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Dissertations / Theses on the topic 'Evolution equations'

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Author: Grafiati

Published: 4 June 2021

Last updated: 29 July 2024

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1

Zangeneh, Bijan Z. "Semilinear stochastic evolution equations." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/31117.

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Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation [formula omitted] where • f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions. • gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H. • Vt is a cadlag adapted process with values in H. • X₀ is a random variable. We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles. Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation.
Science, Faculty of
Mathematics, Department of
Graduate

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2

Aterman, R. D. "Two nonlinear evolution equations." Thesis, University of Oxford, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355778.

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3

Tudor, Jan. "Stochastic flow and evolution equations." Thesis, University of Oxford, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.547462.

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4

Ratter, Mark C. "Grammians in nonlinear evolution equations." Thesis, University of Glasgow, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.264153.

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5

Yilmaz, Halis. "Evolution equations for differential invariants." Thesis, University of Glasgow, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.274288.

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6

Šipčić, Radica 1972. "Generalized long-wave evolution equations." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/49623.

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7

Pogan, Alexandru Alin. "Dichotomy theorems for evolution equations." Diss., Columbia, Mo. : University of Missouri-Columbia, 2008. http://hdl.handle.net/10355/6090.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2008.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 22, 2009) Vita. Includes bibliographical references.

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8

URBANI, CRISTINA. "Bilinear Control of Evolution Equations." Doctoral thesis, Gran Sasso Science Institute, 2020. http://hdl.handle.net/20.500.12571/10061.

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The thesis is devoted to the study of the stabilization and the controllability of the evolution equations $$u'(t) + Au (t) + p (t) Bu (t) = 0$$ by means of a bilinear control $p$. Bilinear controls are coefficients of the equation that multiply the state variable. Multiplicative controls are therefore suitable to describe processes that change their principal parameters in presence of a control. We first present a result of rapid stabilization of the parabolic equations towards the ground state by bilinear control with a doubly exponential rate of convergence. Under stronger hypotheses on the potential $B$, we show results of exact local and global controllability towards the solution of the ground state in arbitrarily small time. We apply these two abstract results to different types of PDE such as the heat equation, or parabolic equations with non-constant coefficients. We then prove local exact controllability of a class of degenerate wave equations relying on a sharp analysis of the spectral properties of the elliptic degenerate operators. We then present a method of constructing multiplicative operators $B$ verifying the sufficient hypotheses required for controllability or stabilization results. This method leads to constructive algorithms of infinite explicit families of such operators $B$. We then prove new controllability results for the Schr{"o}dinger equation with hybrid boundary conditions. We also give applications of our method to parabolic equations leading to results of rapid stabilization, local and global controllability to the ground state which are explicit with respect to the operators $B$.

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9

Guan, Meijiao. "Global questions for evolution equations Landau-Lifshitz flow and Dirac equation." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/22491.

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This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questions regarding the solutions concern existence, uniqueness, stability and singularity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or ground state solution, is a solution which propagates without changing its shape. The class of Landau-Lifshitz equations (including harmonic map heat flow and Schrödinger map equations) arises in the study of ferromagnets (and anti-ferromagnets), liquid crystals, and is also very natural from a geometric standpoint. Harmonic maps are the stationary solutions to these equations. My thesis concerns the problems of singularity formation vs. global regularity and long time asymptotics when the target space is a 2-sphere. We consider maps with some symmetry. I show that for m-equivariant maps with energy close to the harmonic map energy, the solutions to Landau-Lifshitz equations are global in time and converge to a specific family of harmonic maps for big m, while for m =1, a finite time blow up solution is constructed for harmonic map heat flow. A model equation for Schrödinger map equations is also studied in my thesis. Global existence and scattering for small solutions and local well-posedness for solutions with finite energy are proved. The existence of standing wave solutions for the nonlinear Dirac equation is studied in my thesis. I construct a branch of solutions which is a continuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continuity unknown. The ground state solutions of nonlinear Schrodinger equations yield solutions to nonlinear Dirac equations. We also show that this branch of solutions is unstable. This leads to a rigorous proof of the instability of the ground states, confirming non-rigorous results in the physical literature.

