In continuum mechanics, the motion of an incompressible, homogeneous fluid is described by the velocity field and the hydrodynamical pressure. The time evolution of the fluid is governed by the Navier–Stokes system of partial differential equations which describes the balance of mass and momentum. In the classical formulation – which goes back to C.-L. Navier and G.G. Stokes – the relation between the viscous stress tensor and the symmetric gradient of the velocity field is linear (i.e. we have a Newtonian fluid). This system is already quite challenging from a mathematical point of view and has fascinated many mathematicians. However, it can only model fluids with a very simple molecular structure such as water, air and several oils. In order to study more complex fluids one has to deal with a generalized Navier–Stokes system. In this model for generalized Newtonian fluids, the viscosity ν is assumed to be a function of the shear-rate $\dot{\gamma }$. Very popular among rheologists is the power-law model in which the generalized viscosity function is of power-type $\nu \sim {\dot{\gamma }}^{p-2}$ with $p>1$. For a specific fluid physicists can identify this power by experiments. Instead of the Laplacian, as in the classical Navier–Stokes system, the main part of the mathematical model is a power-type nonlinear second order differential operator. So, in addition to the convective term, a second highly nonlinear term appears. The only framework which is available today is the concept of weak solutions. These solutions belong to appropriate Sobolev spaces: Derivatives have to be understood in the sense of distributions and singularities may occur.

The mathematical observation of power-law fluids began in the late sixties in the pioneering work of J.-L. Lions and O.A. Ladyzhenskaya. The cases they could handle were restrictive for real world situations. Nevertheless, it was a breakthrough in the theory of partial differential equations. Since then, there has been huge progress in the mathematical theory of generalized Newtonian fluids. The first systematic study was initiated by the group around J. Nečas in the 1990s. Situations which are realistic for physical and industrial applications could finally be handled. In the 2000s the Lipschitz truncation (the approximation of a Sobolev function by a Lipschitz continuous one done in such a way that both are equal outside of a small set whose size can be controlled) was shown to be a powerful tool in the analysis of generalized Newtonian fluids. A very wide range of non-Newtonian fluids could finally be included in the mathematical theory. The best known bound $p>\frac{6}{5}$ for three dimensional flows of power-law fluids was achieved.

A drawback of the Lipschitz truncation is its nonlinear and nonlocal character. In fact, the property of a function to be solenoidal (that is, divergence-free) is lost by truncating it. So, one has to introduce the pressure function in the weak formulation which results in additional technical difficulties. An advanced pressure decomposition via singular integral operators is necessary in the non-stationary case. An improved version, the “solenoidal Lipschitz truncation”, was developed only recently. It allows the existence of weak solutions to the generalized Navier–Stokes system to be shown without the appearance of the pressure function and therefore highly simplifies the proofs. Moreover, it allows the Prandtl–Eyring fluid model to be studied which was out of reach before. In this model the power-growth is replaced by some logarithmic function: the law $\nu \sim \ln (1+\dot{\gamma })/\dot{\gamma }$ was introduced in 1936 based on a molecular theory. This leads to a limit case in the functional analytical setting of generalized Newtonian fluids. It is not possible to introduce the pressure in the expected function space. Neither can the divergence be corrected.

The aim of this book is to present a complete and rigorous mathematical existence theory for generalized Newtonian fluids – for stationary, non-stationary and stochastic models. The balance laws are formulated in all situations via a generalized Navier–Stokes system. The proofs are presented as self-contained as possible and require from the reader only basic knowledge of nonlinear partial differential equations.

The heart of this book is the construction of the “solenoidal Lipschitz truncation”. It has numerous applications and is of interest for future research beyond the scope of this monograph. The stationary truncation is presented in Chapter 3 and the non-stationary version in Chapter 6. Based on the “solenoidal Lipschitz truncation” the existence of weak solutions to generalized Navier–Stokes equations is shown. The existence proof itself is only slightly more complicated than the classical monotone operator theory and easy to follow.

In Chapter 4 we study the stationary Prandtl–Eyring model in two dimensions. Here, several important tools like the Bogovskiĭ operator and Korn's inequality loose some of their continuity properties. Optimal results, which might allow for a loss of integrability, can be achieved in the framework of Orlicz spaces. They are flexible enough to study fine properties of measurable functions which are required in the Prandtl–Eyring fluid model. We present optimal versions of the Bogovskiĭ operator, Nečas' negative norm theorem and Korn's inequality in this framework in Chapter 2. The background about Orlicz- and Orlicz–Sobolev spaces is revised in Chapter 1.

In Chapter 7 we deal with non-stationary flows of power law fluids. A first step is to approximate the equations by a system whose solutions are known to exist. In order to pass to the limit in the regularization parameter, one has to apply compactness of the velocity, the “solenoidal Lipschitz truncation” and arguments from monotone operator theory.

In the last part of the book we study stochastic partial differential equations in fluid mechanics. Probabilistic models have become more and more important for applications and earned a strong interest amongst mathematicians. They can, for instance, take into account physical uncertainties and model turbulence in the fluid motion. We present first existence theorems for generalized Navier–Stokes equations under random perturbations. The results and methods build a basis for future research on stochastic partial differential equations in the analysis of generalized Newtonian fluids. All probabilistic tools are presented as well. The Chapter can be viewed as an introduction to stochastic partial differential equations from an analytical point of view. Thus, the proofs of the main result given in Chapter 10 are accessible to analysts without prior knowledge in stochastic analysis.