industrial, chemical, geophysical and environmental applications. Thin films driven by external forces are susceptible to instabilities leading to the break-up of the film into fingering-type patterns. These fingering-type patterns are usually undesirable as they lead to imperfections and dry spots. This behaviour has motivated theoreticians to try to understand the behaviour of the flow and the mechanisms by which these instabilities occur. In the physically relevant case when surface tension is large, the film’s free surface exhibits internal layers where there is rapid spatial variation in the film’s curvature over very short lengthscales and away from these internal layers the film’s curvature is almost negligible. This provides the main motivation for this thesis which is to develop adaptive numerical solution techniques for thin film flow equations that fully resolve such internal layers in order to obtain accurate numerical solutions. We consider two thin film flow problems in one and two-dimensions to test the adaptive numerical solution techniques developed in this thesis. The first problem we consider is related to a liquid sheet or drop spreading down an inclined pre-wetted plane due to influence of gravity. The second problem we consider is also related to the spreading of a liquid sheet or drop down an inclined pre-wetted plane including surfactant-related effects in addition to gravity. We follow the r-adaptive moving mesh technique which uses moving mesh partial differential equations (MMPDEs) to adapt and move the mesh coupled to the underlying PDE(s). We show how this technique can accurately resolve the various one and two-dimensional structures observed in the above test problems as well as reduce the computational effort in comparison to numerical solutions using a uniform mesh.

The problem of localised bulging in inflated thin-walled tubes has been studied by many authors. In all these studies, the strain-energy function is expressed only in term of principal stretches. However, there are some applications where the cylindrical tube concerned may have walls thick enough so that the membrane theory may become invalid. One such situation that motivates the present study is the mathematical modeling of aneurysm initiation; in that context the human arteries exhibit noticeable bending stiffness. The effects of bending stiffness on localized bulging are studied using two different approaches. The first approach is related to the continuum-mechanical theory for three-dimensional finite deformations of coated elastic solids formulated by Steigmann and Ogden (1997, 1999). Strain-energy function has been defined in terms of the curvature of the middle surface and the principal stretches. The elasticity of the coating incorporates bending stiffness and generalizes the theory of Gurtin and Murdoch (1975). A bifurcation condition is derived using a weakly non-linear analysis and the near-critical behaviour is determined analytically. A finite difference scheme and a shooting method are formulated to determine the fully non-linear bulging solutions numerically. The second approach is based on the exact theory of finite elasticity, and the tube concerned is assumed to have arbitrary thickness. The exact bifurcation condition is derived and used to quantify the effects of bending stiffness. A two-term asymptotic bifurcation condition that incorporates bending stiffness is also derived. Finally, it is shown that when the axial force is held fixed, the bifurcation pressure is equal to the maximum pressure in uniform inflation. However when the axial stretch is fixed, localized solution is possible even if the pressure does not have a maximum in uniform inflation. This last result is particularly relevant to the continuum-mechanical modelling of the initiation of aneurysms in human arteries.

The work presented in this thesis is a study of homogenisation problems in electromagnetics and elasticity with potential applications to the development of metamaterials. In Chapter 1, I study the leading order frequency approximations of the quasi-static Maxwell equations on the torus. A higher-order asymptotic regime is used to derive a higher-order homogenised equation for the solution of an elliptic second-order partial differential equation. The equivalent variational approach to this problem is studied which leads to an equivalent higher-order homogenised equation. Finally, the derivation of higher-order constitutive laws relating the fields to their inductions is presented. In Chapter 2, I study the governing equations of linearised elasticity where the periodic composite material of interest is made up of a "critically" scaled "stiff" rod framework with the voids in between filled in with a "soft" material which is in high-contrast with the stiff material. Using results from two-scale convergence theory, a well posed homogenised model is presented with features reminiscent of both high-contrast and thin structure homogenised models with the additional feature of a linking relation of Wentzell type. The spectrum of the limiting operator is investigated and the establishment of the convergence of spectra from the initial problem is derived. In the final chapter, I investigate brie y three additional homogenisation problems. In the first problem, I study a periodic dielectric composite and show that there exists a critical scaling between the material parameter of the soft inclusion and the period of the composite. In the second problem, I use of two-scale convergence theory to derive a homogenised model for Maxwell's equations on thin rod structures and in the final problem I study Maxwell's equations in R^3 under a chiral transformation of the coordinates and derive a homogenised model in this special geometry.

