In this dissertation, we present some analysis of Ricci flow on complete noncompact manifolds. The first half of the dissertation concerns the formation of Type-II singularity in Ricci flow on [mathematical equation]. For each [mathematical equation] , we construct complete solutions to Ricci flow on [mathematical equation] which encounter global singularities at a finite time T such that the singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate [mathematical equation]. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on [mathematical equation] whose blow-ups near the origin converge uniformly to the Bryant soliton. In the second half of the dissertation, we fully analyze the structure of the Lichnerowicz Laplacian of a Bergman metric g[subscript B] on a complex hyperbolic space [mathematical equation] and establish the linear stability of the curvature-normalized Ricci flow at such a geometry in complex dimension [mathematical equation]. We then apply the maximal regularity theory for quasilinear parabolic systems to prove a dynamical stability result of Bergman metric on the complete noncompact CH[superscript m] under the curvature-normalized Ricci flow in complex dimension [mathematical equation]. We also prove a similar dynamical stability result on a smooth closed quotient manifold of [mathematical symbols]. In order to apply the maximal regularity theory, we define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties. / text

Inspired by studies on the regularity of solutions to the fractional Navier-Stokes system and the impact of noise on singularity formation in hydrodynamic models, we investigated these issues within the framework of the fractional 1D Burgers equation. Initially, our research concentrated on the deterministic scenario, where we conducted precise numerical computations to understand the dynamics in both subcritical and supercritical regimes. We utilized a pseudo-spectral approach with automated resolution refinement for discretization in space combined with a hybrid Crank-Nicolson/ Runge-Kutta method for time discretization.We estimated the blow-up time by analyzing the evolution of enstrophy (H1 seminorm) and the width of the analyticity strip. Our findings in the deterministic case highlighted the interplay between dissipative and nonlinear components, leading to distinct dynamics and the formation of shocks and finite-time singularities. In the second part of our study, we explored the fractional Burgers equation under the influence of linear multiplicative noise. To tackle this problem, we employed the Milstein Monte Carlo approach to approximate stochastic effects. Our statistical analysis of stochastic solutions for various noise magnitudes showed that as noise amplitude increases, the distribution of blow-up times becomes more non-Gaussian. Specifically, higher noise levels result in extended mean blow-up time and increase its variability, indicating a regularizing effect of multiplicative noise on the solution. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. Although the trends are rather weak, they nevertheless are consistent with the predictions of the theorem of [41]. However, there is no evidence for a complete elimination of blow-up, which is probably due to the fact that the noise amplitudes considered were not sufficiently large. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. / Thesis / Master of Science (MSc)

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.