Weak solutions and the universal iterative process -- Regularity of solutions for non degenerated quasilinear second order elliptic systems of the divergent form with bounded nonlinearities -- Some properties and applications of regular solutions for quasilinear elliptic systems -- Diffeentiability of solutions for second order elliptic systems -- Regularity of solutions for parabolic systems with some applications -- The Navier-Stokes system; strong solutions.

The smoothness of solutions for quasilinear systems is one of the most important problems in modern mathematical physics. This book deals with regular or strong solutions for general quasilinear second-order elliptic and parabolic systems. Applications in solid mechanics, hydrodynamics, elasticity and plasticity are described. The results presented are based on two main ideas: the universal iterative method, and explicit, sometimes sharp, coercivity estimates in weighted spaces. Readers are assumed to have a standard background in analysis and PDEs.