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10

Nguyen, Thieu-Huy. "Functional partial differential equations and evolution semigroups." [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=973911344.

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11

Branding, Volker. "The evolution equations for Dirac-harmonic Maps." Phd thesis, Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2013/6420/.

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This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.
Die vorliegende Dissertation untersucht den Gradientenfluss von Dirac-harmonischen Abbildungen. Dirac-harmonische Abbildungen sind kritische Punkte eines Energiefunktionals, welches aus supersymmetrischen Feldtheorien motiviert ist. Die kritischen Punkte dieses Energiefunktionals koppeln die Gleichung für harmonische Abbildungen mit Spinorfeldern. Viele analytische Eigenschaften von Dirac-harmonischen Abbildungen sind bereits bekannt, ein allgemeines Existenzresultat wurde aber noch nicht erzielt. Diese Dissertation untersucht das Existenzproblem, indem der Gradientenfluss von einer regularisierten Version Dirac-harmonischer Abbildungen untersucht wird. Die Methode des Gradientenflusses kann nicht direkt angewendet werden, da das Energiefunktional für Dirac-harmonische Abbildungen nach unten unbeschränkt ist. Daher wird zunächst eine Regularisierungsvorschrift für Dirac-harmonische Abbildungen eingeführt und dann der Gradientenfluss betrachtet. Kapitel 1 stellt für die Arbeit wichtige Resultate über harmonische Abbildungen/harmonische Spinoren zusammen. Außerdem werden die zur Zeit bekannten Resultate über Dirac-harmonische Abbildungen zusammengefasst. In Kapitel 2 werden Dirac-harmonische Abbildungen im Detail eingeführt, außerdem wird eine Regularisierungsvorschrift präsentiert. Kapitel 3 führt die Evolutionsgleichungen für regularisierte Dirac-harmonische Abbildungen ein. Zusätzlich wird die Evolution von verschiedenen Energien diskutiert. Schließlich wird die Existenz einer Kurzzeitlösung bewiesen. In Kapitel 4 werden die Evolutionsgleichungen für den Fall analysiert, dass die Ursprungsmannigfaltigkeit eine geschlossene Kurve ist. Die Existenz einer Langzeitlösung der Evolutionsgleichungen wird bewiesen. Es wird außerdem gezeigt, dass die Evolutionsgleichungen konvergieren, falls die Regularisierung groß genug gewählt wurde. Schließlich wird diskutiert, ob die Regularisierung wieder entfernt werden kann. Kapitel 5 schlussendlich untersucht die Evolutionsgleichungen für den Fall, dass die Ursprungsmannigfaltigkeit eine geschlossene Riemannsche Spin Fläche ist. Es wird die Existenz einer global schwachen Lösung bewiesen, welche bis auf endlich viele Singularitäten glatt ist. Die Lösung konvergiert im schwachen Sinne gegen eine regularisierte Dirac-harmonische Abbildung. Auch hier wird schließlich untersucht, ob die Regularisierung wieder entfernt werden kann.

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12

Al-Hussein, Abdul Rahman. "Backward stochastic evolution equations in infinite dimensions." Thesis, University of Warwick, 2002. http://wrap.warwick.ac.uk/59378/.

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13

Pytka, Mario Antonio. "Testing nonlinear evolution equations for exact solubility." Thesis, University of Leeds, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236765.

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14

Quiroga, Gonzáles Cruz Sonia, Juan Límaco, and Rioco K. Barreto. "The penalty method and beam evolution equations." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96079.

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15

Achache, Mahdi. "Maximal regularity for non-autonomous evolution equations." Thesis, Bordeaux, 2018. http://www.theses.fr/2018BORD0026/document.

