This Lecture Course is devoted to the modern mathematical theory of complete fluid systems - the fluid flows satisfying the basic physical principles of conservation of mass, momentum, and energy. Our approach is based on the concept of weak solution introduced in the monograph [M].

The main topics include:

• Basic equations of mathematical fluid dynamics, the Navier-Stokes-Fourier system
• Weak vs. strong formulation, basic properties, advantages, shortcomings
• Nonlinear equations and a priori bounds, global dissipation inequality and related questions
• Relative entropies, weak-strong uniqueness, long-time behavior of complete fluid systems
• The property of (weak) sequential stability
• Principal ideas of the mathematical theory of global-in-time large data solutions, the effective viscous pressure, commutator estimates, div-curl lemma
• More advanced topics related to complete fluid systems, model reduction and singular limits

Bibliography:
[M] E. Feireisl and A. Novotny: Singular limits in thermodynamics of viscous fluids. Birkhäuser-Verlag, Basel, 2009.