Patterns occur naturally in many physical and biological systems. By pattern, we mean a structure which has a complicated spatial dependence, but retains its shape as time passes. Prototypical examples are water waves, traveling pulses in neurons, convection cells, and tropical cyclones. This dissertation is concerned with the mathematical analysis of such patterns when they are subjected to random fluctuations in the environment, which we refer to as noise. The key questions which we address are stability, noise-induced motion, and long-time behavior of patterns. ... Read more

This thesis considers the thin-film equation ∂_tu = −∂_x (|u|ⁿ∂³x u) with respect to time t ≥ 0 and one dimensional space x ∈ R where n > 0. A special case of the thin-film equation is when the initial condition is u₀(x,0) = cδ(x). A solution with this initial condition is called a source type solution. A source type solution describes how a viscous droplet spreads over a solid flat surface with volume c > 0. Source type solutions are expected to have a self-similar form with u(x,t) = t^{-α}f (μ), μ = xt^{-α} and α = 1/(n+4) which reduces the equation into an ordinary boundary-value problem (|f (μ)|ⁿ f ′′′(μ))′ = α(μf (μ))′ with μf (μ)→0 as μ→±∞ and
∫_{-∞}^{∞} f (μ)dμ = c. A solution of this boundary-value problem is called a self-similar solution. This thesis presents a detailed discussion of the paper in 1992 by Bernis, Peletier & Williams on the existence and uniqueness of self-similar solutions to the thin-film equation together with its qualitative properties. Here, existence will be proven by using a shooting method. Additionally, the thin-film equation will be derived from the Navier-Stokes equations using a lubrication approximation. Furthermore, a numerical construction of the
self-similar solution is presented to visualize its behavior. The results demonstrate that the solution exhibits key qualitative features such as compact support and conservation of mass. ... Read more

In this thesis the question of existence and uniqueness of solutions to stochastic thin-film equations is investigated. The latter refers to a class of fourth-order, quasilinear, degenerate parabolic stochastic partial differential equations with (possibly nonlinear) gradient noise, which describe the evolution of a thin liquid film driven by surface tension and thermal fluctuations. Difficulties in their analysis arise due to the subtle interplay between the gradient noise term and the degenerate parabolic operator as well as the absence of a comparison principle for fourth-order equations.

Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.

Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits. ... Read more

The (nonlinear) behaviour of laser beams can be described with Nonlinear Schrödinger equations (NLS). The purpose of this thesis is to shed light on two mathematical papers that give existence and uniqueness results for an NLS equation called the soliton equation. The contribution is to expand the level of detail with which the analysis and some proofs are presented, and to unify the notation where possible. ... Read more

In this thesis we consider orbital stability of certain patterns in stochastic partial differential equations. We study two examples: a rotating wave in a two-dimensional reaction-diffusion equation and a soliton in a parametrically forced nonlinear Schrödinger equation. In both cases, we show that, for small noise, solutions to the stochastic equations remain close to a version of the pattern which is shifted according to some stochastic phase. We give explicit expressions for this phase, and show that it is optimal to first order in the strength of the noise.

To show stability, we construct a multiscale expansion of the solution around an arbitrarily shifted version of the pattern, and show that this expansion is accurate to second order. From this expansion an obvious candidate for the correct phase shift arises. For technical reasons we then construct a sequence of approximations to this phase shift, which is necessary to show the multiscale expansion around the correctly shifted pattern. We then combine this expansion with a deterministic stability result to get stochastic stability.

Finally, we take first steps towards formulating and proving the same results in a more general setting, where the pattern shift is represented by the action of a Lie group. We obtain some estimates necessary for the multiscale expansion, find the correct phase, and formulate necessary assumptions for the stability to hold. ... Read more

In this thesis we consider the incompressible and stationary Stokes problem with Navier-slip boundary conditions on an infinite two-dimensional wedge with opening angle θ. As is common for differential equations on domains with corners, the problem is decomposed into a singular expansion near the corner (polynomial problem) and a regular remainder (smooth problem). We prove existence and uniqueness of solutions to the smooth problem related to the Stokes equation which is given by -PΔu = f, where P is the Helmholtz projection. By means of the Lax-Milgram theorem it is found that this problem has a unique strong solution in a certain class of weighted Sobolev spaces if the opening angle θ is small enough. Direct application of the Lax-Milgram theorem would normally only yield a weak solution. However, by introducing additional bilinear forms we gain control on all second order derivatives and therewith obtain a strong solution. Finally, we touch upon the time-dependent Stokes problem and the polynomial problem. ... Read more