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Cette thèse est dédiée a l''etude de certaines propriétés des équations d' évolutions non-autonomes $u'(t)+A(t)u(t)=f(t), u(0)=x.$ Il s'agit précisément de la propriété de la régularité maximale $L^p$: étant donnée $fin L^{p}(0,tau;H)$, montrer l'existence et unicité de la solution $u in W^{1,p}(0,tau;H)$. Ce problème a 'et'e intensivement étudie dans le cas autonome, i.e., $A(t)=A$ pour tout $t$. Dans le cas non-autonome, le problème a été considéré par J.L.Lions en 1960. Nous montrons divers résultats qui étendent tout ce qui est connu sur ce problème. On suppose ici que la famille des opérateurs $(mathcal{A}(t))_{tin [0,tau]}$ est associée à des formes quasi-coercives, non autonomes $(fra(t))_{t in [0,tau]}.$ Nous considérons également le problème de régularité maximale pour les d'ordre 2 (équations des ondes). Plusieurs exemples et applications sont considérés
This Thesis is devoted to certain properties of non-autonomous evolution equations $u'(t)+A(t)u(t)=f(t), u(0)=x.$ More precisely, we are interested in the maximal $L^p$-regularity: given $fin L^{p}(0,tau;H),$ prove existence and uniqueness of the solution $u in W^{1,p}(0,tau;H)$. This problem was intensively studied in the autonomous cas, i.e., $A(t)=A$ for all $t.$ In the non-autonomous cas, the problem was considered by J.L.Lions in 1960. We prove serval results which extend all previously known ones on this problem. Here we assume that the familly of the operators $(mathcal{A}(t))_{tin [0,tau]}$ is associated with quasi-coercive, non-autonomous forms $(fra(t))_{t in [0,tau]}.$ We also consider the problem of maximal regularity for second order equations (the wave equation). Serval examples and applications are given in this Thesis

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16

Sorensen, Julian Karl. "White noise analysis and stochastic evolution equations." Title page, contents and abstract only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phs713.pdf.

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17

AFFILI, ELISA. "EVOLUTION EQUATIONS WITH APPLICATIONS TO POPULATION DYNAMICS." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/820854.

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The main topic of this thesis is the analysis of evolution equations reflecting issues in ecology and population dynamics. In mathematical modelling, the impact of environmental elements and the interaction between species is read into the role of heterogeneity in equations and interactions in coupled systems. In this direction, we investigate three separate problems, each corresponding to a chapter of this thesis. The first problem addresses the evolution of a single population living in a periodic medium with a fast diffusion line; this corresponds to the study of a reaction-diffusion system with equations in different dimensions. We derive results on asymptotic behaviour through the study of some generalised principal eigenvalues. We find that the road has no impact on the survival chances of the population, despite the deleterious effect expected from fragmentation. The second investigation regards a model describing the competition between two populations in a situation of asymmetrically aggressive interactions; this consists of a system of two ODEs. The evolution progresses through two possible scenarios, where only one population survives. Then, the interpretation of one of the parameters as the aggressiveness of the attacker population naturally raises questions of controllability. We characterise the set of initial conditions leading to the victory of the attacker through a suitable (possibly time-dependant) strategy. The third and last part of this thesis analyses the time decay of some evolution equations with classical and fractional time derivatives. Depending on the type of derivative and some degree of non-degeneracy of the spatial operator, quantitative polynomial or exponential estimates are entailed.

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18

Porter, Annabelle Louise. "The evolution of equation-solving: Linear, quadratic, and cubic." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3069.

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This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. It uses a historical perspective to highlight many of the concepts underlying modern equation solving.

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19

Mai, Thanh Tan [Verfasser]. "Stochastic partial differential equations corresponding to time-inhomogeneous evolution equations / Thanh Tan Mai." München : Verlag Dr. Hut, 2012. http://d-nb.info/1029399719/34.

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20

Kerce, James Clayton. "Geometric problems relating evolution equations and variational principles." Diss., Georgia Institute of Technology, 2000. http://hdl.handle.net/1853/28739.

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21

Mohamad, Mohd Nor Bin. "Travelling wave solutions for some nonlinear evolution equations." Thesis, University of Newcastle Upon Tyne, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.238938.

