The Convex Integration Paradigm in Stochastic Fluid Dynamics

This dissertation investigates the pivotal role of convex integration in the analysis of stochastic partial differential equations in the realm of fluid dynamics, with particular emphasis on the incompressible Navier--Stokes equations and shear-thinning fluid flows. <br /><br /> The method of convex integration, discussed in Part I, is a highly flexible technique in the analysis of stochastic partial differential equations, enabling the construction of solutions in the analytically weak and probabilistically strong sense. Through an iterative process, it introduces controlled perturbations into the underlying equation that progressively refine approximate solutions while preserving key constraints. This approach has been particularly instrumental in revealing non-uniqueness phenomena and irregular behavior in fluid models, offering deep insights into the mathematical structure of turbulence and anomalous dissipation. <br /><br /> In Part II, we consider the stochastic Navier--Stokes equations in three dimensions with linear multiplicative stochastic forcing and certain initial data. Prescribing the kinetic energy within the framework of convex integration enables the construction of analytically weak and probabilistically strong solutions with deterministic initial values, defined up to an arbitrarily large stopping time. The selection of different energy profiles result in non-uniqueness of solutions. If the underlying energies are additionally chosen to be equal over a small time interval close to time zero, the corresponding solutions will coincide on that interval as well. <br /><br /> Part III is dedicated to the power-law equations in $d\geq 3$ dimensions with power indices ranging from $1$ to $\frac{2d}{d+2}$, driven by an additive noise of trace-class. In contrast to the first result, we employ advanced convex integration techniques incorporating expectations to overcome the limitation of stopping times and achieve global-in-time bounds. Along with a Krylov--Bogoliubov argument, this leads to the stationarity of solutions, while Krein--Milman's theorem even ensures their ergodicity. Additionally, a novel energy-related functional is introduced into the convex integration scheme, enabling not only the construction of ergodic solutions but also of Leray--Hopf solutions, i.e. solutions that adhere to an energy inequality. Non-uniqueness of these ergodic Leray--Hopf solutions follows again through the selection of different auxiliary energies. <br /><br /> Finally, Part IV provides an overview of hemodynamics, focusing on the applied aspects of fluid dynamics in biological systems, particularly in the cardiovascular system, where the Navier--Stokes equations and shear-thinning fluids play crucial roles in providing a more accurate description of blood flow. Rather than delving into specific models, this section explores the physiological and biomechanical principles that motivate their development, such as the complex rheology of blood and the influence of vessel structures. Given its medical relevance, this part bridges theoretical results with real-world applications in biomechanics and clinical research.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2025

Author dc:creator

• Berkemeier, Stefanie Elisabeth

Repository record source_url

https://pub.uni-bielefeld.de/record/3005214

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:3005214