We study the initial value problem for the defocusing nonlinear wave equation with cubic nonlinearity F(u)=|u|^2u in the energy-supercritical regime, that is dimensions d\geq 5. We prove that solutions to this equation satisfying an a priori bound in the critical homogeneous Sobolev space exist globally in time and scatter in the case of spatial dimensions d\geq 6 with general (possibly non-radial) initial data, and in the case of spatial dimension d=5 with radial initial data. / text

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22863号 / 情博第742号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 准教授 柴山 允瑠, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.

Thesis (Ph.D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains ix, 176 p.; also includes graphics. Includes bibliographical references (p. 172-176). Available online via OhioLINK's ETD Center

In dieser Arbeit entwickeln und untersuchen wir ein numerisches Schema für die eindimensionale Wellengleichung mit Hysterese für unterschiedliche Arten von Randbedingungen. Diese Gleichung ist ein Modell für die Longitudinal- oder Torsionsschwingungen eines homogenen Stabes unter dem Einfluß einer uniaxialen äußeren Kraftdichte, wobei wir ein elastoplastisches Materialgesetz annehmen. Hysterese-Operatoren sind ratenunabhängige Volterra-Operatoren, die Zeitfunktionen in Zeitfunktionen abbilden. Mit ihnen lassen sich Gedächtniseffekte modellieren, wie sie zum Beispiel in der Elastoplastizität oder im Ferromagnetismus auftauchen. Zunächst führen wir Hysterese-Operatoren allgemein ein und analysieren dann eine spezielle Klasse von Hysterese-Operatoren, die Prandtl-Ishlinskii-Operatoren. Wir untersuchen ihre Gedächtnisstruktur und erklären, wie sich die Operatoren numerisch auswerten lassen. Dazu stellen wir zwei verschiedene Approximationsansätze vor. Wir führen aus, wie sich die approximierenden Operatoren implementieren lassen und leiten lineare und quadratische Fehlerabschätzungen her. Zur numerischen Lösung des gekoppelten Systems aus der Wellengleichung mit einem Hysterese-Operator führen wir ein implizites Differenzenschema mit Gedächtnis ein. Für eine Klasse von Hysterese-Operatoren zeigen wir die Existenz und Eindeutigkeit der Lösung des numerischen Schemas, beweisen mit Hilfe von Kompaktheitsschlüssen und einem Monotonieargument die Konvergenz des Verfahrens und leiten eine Fehlerabschätzung der Ordnung 1/2 her. Wir diskutieren, wie das vorgestellte Verfahren auf die Prandtl-Ishlinskii-Operatoren angewendet werden kann. / In this thesis we develop and investigate a numerical scheme for the one-dimensional wave equation with hysteresis for different kinds of boundary conditions. This equation can be regarded as a model for the longitudinal or torsional oscillations of a homogeneous bar under the influence of an uniaxial external force density assuming an elastoplastic material law. Hysteresis operators are rate-independent Volterra operators mapping time functions to time functions. This kind of operator can be used to model memory effects as they appear in elastoplasticity or ferromagnetism, for example. We first give an introduction to the general concept of hysteresis operators before we analyze a special class of hysteresis operators called Prandtl-Ishlinskii operators. We investigate their memory structure and explain how the operators can be evaluated numerically. To that end we present two different kinds of approximation schemes. We point out how the approximating operators can be implemented and we derive linear and quadratic error estimates. For the numerical solution of the coupled system of the wave equation with a hysteresis operator we introduce an implicit difference scheme with memory. For a class of hysteresis operators we show the existence and uniqueness of the numerical solution. We prove the convergence of the scheme by compactness and monotonicity arguments. We derive an error estimate of order 1/2. We discuss the application of the method presented to Prandtl-Ishlinskii operators.

The PhD thesis deals with global in time existence results and blow-up result for a semilinear wave model with scale-invariant damping and mass. Since the time-dependent coefficients for the considered model make somehow the damping and the mass a threshold term between effective and non-effective terms, it turns out that a fundamental role in the description of qualitative properties of solutions to this semilinear model and to the corresponding linear homogeneous Cauchy problem is played by the multiplicative constants appearing in those coefficients. For coefficients that make the damping term dominant, we can use the standard approach for the classical damped wave model with L^2 − L^2 estimates and the so-called test function method. On the other hand, when the interaction among those coefficients is balanced, then, it is possible to observe how typical tools for hyperbolic models, as for example Kato’s lemma, provide sharp global in time existence results and sharp blow-up results for super- and sub-Strauss type exponents, respectively.

Nous étudions l’analyse asymptotique en relativité générale sous deux aspects: le peeling et le scattering (diffusion) conforme. Le peeling est construit pour les champs scalaires linéaire et non-linéaires et pour les champs de Dirac en espace-temps de Kerr (qui est non-stationnaire et à symétrie simplement axiale), généralisant les travaux de L. Mason et J-P. Nicolas (2009, 2012). La méthode des champs de vecteurs (estimations d’énergie géométriques) et la technique de compactification conforme sont développées. Elles nous permettent de formuler les définitions du peeling à tous ordres et d’obtenir les données initiales optimales qui assurent ces comportements. Une théorie de la diffusion conforme pour les équations de champs sans masse de spîn n/2 dans l’espace-temps de Minkowski est construite.En effectuant les compactifications conformes (complète et partielle), l’espace-temps est complété en ajoutant une frontière constituée de deux hypersurfaces isotropes représentant respectivement les points limites passés et futurs des géodésiques de type lumière. Le comportement asymptotique des champs s’obtient en résolvant le problème de Cauchy pour l’équation rééchelonnée et en considérant les traces des solutions sur ces bords. L’inversibilité des opérateurs de trace, qui associent le comportement asymptotique passé ou futur aux données initiales, s’obtient en résolvant le problème de Goursat sur le bord conforme. L’opérateur de diffusion conforme est alors obtenu par composition de l’opérateur de trace futur avec l’inverse de l’opérateur de trace passé. / This work explores two aspects of asymptotic analysis in general relativity: peeling and conformal scattering.On the one hand, the peeling is constructed for linear and nonlinear scalar fields as well as Dirac fields on Kerr spacetime, which is non-stationary and merely axially symmetric. This generalizes the work of L. Mason and J-P. Nicolas (2009, 2012). The vector field method (geometric energy estimates) and the conformal technique are developed. They allow us to formulate the definition of the peeling at all orders and to obtain the optimal space of initial data which guarantees these behaviours. On the other hand, a conformal scattering theory for the spin-n/2 zero rest-mass equations on Minkowski spacetime is constructed. Using the conformal compactifications (full and partial), the spacetime is completed with two null hypersurfaces representing respectively the past and future end points of null geodesics. The asymptotic behaviour of fields is then obtained by solving the Cauchy problem for the rescaled equation and considering the traces of the solutions on these hypersurfaces. The invertibility of the trace operators, that to the initial data associate the future or past asymptotic behaviours, is obtained by solving the Goursat problem on the conformal boundary. The conformal scattering operator is then obtained by composing the future trace operator with the inverse of the past trace operator.