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22

Morrison, Alan James. "Soliton solutions of some novel nonlinear evolution equations." Thesis, University of Strathclyde, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.248782.

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23

Hanaç, Esen. "The large-time solution of nonlinear evolution equations." Thesis, University of Birmingham, 2015. http://etheses.bham.ac.uk//id/eprint/6091/.

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In this thesis we use the method of matched asymptotic coordinate expansions to examine in detail the structure of the large-time solution of a range of initial-value and initial-boundary value problems based on Burgers' equation or the related Burgers-Fisher equation. The normalized nonlinear partial differential equations considered are: (i) Burgers' equation Ut + UUx - Uxx = 0. (ii) Burgers-Fisher equation Ut + kuux = Uxx + u( 1 - u). Here x and t represent dimensionless distance and time, respectively, while k (≠ 0) is a constant. In particular, we are interested in the emergence of coherent structures (for example: expansion waves, stationary states and travelling waves) in the large-time solution of the problems considered.

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24

Hoang, Duc-Trung. "Controllability and observability of non autonomous evolution equations." Thesis, Bordeaux, 2018. http://www.theses.fr/2018BORD0083/document.

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Cette thèse est consacrée à la contrôlabilité et à l’observabilité de l’équation d’évolution non autonome. Dans la première partie, nous donnons un aperçu de la théorie du contrôle ainsi que quelques résultats classiques sur le contrôle des systèmes autonomes et non autonomes. En fait, nous rappellerons les techniques de la théorie des semi-groupes, théorie de l’évolution familiale, théorie de la dualité et de l’opérateur. Dans la deuxième partie, nous sommes intéressés à étudier le problème de contrôle pour les systèmes EDP définis sur des domaines dépendant du temps. Nous développons de nouvelles techniques pour obtenir les résultats sur l’observabilité exacte des équations de l’onde et de Schrödinger 1D, puis par dualité nous établissons la contrôlabilité exacte du système adjoint. Le dernier résultat est une généralisation des tests de Hautus pour l’observabilité du système d’évolution non autonome. Notre méthode peut s’appliquer aux équations de Schrödinger et à l’équation d’onde avec des potentiels dépendant du temps
This thesis is devoted to the controllability and observability of nonautonomous evolution equation. In the first part, we give an overview on control theory as well as some classical results on control of both autonomous and nonautonomous systems. In fact, we will recall the technique in semigroup theory, evolution familys theory, duality theory and operator theory. In the second part, we are interested to investigate the control problem for PDEs systems defined on time-dependent domains. We develope some new techniques to obtain the results on exact observability for one dimensional wave and Schrödinger equations, then by duality we establish exact controllability of adjoint system. The last result is a generalization of Hautus tests for observability of non- autonomous evolution system.Our method can apply for Schrodinger equations with time dependent potentials and to a damped wave-equation with time-dependent damping

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25

Pinto, João Teixeira. "Slow motion manifolds for a class of evolutionary equations." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/29342.

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26

Kok, Tayfun. "Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equation." Thesis, University of York, 2017. http://etheses.whiterose.ac.uk/18070/.

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The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.

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27

Bocharov, Boris. "Stochastic evolution inclusions." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/3772.

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This work is concerned with an evolution inclusion of a form, in a triple of spaces \V -> H -> V*", where U is a continuous non-decreasing process, M is a locally square-integrable martingale and the operators A (multi-valued) and B satisfy some monotonicity condition, a coercivity condition and a condition on growth in u. An existence and uniqueness theorem is proved for the solutions, using semi-implicit time-discretization schemes. Examples include evolution equations and inclusions driven by square integrable Levy martingales.

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28

Alsulami, Saud M. A. "On Evolution Equations in Banach Spaces and Commuting Semigroups." Ohio University / OhioLINK, 2005. http://www.ohiolink.edu/etd/view.cgi?ohiou1126042587.

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29

Lee, Haewon. "Nonlinear evolution equations and optimization problems in Banach spaces." Ohio : Ohio University, 2005. http://www.ohiolink.edu/etd/view.cgi?ohiou1127498683.

