On Some Analytical Questions Concerning the Dynamics of Newtonian and non-Newtonian Incompressible Fluids

The purpose of this thesis is to present several results concerning different topics and problems in Fluid Dynamics: more specifically, the well-posedness of problems involving Newtonian fluids and a particular class of non-Newtonian fluids, the power-law fluids, is investigated. It is based on the results presented in [25, 26, 46, 45] and is organized into four chapters: • In Chapters 1 and 2, we study the well-posedness of a parabolic p-Laplacian system with a convective term, derived from the power-law system in the subquadratic case (p < 2), by replacing the symmetric gradient with the full gradient and eliminating the pressure term. It should be noted that these modifications make the results less relevant from a Fluid Dynamics perspective, since the corresponding constitutive law does not comply with the principle of material invariance. Nevertheless, they are useful to better delimit the expectations for possible results in the fluid dynamics context. We estabilish existence and a maximum principle for regular solutions (for p∈ 3 2 ,2 ) and weak solutions (for p∈(1,2)) for an initial datum v◦(x) ∈L∞(Ω); for regular solutions we analyze the property of extinction in a finite time under suitable smallness assumptions on the initial datum. Moreover, for v◦(x) ∈L∞(Ω) ∩W1,2 0 (Ω), we are able to prove the uniqueness of regular solutions for p∈ 5 3 ,2. • In Chapter 3, we focus on a different problem concerning power-law fluids: the initial value problem in a spatially periodic domain, with the goal of constructing a weak solution satisfying an energy equality. The result is a weak solution that satisfies a sort of energy equality, i.e., an energy equality with additional dissipation. • InChapter4, weinvestigateaprobleminthecontextofNewtonianfluidtheory, focusingon theStokessysteminthehalf-spacewithintheframeworkofweightedLebesguespaces. This analysis is motivated by the long-term goal of understanding the Navier–Stokes equations in weighted settings, not only in the half-space but also in exterior domains. In particular, we introduce a class of weights defined as the products of powers of distances from fixed points, and we establish existence, uniqueness, and regularity results for strong solutions to the Stokes problem in the half-space. Moreover, in the Appendix we present some background material, and a comprehensive bibliog- raphy of the relevant literature is provided.