This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

How can we model global phenomenon based on local interactions? Agent-Based (AB) models are local rule-based discrete method that can be used to simulate complex interactions of many agents. Unfortunately, the relative ease of implementing the computational model is often counter-balanced by the difficulty of performing rigorous analysis to determine emergent behaviors. Calculating existence of fixed points and their stability is not tractable from an analytical perspective and can become computationally expensive, involving potentially millions of simulations. To construct meaningful analysis, we need to create a framework to approximate the emergent, global behavior. Our research has been devoted to developing a framework for approximating AB models that move via random walks and undergo state transitions. First, we developed a general method to estimate the density of agents in each state for AB models whose state transitions are caused by neighborhood interactions between agents. Second, we extended previous random walk models of instantaneous state changes by adding a cumulative memory effect. In this way, our research seeks to answer how memory properties can also be incorporated into continuum models, especially when the memory properties effect state changes on the agents. The state transitions in this type of AB model is primarily from the agents’ interaction with their environment. These modeling frameworks will be generally applicable to many areas and can be easily extended.

Doctor of Philosophy / Department of Mathematics / Ivan Blank / We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.

In this thesis, we develop multiscale models for particle simulations in population dynamics. These models are characterised by prescribing particle motion on two spatial scales: microscopic and macroscopic.At the microscopic level, each particle has its own mass, position and velocity, while at the macroscopic level the particles are interpolated to a continuum quantity whose evolution is governed by a system of transport equations.This way, one can prescribe various types of interactions on a global scale, whilst still maintaining high simulation speed for a large number of particles. In addition, the interplay between particle motion and interaction is well tuned in both regions of low and high densities. We analyse links between models on these two scales and prove that under certain conditions, a system of interacting particles converges to a nonlinear coupled system of transport equations.We use this as a motivation to derive a model defined on both modelling scales and prescribe the intercommunication between them. Simulation takes place in inhomogeneous domains with arbitrary conditions at inflow and outflow boundaries. We realise this by modelling obstacles, sources and sinks.Integrating these aspects into the simulation requires a route planning algorithm for the particles. Several algorithms are considered and evaluated on accuracy, robustness and efficiency. All aspects mentioned above are combined in a novel open source prototyping simulation framework called Mercurial. This computational framework allows the design of geometries and is built for high performance when large numbers of particles are involved. Mercurial supports various types of inhomogeneities and global systems of equations. We apply our framework to simulate scenarios in crowd dynamics.We compare our results with test cases from literature to assess the quality of the simulations. / <p>Master Thesis in Industrial and Applied Mathematics</p>

In this thesis we derive and solve mathematical models for surfactant systems with differing adsorption structures at interfaces. The first part of this thesis considers two dynamic experimental set-ups for which we derive the associated mathematical surfactant–fluid description. Firstly we consider the behaviour of a weakly interacting polymer–surfactant solution under the influence of a steady straining flow. We reduce the model using asymptotic methods to predict the regimes under which we observe phase transitions of the species in the system and show how the bulk dynamics couple to the surfactant adsorption. Secondly we model an experiment to observe the desorption kinetics of a surfactant monolayer, designed to emulate the 'rinse mechanism' used for the removal of surfactant-containing products using water. Through the comparison of our model with experimental data we derive a semi-empirical relationship that describes the variation in depth of a near-surface diffusive boundary layer with the reduced Peclet number. We then employ a combination of asymptotic and numerical techniques that validate this result. The second part of this thesis is concerned with surfactant systems that exhibit more pronounced adsorption at the interface due to the surfactant monomers no longer arranging themselves in a single layer, as is typically the case, but rather in multiple layers. Such self-assembled structures are commonly referred to as multilayers. We derive a simplified model that describes the rearrangement of surfactant within the multilayer structure and draw comparisons between the features of our model and experimental observations. We consider an extension of the theory to the situation of multilayer formation between two adsorbing interfaces, which is governed by an implicit free-boundary problem. We also consider incorporation of bulk solution effects, such as the addition of an electrolyte. Finally, we draw our conclusions and suggest further theoretical and experimental work related to the models presented in this thesis.

