Mathematics of Open Fluid Systems

The goal of this monograph is to develop a mathematical theory of open fluid systems in the framework of continuum thermodynamics. Part I discusses the difference between open and closed fluid systems and introduces the Navier-Stokes-Fourier system as the mathematical model of a fluid in motion that will be used throughout the text. A class of generalized solutions to the Navier-Stokes-Fourier system is considered in Part II in order to show existence of global-in-time solutions for any finite energy initial data, as well as to establish the weak-strong uniqueness principle. Finally, Part III addresses questions of asymptotic compactness and global boundedness of trajectories and briefly considers the statistical theory of turbulence and the validity of the ergodic hypothesis.

Presents a mathematical theory of open fluid systems in the context of continuum thermodynamics Develops a concept of weak solutions based on entropy rather than the energy balance equation Considers the statistical theory of turbulence and the validity of the ergodic hypothesis

Résumé
"This book deals with issues related to the delicate problem of open fluid systems, where by open system we mean a system in which an exchange of matter and/or energy can take place with the surrounding environment. The topic is extremely current and relevant ... . it is unique in its kind and can represent the starting point for studies and reflections in the field of open systems." (Francesca Brini, Mathematical Reviews, August, 2023)

"This book may be useful to scientists working in the field of hydrodynamics, and the authors were successful in putting their brick into the building of fluid mechanics." (Aleksey Syromyasov, zbMATH 1504.76003, 2023)

Contenu
Part I: Modelling.- Mathematical Models of Fluids in Continuum Mechanics.- Open vs. Closed Systems.- Part II: Analysis.- Generalized Solutions.- Constitutive Theory and Weak-Strong Uniqueness Revisited.-Existence Theory, Basic Approximation Scheme.- Vanishing Galerkin Limit and Domain Approximation.-Vanishing Artificial Diffusion Limit.- Vanishing Artificial Pressure Limit.- Existence Theory - Main Results.-Part III: Qualitative Properties.- Long Time Behavior.- Statistical Solutions, Ergodic Hypothesis, and Turbulence.- Systems with Prescribed Boundary Temperature.