Passive Scalar Transport by Non-Smooth Incompressible Fluids: Mixing and Vanishing Viscosity

Abstract

This thesis explores fundamental questions in fluid dynamics through rigorous mathematical analysis of the passive scalar transport model. Our investigation centers on the behaviour of fluid flows characterised by vector fields of lower regularity—a crucial feature in understanding turbulent dynamics. Through careful examination of these flows in various function spaces, particularly Sobolev spaces, we develop new analytical tools and insights into three key areas: well-posedness, regularity, and solution selection. The first major contribution introduces a novel weak compactness technique that yields improved quantitative estimates for the transport equation. This approach leads to several significant advances, including enhanced classical mixing estimates with exponential lower bounds, propagation of mild logarithmic fractional regularity, and state-of-the-art weak stability estimates for transport along Sobolev vector fields. Most notably, we establish the first quantitative stability estimate for transport along vector fields with bounded variation, marking progress on the challenging p=1 case of Bressan's conjecture. Our second principal contribution extends to the analysis to the transport-diffusion equation, where we develop techniques beyond standard energy estimates. By combining mild solutions, weak convolution estimates, and maximal regularity methods, we establish new results under the Ladyzhenskaya-Prodi-Serrin integrability condition on the vector. These methods effectively capture the interplay between transport and diffusion on regularisation, leading to improved uniqueness and regularity results. The final contribution challenges conventional approaches to solution selection through vanishing diffusion limits. Through explicit constructions, we demonstrate that the vanishing diffusion approach fails to consistently select physically meaningful solutions for the passive scalar transport model. Our results show that this method can produce solutions violating basic thermodynamic principles, including time-arrow reversal—a finding that questions traditional approaches to solution selection in fluid dynamics. These contributions advance our understanding of irregular fluid flows while raising important questions about current mathematical frameworks in fluid mechanics. The thesis concludes by identifying critical open problems, particularly regarding the well-posedness of turbulent fluid flows and the development of alternative approaches to solution selection.