This thesis treats the thin-film equation which models the film height $h$ for a viscous film in the complete wetting regime. We show existence and uniqueness to the thin-film equation with mobility m(h) = hⁿ and mobility exponent n∈ (1,3/2)∪ (3/2,3). The thin-film equation is rewritten as an abstract Cauchy problem and usage of semi-group theory yields maximal L^p-regularity for the linearized problem. With a fixed point argument, analogous to the one used by Giacomelli, Gnann, Knüpfer and Otto in their 2014 paper, the nonlinear problem is treated. Under a smallness condition on the initial value to a suitably transformed version of the thin-film equation, we obtain a solution in L^p(0,∞;H_{k-2,α-1/2})∩ \dot{W}^1,p(0,∞;H_{k+2,α+1\2}), where the H-spaces denote weighted Sobolev spaces. The novelty of this work lies in the usage of L^p-spaces in time, where the existing literature only deals with L²-spaces. It is found that the L^p setting allows for treatment of all n∈ (1,3/2)∪ (3/2,3). ... Read more

This thesis considers solutions to the discrete Nagumo equation u˙ n = d(un−1 − 2un + un+1) + f(un), n ∈ Z. For sufficiently large d, the solutions are of the form un(t) = U(n + ct) with c > 0. This thesis contains the proof of existence of traveling wave solutions of the discrete Nagumo equations and originates from Bertram Zinner’s article ”Existence of Traveling Wavefront Solutions for the Discrete Nagumo equation” [Zin90]. In the first chapter, all the prerequisite knowledge needed to understand the proof, such as Brouwer’s fixed point theorem, is presented. The proof starts by considering f(un) as a linear function and thus simplifying the problem. The simplified problem is converted into a fixed point problem by considering a Poincar´e map which can be solved using Brouwer’s fixed point theorem. Finally, the proof ends by confirming that the solutions of the approximated, simplified problem have a limit point which corresponds to the traveling wave solutions of the discrete Nagumo equation ... Read more

The porous medium equation $\dv{t}u=\dv{x}(k(u)\dv{x}u)$ is a non-linear degenerate parabolic partial differential equation. Consequently, existence and uniqueness of its solutions is not immediately evident.
This bachelor thesis presents a detailed discussion of Atkinson's and Peletier's 1971 article ``Similarity profiles of flows through porous media" on existence and uniqueness of self-similar solutions to the porous medium equation.
First, in chapter 2 the general version of the porous medium equation along with some applications will be discussed. Then, in chapter 3 the proofs and statements of the Picard-Lindelöf theorem, Peano's existence theorem and Gronwall's inequality will be presented. These standard theorems concern differential equations and will be used in the next chapter. Finally, in chapter 4 Atkinson's and Peletier's article will be worked out in detail. ... Read more

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R).
The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing. ... Read more

This master's thesis introduces a new $p$-dependent coercivity condition through which $L^p(\Omega; L^2([0, T]; X))$ estimates can be obtained for a large class of SPDEs in the variational framework. Using these estimates, we obtain existence and uniqueness results by using a Galerkin approximation argument. The framework that is built is applied to many SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burger's equation and Navier-Stokes in 2D. Furthermore, we obtain known results for systems of SPDEs and higher order SPDEs using our unifying coercivity condition. We also obtain first steps towards a theory of higher order regularity of stochastic heat equations. ... Read more

This thesis considers the thin-film equation in partial wetting. The mobility in this equation is given by h³+λ^3-nhⁿ, where h is the film height, λ is the slip length and n is the mobility exponent. The partial wetting regime implies the boundary condition dh/dz>0 at the triple junction. The asymptotics as h↓0 are investigated. This is done by using a dynamical system for the error between the solution and the microscopic contact angle. Using the linearized version of the dynamical system, values for n when resonances occur are found. These resonances lead to a different behaviour for the solution as h↓0, so the asymptotics are found to be different for different values of n. Together with the asymptotics for h→∞ as found in [Giacomelli et al., 2016], the solution to the thin-film equation in partial wetting can be characterized. Also, via this solution, the relation between the microscopic and macroscopic contact angles can be analyzed. From the main result of this thesis, it can be seen that the macroscopic Tanner law for the contact angle depends smoothly on the microscopic contact angle. ... Read more