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Gradient flows of energy functionals on the space of probability measures with Wasserstein metric has proved to be a strong tool in studying certain mass conserving evolution equations. Such gradient flows provide an alternate formulation for the solutions of the corresponding evolution equations. An important condition, which is known to guarantees existence, uniqueness, and continuous dependence on initial data is that the corresponding energy functional be displacement convex. We introduce a relaxed notion of displacement convexity and we show that it still guarantees short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals which are not displacement convex in the standard sense. This extends the applicability of the gradient flow approach to larger family of energies. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.

Gradient flows of energy functionals on the space of probability measures with Wasserstein metric has proved to be a strong tool in studying certain mass conserving evolution equations. Such gradient flows provide an alternate formulation for the solutions of the corresponding evolution equations. An important condition, which is known to guarantees existence, uniqueness, and continuous dependence on initial data is that the corresponding energy functional be displacement convex. We introduce a relaxed notion of displacement convexity and we show that it still guarantees short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals which are not displacement convex in the standard sense. This extends the applicability of the gradient flow approach to larger family of energies. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.

Abstract available on page 4 of the document / Thesis (PhD (Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Possibilities of generating plasma turbulence to provide control of space weather processes have been of particular interest in recent years. Such turbulence can be created by chemical released into a magnetized background plasma. The released plasma clouds are heavy ions which have ring velocity distribution and large free energy to drive the turbulence. An electromagnetic hybrid (fluid electrons and particle ions) model incorporating electron inertia is developed to study the generation and nonlinear evolution of this turbulence. Fourier pseudo-spectral methods are combined with finite difference methods to solve the electron momentum equations. Time integration is accomplished by a 4th-order Runge-Kutta scheme or predicator-corrector method. The numerical results show good agreement with theoretical prediction as well as provide further insights on the nonlinear turbulence evolution. Initially the turbulence lies near harmonics of the ring plasma ion cyclotron frequency and propagates nearly perpendicular to the background magnetic field as predicted by the linear theory. If the amplitude of the turbulence is sufficiently large, the quasi-electrostatic short wavelength ion cyclotron waves evolve nonlinearly into electromagnetic obliquely propagating shear Alfven waves with much longer wavelength. The results indicate that ring densities above a few percent of the background plasma density may produce wave amplitudes large enough for such an evolution to occur. The extraction of energy from the ring plasma may be in the range of 10-15% with a generally slight decrease in the magnitude as the ring density is increased from a few percent to several 10's of percent of the background plasma density. Possibilities to model the effects of nonlinear processes on energy extraction by introducing electron anomalous resistivity are also addressed. Suitability of the nonlinearly generated shear Alfven waves for applications to scattering radiation belt particles is discussed. / Master of Science

Orientador: Marcelo da Silva Montenegro / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-19T06:20:04Z (GMT). No. of bitstreams: 1 Ottoboni_RafaelRodrigo_D.pdf: 3423937 bytes, checksum: 451302f125ad3220b825af6cd34d4b52 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho apresentamos resultados sobre o fluxo de curvatura média, Gauss e harmônica de superfícies de revolução sujeito a condições de fronteira do tipo Dirichlet, Neumann ou singular. Soluções de alguns dos fluxos de curvatura com alguma destas condições de fronteira ou se anulam em tempo finito ou existem globalmente no tempo convergindo a um segmento de reta / Abstract: In this thesis we present results on mean curvature flow, Gaussian curvature flow and harmonic mean curvature flow subject to boundary conditions of Dirichlet type, Neumann or singular. Solutions to some of curvature flows with some of these boundary conditions quench in finite time or exist globally in time and converge to a straight line / Doutorado / Matematica / Doutor em Matemática

Dans le cadre de la thèse des modèles physico-mathématiques pour des microsystèmes électromécaniques avec une permittivité générale sont développés et analysés par des méthodes mathématiques modernes du domaine des équations aux dérivées partielles. En particulier ces systèmes sont à frontière libre et pour conséquence difficiles à traiter. Des méthodes numériques ont été développées pour valider les résultats analytiques obtenus. / In the framework of this thesis physical/mathematical models for microelectromechanical systems with general permittivity have been developed and analysed with modern mathematical methods from the domain of partial differential equations. In particular these systems are moving boundary problems and thus difficult to handle. Numerical methods have been developed in order to validate the obtained analytical results.