Theoretical and numerical analysis of p-Navier-Stokes equations with inhomogeneous boundary conditions

This thesis studies the weak existence and finite element approximation of the incompressible p-Navier–Stokes equations for generalized Newtonian fluids with nonlinear (p,δ)-type constitutive laws. Emphasis is placed on inhomogeneous Dirichlet and mixed Dirichlet–Neumann boundary conditions, which are common in engineering but less understood theoretically. Using lifting techniques, pseudomonotone operator theory, and Lipschitz truncation, we establish existence of weak solutions for steady and unsteady flows, including shear-thinning regimes and large data cases, and derive stability conditions for Neumann terms. A compatible inf-sup stable finite element scheme is proposed, including an in-depth discussion of interpolation of Dirichlet boundary data. We then prove existence, convergence, and a priori error estimates for discrete solutions.