In this thesis, basic concepts of populational models are studied from a theoretical point of view, especially the long term behaviours. All models are at least time dependent with additional age structure, spatial structure. The last model which is an extension of the von Foerster equation, is dependent on all o f these structures and have a long-term solution for large values of time. Modeling population is a frequent subject in modern biology. It is hard to create a model that appears as realistic as possible. First one might consider that a population size is governed by the current size of the population, along with rates of how each individual contributes (give birth), so that the population increases. and how frequent an individual dies, causing the population to decrease in size. However these sort of models can only describe the size of population in a shorter span of time.

We study a nonlinear kinetic model describing the interactions of particles in a granular medium, i.e. inelastic systems where kinetic energy is not conserved due to internal friction. Examples of particles that fall into this category are sand, ground coffee and many others. Originally studied by Benedetto, Caglioti and Pulvirenti in the one-dimensional setting (RAIRO Model. Math. Anal. Numer, 31(5): 615-641, (1997)) the original model contained inconsistencies later accounted for and corrected by invoking a mollifier (Modelisation Mathematique et Analyse Numerique, M2AN, Vol. 33, No 2, pp. 439–441 (1999)). This thesis approximates the generalized model presented by Agueh (Arch. Rational Mech., Anal. 221, pp. 917-959 (2016)) with the added assumption of a spatial mollifier present in the kinetic equation. In dimension d ≥ 1 this model reads as ∂tf + v · ∇xf = divv(f([ηα∇W] ∗(x,v) f)) where f is a non-negative particle density function, W is a radially symmetric class C2 velocity interaction potential, and and ηα is a mollifier. A physical interpretation of this approximation is that the particles are spheres of radius α > 0 as opposed to the original assumption of being point-masses. Properties lost by this approximation and macroscopic quantities that remain conserved are discussed in greater detail and contrasted. The main result of this thesis is a proof of the weak global existence and uniqueness. An argument utilizing the tools of Optimal Transport allows simple construction of a weak solution to the kinetic model by transporting an initial measure under the characteristic flow curves. Concluding regularity arguments and restrictions on the velocity interaction potential ascertain that global classical solutions are obtained. / Graduate

In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation in an open sphere around the origin, with constant internal and external damping coefficients and nonlinear term of the form G' (w) = w ^p, with p an odd number greater than 1. We prove that our scheme is consistent of quadratic order, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to study the effects of internal and external damping.

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.

We consider the flow of an ideal gas with internal friction and heat conduction in a layer between a fixed plane and an upper free boundary. We describe the top free surface as the graph of a time dependent function. This forces us to exclude breaking waves on the surface. For this and other reasons we need to confine ourselves to flow close to a motionless equilibrium state which is fairly easy to compute. The full equations of motion, in contrast to that, are quite difficult to solve. As we are close to an equilibrium, a linear system of equations can be used to approximate the behavior of the nonlinear system. Analytic, strongly continuous semigroups defined on a suitable Banach space X are used to determine the behavior of the linear problem. A strongly continuous semigroup is a family of bounded linear operators {T(t)} on X where 0 ≤ t < infinity satisfying the following conditions. 1. T(s+t)=T(s)T(t) for all s,t ≥ 0 2. T(0)=E, the identity mapping. 3. For each x ∈ X, T(t)x is continuous in t on [0,infinity). Then there exists an operator A known as the infinitesimal generator of such that T(t)=exp (tA). Thus, an analytic semigroup can be viewed as a generalization of the exponential function. Some estimates about the decay rates are derived using this theory. We then prove the existence of long term solutions for small initial values. It ought to be emphasized that the decay is not an exponential one which engenders significant difficulties in the transition to nonlinear stability.