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30

Nabti, Abderrazak. "Non linear, non-local evolution equations : theory and application." Thesis, La Rochelle, 2015. http://www.theses.fr/2015LAROS032.

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Cette thèse concerne l’étude qualitative (existence locale, existence globale, explosion en temps fini) de quelques équations de Schrödinger non-linéaires non-locales. Dans le cas où les solutions explosent en temps fini, l’estimation du temps maximal d’existence des solutions sera présentée. Le chapitre 1 concerne l’étude d’une équation de Schrödinger non-linéaire sur RN. On s’intéresse à l’existence locale d’une solution pour toute condition initiale donnée dans L2(RN). De plus, on montre que la norme-L2 de la solution explose en temps fini T < 1. Les démonstrations reposent essentiellement sur le théorème de point fixe de Banach et les estimations de Strichartz, et aussi sur le choix convenable de la fonction test dans la formulation faible du problème. Dans le chapitre 2, on considère une équation de Schrödinger non-linéaire non-locale en temps, et on démontre que les solutions de notre problème explosent en temps fini ; ensuite on obtient des conditions nécessaires d’existence globale. Finalement, on obtient une borne inférieure du temps maximal d’existence de la solution. Le chapitre 3 porte sur la non-existence de solutions d’une équation de Schrödinger non-linéaire posée dans RN. Dans un premier temps, sous certaines conditions sur la donnée initiale, on montre qu’il n’existe pas de solution faible globale ; puis on donne une estimation du temps maximal d’existence de la solution. Enfin, on établit des conditions d’existence locale, ou globale de l’équation considérée. En plus, on généralise les résultats précédents au cas d’un système 2 _ 2. Le dernier chapitre traite une équation de Schrödinger non-linéaire non-locale en temps sur le groupe de Heisenberg H. En utilisant la méthode de la fonction test, on démontre que l’équation n’admet pas de solution faible globale. De plus, on obtient, sous certaines conditions sur les données initiales, une estimation inférieure du temps maximal d’existence de la solution
Our objective in this thesis is to study the existence of local solutions, existence global and blow up of solutions at a finite time to some nonlinear nonlocal Schrödinger equations. In the case when a solution blows-up at a finite time T < 1, we obtain an upper estimate of the life span of solutions. In the first chapter, we consider a nonlinear Schrödinger equation on RN. We first prove local existence of solution for any initial condition in L2 space. Then we prove nonexistence of a nontrivial global weak solution. Furthermore, we prove that the L2-norm of the local intime L2-solution blows up at a finite time. The second chapter is dedicated to study an initial value problem for the nonlocal intime nonlinear Schrödinger equation. Using the test function method, we derive a blow-up result. Then based on integral inequalities, we estimate the life span of blowing-up solutions. In the chapter 3, we prove nonexistence result of a space higher-order nonlinear Schrödinger equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided. Next, we extend our results to the 2 _ 2-system. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Finally, we consider a nonlinear nonlocal in time Schrödinger equation on the Heisenberg group. We prove nonexistence of non-trivial global weak solution of our problem. Furthermore, we give an upper bound of the life span of blowing up solutions

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31

Cordero, Carrión Isabel. "Evolution formalisms of Einstein equations: Numerical and Geometrical Issues." Doctoral thesis, Universitat de València, 2009. http://hdl.handle.net/10803/31814.

