# Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Fluid mechanics has a long history since Greek philosopher Archimedes discovered his famous law of forces acting on bodies in motionless fluid. A good understanding of its principles and its mathematical formulation and properties is crucial in many branches of contemporary science and technology. Not only its modern foundation relies on mathematics but also many branches of mathematics were developed or even emerged through the research in fluid mechanics or through the mathematical formulation of problems of fluid mechanics. The examples include the theory of functions of one complex variable, topology, dynamical systems, differential equations, differential geometry, probability theory, and functional analysis, to name only a few.

Mathematics has been always playing a key role in the research on fluid mechanics. Many imminent problems in various branches of mathematics have their origin or can be interpreted as problems of fluid mechanics although in many cases the community of pure mathematicians fails to notice this fact.

The purpose of this handbook is to provide a synthetic review of the state of the art in the theory of viscous fluids, present fundamental notions, formulate problems of fluid mechanics representing the development of the theory during last several decades, and show the methods and mathematical tools for their resolution. Since the field of mathematical fluid mechanics is huge, it is impossible to cover all topics. In this handbook, we focus on athematical analysis in mechanics of viscous Newtonian fluids. The first part consisting of two chapters is devoted to derivation of basic equations by physical modeling. The second part is devoted to mathematical analysis of incompressible fluids, while the third part is dealing with the mathematical theory of viscous compressible fluids. There are many topics that are not covered by the handbook. In particular, this is the case of numerical analysis of the equations which would deserve by itself an independent volume.
The handbook reviews important problems and notions that marked the develop-ment of the theory. It explains the methods and techniques that may be used for their resolution. We hope that it will be useful not only to mathematicians who work on the future development of the theory but also to physicists and engineers who need to know the tools of mathematical analysis for developing applications.
Continuum mechanics and thermodynamics are based on the idea of continuously distributed matter and other physical quantities. Originally, continuum mechanics was equated with hydrodynamics, aerodynamics, and elasticity. The scope of the study was the motion of water and air and the deformation of some special solid substances. However, the concept of continuous medium has been shown to be useful and extremely viable even in the modelling of the behavior of much more complex systems such as polymeric solutions, granular materials, rock and land masses, special alloys, and many others. Moreover, the range of physical processes modelled in the continuum framework nowadays goes beyond purely mechanical processes. Processes such as phase transitions or growth and remodeling of biological tissues are routinely approached in the setting of continuum thermodynamics.
Finally, the systems studied in continuum thermodynamics range from the very small ones studied in microfluidics (see, e.g., Squires and Quake [89]) up to the gigantic ones studied in planetary science; see, for example, Karato and Wu
It seems that the laws governing the motion of acontinuous medium must be extremely complicated in order to capture such a wide range of physical systems and phenomena of interest. This is only partially true. In principle, the laws governing the motion of a continuous medium can be seen as reformulations and generalizations/counterparts of the classical physical laws (the laws of Newtonian physics of point particles and the laws of classical thermodynamics). Surprisingly, the continuum counterparts of the classical laws are relatively easy to derive. These laws are – in the case of a single continuum medium – the balance equations for
the mass, momentum, angular momentum, and energy and the evolution equation for the entropy. The balance equations are supposed to be universally valid for any continuous medium, and their derivation is briefly discussed in Sect. 2.The critical and the most difficult part in formulating the system of governing equations for a given material is the specification of the response of the material to the given stimuli. The sought stimulus-response relation can be, for example, a relation between the deformation and the stress or a relation between the heat flux and the temperature gradient. The task of finding a description of the material response is the task of finding the so-called constitutive relations.

Apparently, the need to specify the constitutive relation for a given material calls for an investigation of the microscopic structure of the material. The reader who is interested in examples of the derivation of constitutive relations from microscopic theories is referred to Bird et al. [4] or Larson [52] to name a few. However, the investigation of the microscopic structure of the material is not the only option.
The constitutive relations can be specified staying entirely at the phenomenological level. Here the henomenological level means that it is possible to deal only with henomena directly accessible to the experience and measurement without trying to interpret the phenomena in terms of ostensibly more fundamental (microscopic)physical theories. Indeed, the possible class of constitutive relations is in fact severely restricted by physical requirements stemming, for example, from the requirement on Galilean invariance of the governing equations, perceived symmetry of the material, or the second law of hermodynamics. As it is apparent from the discussion in Sect. 4, such restrictions allow one to successfully specify constitutive relations