University thesis:
Dissertation, Universität Freiburg, 2021
Footnote:
Description:
Abstract: The celebrated classical theory of pseudo-monotone operators, whose founding fathers include G.J. Minty, F.E. Browder, and H. Brézis, has for decades been regarded as the “Swiss Army knife” or “WD–40 spray” of non-linear functional analysis. Probably one of its most famous applications is the weak solvability of the unsteady p–Navier–Stokes equations. The latter describe the unsteady motion of an incompressible non–Newtonian fluid of constant density, where p is a constant, often referred to as the so-called power-law index, in which many characteristics of the fluid are encoded.<br>In laboratory experiments, it was discovered, however, that in a number of non–Newtonian fluid flow problems, the power-law index p cannot be a fixed constant but is, rather, variable-dependent. Concretely, these non–Newtonian fluid flow problems involved so-called electrorheological fluids. These fluids are characterized by their ability to undergo significant changes in their mechanical properties when exposed to an electro-magnetic field. The first observation of this exceptional behavior of electrorheological fluids is credited to the engineer W. Winslow in 1949.<br>Owing to the variable dependence of the power-law index p(·, ·) in electrorheological fluids, the classical model of the unsteady p–Navier–Stokes equations is no longer applicable. This motivated K.R. Rajagopal and M. Růžička to develop a mathematical model describing the physics of electrorheological fluids, the so-called unsteady p(·, ·)–Navier–Stokes equations, where now p(·, ·) is not a fixed constant but a function. Until now, however, it has not been possible to demonstrate the weak solvability using the theory of pseudo-monotone operators. This is mainly due to the lack of a suitable mathematical environment that allows to extend the theory of pseudo-monotone operators. The elaboration of this – still missing – environment and likewise of the corresponding extensions of the theory of pseudo-monotone operators is the central mission of the present work.<br>To begin with, we will consider separately the influence of the symmetric gradient on the unsteady p(·, ·)–Navier–Stokes equations and first establish a suitable mathematical framework for the treat- ment of the unsteady p(·, ·)–Navier–Stokes equations reduced by the incompressibility constraint. This will lead to so-called variable exponent Bochner–Lebesgue spaces with a symmetric gradient structure, which we will meticulously analyze for their function space properties.<br>Subsequently, we will incorporate the incompressibility constraint into our investigations and establish the appropriate mathematical framework that will enable us to attack the weak solvability of the unsteady p(·, ·)–Navier–Stokes equations through the usage of pseudo-monotonicity methods. To this end, we will introduce so-called solenoidal variable exponent Bochner–Lebesgue spaces, subspaces of variable exponent Bochner–Lebesgue spaces with a symmetric gradient structure in which an incompressibility constraint is additionally encoded.<br>As the classical theory of pseudo-monotone operators is no longer applicable in the mathematical framework of this work, we will develop suitable extensions of this theory. These extensions include the generalized notions of pseudo-monotonicity and coercivity, (C0–)Bochner pseudo-monotonicity and (C0–)Bochner coercivity, respectively, as well as the so-called Hirano–Landes approach, which allows us to specify general and easily verifiable conditions for these new concepts. Ultimately, these extensions will culminate in an abstract existence result that will be applicable to a number of problems, such as – most importantly – the unsteady p(·, ·)–Navier–Stokes equations.<br>Moreover, we will address the reconstruction of pressure, which was initially excluded from our analysis due to the encoding of the incompressibility constraint in the energy spaces, and the closely related question of possible extensions of the L∞– and (solenoidal) Lipschitz truncation techniques.<br>Since the developed extensions of the theory of pseudo-monotone operators, in particular, the abstract existence result, are applicable only to the case of Lipschitz domains and of dilatant fluids, i.e., when p(·, ·) > 2, but especially for applications pseudo-plastic fluids, i.e., when p(·, ·) < 2, are of particular interest, since a large number of fluids fall within this category, we will, furthermore, develop an existence theory applicable to non–Lipschitz domains and pseudo-plastic fluids