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The topic treated along this thesis is the theoretical and numerical study of formalisms of Einstein equations, with the final aim of applications to black holes and gravitational waves. The General Relativity theory of Einstein (1915) postulated that light and trajectories of all particles are curved by the geometry of spacetime. Schwarzschild a few months later and Kerr in 1963 found solutions which describe non-rotating and rotating black holes. From an astrophysical point of view, a stellar black hole can be seen as the final result of some kind of collapse of massive stars or merger of compact binaries objects. One of the predicted consequences of General Relativity, not detected yet, is the existence of gravitational waves. This is the only direct method for detecting black holes. These waves can be viewed as ripples in the curvature of spacetime caused by non-spherically symmetric accelerations of matter. The first indirect detection was in 1974 by Hulse and Taylor, and they were awarded the Nobel. Huge experimental, theoretical and numerical efforts have been carried out in the last forty years, from the resonant bars of Weber to the future space-based interferometers as LISA. The General Relativity theory describes scenarios involving strong gravitational fields and velocities close to light velocity. The different formalisms lead to write Einstein equations as a set of partial differential equations. We must recognize the capability of the most used ones, as the so-called BSSN (Baumgarte-Shapiro-Shibata-Nakamura), crucial in the recent simulations of binary black holes. One of the recent formalisms is the FCF (Fully Constrained Formalism), which will be object of study along the thesis. In FCF, Einstein equations are written as a set of elliptic-hyperbolic equations, where the constraints are solved in each time step. It is a natural generalization of the relativistic approximation CFC (Conformally Flat Condition), used in many astrophysical applications. The theoretical work done in the thesis is very important, as the proof of the local existence of maximal slicings in spherically symmetric spacetimes. Moreover, the resulting equations in FCF have been studied mathematically. On one hand, the introduction of a new vector allows rewriting the elliptic equations such that local uniqueness is guaranteed and the equations form a hierarchical system. This is a very important in order to guarantee the well-posedness of the whole system. Numerical problems appear as consequence of the theoretical ones, and it was no possible to compute the migration test of a rotating neutron star and the spherical and rotational collapse to a black hole in the CFC approximation (and, so, in the FCF). The hyperbolic properties of the evolution system have also been studied. The explicit expressions of the eigenvalues are very useful in the study of inner boundary conditions of trapping horizons in which the singularity is removed from the numerical grid. The numerical work done in the thesis has as objective the extension of the CoCoNuT code to the FCF, in order to simulate non-vacuum dynamical spacetimes, including magnetic fields. We have performed the evolution of Teukolsky waves, analytical solution in vacuum and in linear regime, and the evolution of stationary rotating and perturbed rotating neutron stars. The next step will be the extraction of the gravitational signal in astrophysical scenarios and to compare the results with other approximations, as the quadrupole formula.
El tema de la tesis es el estudio teórico y numérico de los formalismos de las ecuaciones de Einstein, con aplicaciones a la formación de agujeros negros y generación de ondas gravitatorias. La teoría de la Relatividad General de Einstein (1915) postulaba que la luz y las trayectorias de las partículas eran curvadas por la geometría del espacio tiempo. Schwarzschild (1915) y Kerr (1963) encontraron las soluciones que describen agujeros negros estático y en rotación. Desde un punto de vista astrofísico, un agujero negro estelar es el resultado de algunos tipos de colapso o la fusión de binarias de objetos compactos. Las ondas gravitatorias, predichas por la Relatividad General, aún no detectadas, son el único método directo para detectar agujeros negros. Son arrugas en la curvatura del espacio-tiempo. La primera detección indirecta por Hulse y Taylor (1974) les valió el Nobel. Enormes esfuerzos experimentales se han llevado a cabo en los últimos cuarenta años, desde las barras resonantes de Weber hasta los futuros observatorios espaciales como LISA. La Relatividad General describe escenarios que involucran campos gravitatorios intensos y velocidades próximas a la de la luz. En los diferentes formalismos las ecuaciones de Einstein se escriben como diferentes sistemas de ecuaciones en derivadas parciales. BSSN ha sido crucial en las recientes simulaciones de binarias de agujeros negros. FCF, introducido recientemente, ha sido objeto de estudio en la tesis. Las ligaduras se resuelven en cada paso de tiempo y es una generalización natural de la aproximación relativista CFC. El trabajo teórico realizado es muy importante: la prueba de la existencia local de foliaciones maximales en espacios-tiempo con simetría esférica; la introducción de un campo vectorial en las ecuaciones elípticas de FCF, que permite garantizar la unicidad local; el estudio de la hiperbolicidad de las ecuaciones de evolución en FCF, con aplicación a horizontes atrapados de agujeros negros. El trabajo numérico se centra en la extensión del código numérico CoCoNuT a la formulación FCF, para poder simular espacios-tiempo dinámicos con materia, incluyendo campos magnéticos. Varios tests satisfactorios permiten pensar en la extracción de la radiación gravitatoria en escenarios más complejos.