In this thesis we study two problems related to the Teichm�uller harmonic map flow, a flow introduced in [21], which aims to deform maps from closed surfaces into closed Riemannian targets of general dimension into branched minimal immersions. It arises as a gradient flow for the energy functional when one varies both the map and the domain metric. We first consider weak solutions of the flow that exist for all time, with metric degenerating at in�nite time. It was shown in [25] that for such solutions one can extract a so-called sequence of almost-minimal maps, which subconverges to a collection of branched minimal immersions (or constant maps). We further improve this compactness theory, in particular showing that no loss of energy can happen, after accounting for all developing bubbles. We also construct an example of a smooth flow where the image of the limit branched minimal immersions is disconnected. These results were obtained in [13], joint with Melanie Ruping and Peter Topping. Secondly, we study limits of the coupling constant n, which controls the relative speed of the metric evolution and the map evolution along the flow. We show that when n | 0, corresponding to slowing down the metric evolution, one obtains the classical harmonic map flow as a limit of the Teichm�uller harmonic map flow when the target N has nonpositive sectional curvature. Finally, we let N | 0 and simultaneously rescale time, 'fixing' the speed at which the metric evolves and accelerating the evolution of the map component. We show that in this setting the Teichm�uller harmonic map flow converges to a flow through harmonic maps, if one assumes the target N to have strictly negative sectional curvature everywhere and the initial map to not be homotopic to a constant map or a map to a closed geodesic in the target.

The starting point of the research developed in this thesis was the work done by Arek Goetz in his PhD thesis [Dynamics of Piecewise Isometrics, University of Illinois, 1996). Following his dissertation, we have considered the simple example of a piecewise rotation in two convex atoms defined in the whole plane (now commonly known as the Goetz map) as our main source of motivation. The first main achievement of our work was the construction of a new family of polygonal sets which are, in fact, global attractors. These examples are very similar in nature to the Sierpinski-gasket triangle presented in Goetz [1998c]. The next natural step was to argue that the definition of a piecewise isometric attractor is not entirely suitable in this context due to the lack of uniqueness. Under a new definition of attractor, some results are proved involving the properties of quasi-invariance and regularity existing in all examples available in the literature on the Goetz map. Following that, we attempt to generalise the results of Goetz regarding periodic cells and periodic points to unbounded spaces. We prove that there is a fundamental discrepancy between piecewise rotations in odd and even dimensions. In the odd-dimensional case the existence of periodic points is rare; hence those that exist must be unstable under almost all perturbations, whereas in even dimensions periodic points are stable for a prevalent set of piecewise rotations. Furthermore, if a piecewise rotation is such that the free monoid generated by the linear parts of the induced rotations does not contain the identity map then it follows trivially that all diverging orbits must be irrationally coded. This implies, together with a result on the coding of the open connected components in the complement of the closure of the exceptional set, that there exist examples of piecewise isometries possessing irrational cells with positive measure. This scenario was not considered previously in the results proved by Goetz. Using a common recurrence argument and Goetz's characterisation of the closure of the exceptional set (see for instance, Goetz [2001]) we prove that every recurrent point (i.e., such that w(x) ? ?) must be rationally coded. Given an invertible piecewise isometry in a compact space we also show that the closure of the exceptional set equals that of its inverse. This sustains the common idea that forward and backward iterates of the discontinuity yield similar graphics. In the context of the Goetz map we have also investigated the appearance of symmetric patterns when plotting the closure of the exceptional set. Although the Goetz map is by its nature discontinuous, the existence of symmetry is still possible under the broader framework of almost-everywhere-symmetry. Finally, we briefly note that the local symmetry properties of symmetric patterns arising in the invertible Goetz map are in part due to the existence of a reversing-symmetry, which generates piecewise continuous reversing symmetries under iteration of the original Goetz map.

One topic of this thesis are products of two Eisenstein series. First we investigate the subspaces of modular forms of level N that are generated by such products. We show that of the weight k is greater than 2, for many levels, one can obtain the whole of M[subspace]k(N) from Eisenstein series and products of two Eisenstein series. We also provide a result in the case k=2 and treat some spaces of modular forms of non-trivial nebentypus. We then analyse the L-functions of products of Eisenstein series. We reinterpret a method by Rogers-Zudilin and use it in two applications, the first concerning critical L-values of a product of two Eisenstein series, and the second special values of derivatives of L-functions. The last part of this thesis deals with the theory of Eichler-cohomology for arbitrary real weights, which was first developed by Knopp in 1974. We establish a new approach to Knopp's theory using techniques from the spectal theory of automorphic forms, reprove Knopp's main theorems, and also providea vector-valued version of the theory.

The thesis deals with dimension theory and ergodic theory. We are interested in applying thermodynamic formalism to give explicit values. Mainly we study dimension of sets with different ergodic averages. An extension to the case of level sets for Gibbs measures of hyperbolic dynamical system are investigated. This leads to very accurate numerical averages.

This thesis consists of three sections, each concerned with the study of the mixing properties of certain classes of measurable transformations of Lebesgue spaces. While in section 1 we consider the class of measure preserving endomorphisms of a fixed measure space, in sections 2 and 3 we restrict our attention to a class of piecewise monotone increasing and continuous functions of the unit interval, together with their corresponding 'natural' invariant measures.