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Li, Linzhong. "Numerical study of nonlinear evolution equations, using compact differencing." Thesis, University College London (University of London), 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.286370.

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Zhao, Q. "Optimal control and asymptotics of stochastic delay evolution equations." Thesis, University of Liverpool, 2017. http://livrepository.liverpool.ac.uk/3014620/.

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Lee, Haewon. "Nolinear Evolution Equations and Optimization Problems in Banach Spaces." Ohio University / OhioLINK, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1127498683.

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Sapountzoglou, Niklas [Verfasser], and Petra [Akademischer Betreuer] Wittbold. "Doubly nonlinear evolution equations / Niklas Sapountzoglou ; Betreuer: Petra Wittbold." Duisburg, 2020. http://d-nb.info/1206538287/34.

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Nguyen, Thi. "On the Evolution of Virulence." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/91.

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The goal of this thesis is to study the dynamics behind the evolution of virulence. We examine first the underlying mechanics of linear systems of ordinary differential equations by investigating the classification of fixed points in these systems, then applying these techniques to nonlinear systems. We then seek to establish the validity of a system that models the population dynamics of uninfected and infected hosts---first with one parasite strain, then n strains. We define the basic reproductive ratio of a parasite, and study its relationship to the evolution of virulence. Lastly, we investigate the mathematics behind superinfection.

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Lam, Chun-kit, and 林晉傑. "The dynamics of wave propagation in an inhomogeneous medium: the complex Ginzburg-Landau model." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40887881.

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Lam, Chun-kit. "The dynamics of wave propagation in an inhomogeneous medium the complex Ginzburg-Landau model /." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40887881.

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Yip, Lai-pan. "Nonlinear and localized modes in hydrodynamics and vortex dynamics." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/hkuto/record/B39316919.

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Yip, Lai-pan, and 葉禮彬. "Nonlinear and localized modes in hydrodynamics and vortex dynamics." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B39316919.

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Benedikter, Niels [Verfasser]. "Effective Evolution Equations from Many-Body Quantum Mechanics / Niels Benedikter." Bonn : Universitäts- und Landesbibliothek Bonn, 2014. http://d-nb.info/1052061079/34.

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Kerbal, Sebti. "Existence of optimal controls for second-order nonlinear evolution equations." Thesis, University of Ottawa (Canada), 1993. http://hdl.handle.net/10393/6919.

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In this thesis we study the question of existence of optimal controls for systems governed by second order nonlinear evolution equations. Let I = ($0,\ T$), $(X, H, X\sp\*)$ be an evolution triple, with compact embedding $X\to H\to X\sp\*$ and Y a separable, reflexive Banach space, modeling the control space. Here $X\sp\*$ denote the dual of the Banach space X. Let $t\to U(t)$ be a measurable set-valued map with values $U(t)\in 2\sp{Y}.$ For admissible controls, we introduce the class ${\cal U}\sb{ad}$ given by $U\sb{ad}\equiv\{ u: I\mapsto Y$, strongly measurable, and $u(t)\in U(t) a.e.\}$. We consider the following Lagrange type optimal control problem: $$\left\{\eqalign{&J(x,u) = f\sbsp{0}{T} L(t,x (t), \dot x(t), u(t))dt\ \to\inf\cr&subject\ to\ the\ following\ state\ and\ control\ constraints:\cr&\ddot x(t) + A(t,\dot x(t) + Bx(t)) = f(t,x(t))u(t),\cr& x(0) =x\sb0\in X, \dot x(0) = x\sb1\in H, u(t)\in U(t)\ a.e.\cr}\right\}(P)$$ To establish the existence of an optimal pair $\{$x,u$\}$ for the problem (P), an appropriate hypotheses on the data have been introduced and some apriori bounds for the admissible trajectories of (P) have been derived.

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Lundberg, Staffan. "On adjoint symmetries and reciprocal Bäcklund transformations of evolution equations." Licentiate thesis, Luleå tekniska universitet, Matematiska vetenskaper, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-26553.

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The aim of this Licentiate Thesis is to discuss special transformations and so-called adjoint symmetries of nonlinear partial differential equations. Nonlinear partial differential equations play an important role in the description of many physical phenomena. In order to understand the phenomena, modelled by the equations mentioned above, it is therefore necessary to obtain and analyze the solutions and the conservation laws of these equations. In this Thesis we investigate some methods to obtain conservation laws and transformations between nonlinear partial differential equations and moreover to classify nonlinear partial differential equations with respect to those methods.The main emphasis is on adjoint symmetries and transformations of evolution equations. In particular we study the adjoint symmetries and the construction of reciprocal Bäcklund transformations for evolution equations.
Godkänd; 2009; 20090115 (lund); Licentiatseminarium för avläggande av teknologie licentiatexamen. Examinator: Docent Marianna Euler, Luleå tekniska universitet Tid: Fredag den 20 mars 2009 kl 10.15 Plats: D2222, Luleå tekniska universitet

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Ferguson, James. "Geometric structures on the target space of Hamiltonian evolution equations." Thesis, Connect to e-thesis, 2008. http://theses.gla.ac.uk/206/.

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Thesis (Ph.D.) - University of Glasgow, 2008.
Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, 2008. Includes bibliographical references. Print version also available.

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Zhang, Yongjin [Verfasser]. "Model order reduction for parameterized nonlinear evolution equations / Yongjin Zhang." Magdeburg : Universitätsbibliothek, 2016. http://d-nb.info/1113687304/34.

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Baroun, Mahmoud [Verfasser]. "Asymptotic Behavior and Observability of Semilinear Evolution Equations / Mahmoud Baroun." Wuppertal : Universitätsbibliothek Wuppertal, 2012. http://d-nb.info/1029845085/34.

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Treharne, Philip Arthur. "Boundary value problems for linear and integrable nonlinear evolution equations." Thesis, University of Cambridge, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.614768.

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Lundberg, Staffan d1952. "On adjoint symmetries and reciprocal Bäcklund transformations of evolution equations /." Luleå : Department of Mathematics, Luleå University of Technology, 2009. http://pure.ltu.se/ws/fbspretrieve/2505404.

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Yu, Edmund Po-ning. "Evolution equations for magnetic islands in a reversed field pinch." Access restricted to users with UT Austin EID, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3037030.

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von, Glehn Ingrid. "A closest point penalty method for evolution equations on surfaces." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:29385f90-b927-4151-b5df-cf877cef00ef.

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This thesis introduces and analyses a numerical method for solving time-dependent partial differential equations (PDEs) on surfaces. This method is based on the closest point method, and solves the surface PDE by solving a suitably chosen equation in a band surrounding the surface. As it uses an implicit closest point representation of the surface, the method has the advantages of being simple to implement for very general surfaces, and amenable to discretization with a broad class of numerical schemes. The method proposed in this work introduces a new equation in the embedding space, which satisfies a key consistency property with the surface PDE. Rather than alternating between explicit time-steps and re-extensions of the surface function as in the original closest point method, we investigate an alternative approach, in which a single equation can be solved throughout the embedding space, without separate extension steps. This is achieved by creating a modified embedding equation with a penalty term, which enforces a constraint on the solution. The resulting equation admits a method of lines discretization, and can therefore be discretized with implicit or explicit time-stepping schemes, and analysed with standard techniques. The method can be formulated in a straightforward way for a large class of problems, including equations featuring variable coefficients, higher-order terms or nonlinearities. The effectiveness of the method is demonstrated with a range of examples, drawing from applications involving curvature-dependent diffusion and systems of reaction-diffusion equations, as well as equations arising in PDE-based image processing on surfaces.

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