This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier–Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner–Lebesgue spaces is not applicable. As a substitute for Bochner–Lebesgue spaces, variable Bochner–Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier–Stokes equations under general assumptions. Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory and non-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.

This volume brings together four contributions to mathematical fluid mechanics, a classical but still highly active research field. The contributions cover not only the classical Navier-Stokes equations and Euler equations, but also some simplified models, and fluids interacting with elastic walls. The questions addressed in the lectures range from the basic problems of existence/blow-up of weak and more regular solutions, to modeling and aspects related to numerical methods. This book covers recent advances in several important areas of fluid mechanics. An output of the CIME Summer School "Progress in mathematical fluid mechanics" held in Cetraro in 2019, it offers a collection of lecture notes prepared by T. Buckmaster, (Princeton), S. Canic (UCB) P. Constantin (Princeton) and A. Kiselev (Duke). These notes will be a valuable asset for researchers and advanced graduate students in several aspects of mathematicsl fluid mechanics.

The objectives of this monograph are to present some topics from the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realizations of those problems in appropriate function spaces. A highlight of this presentation is the large number and variety of examples introduced to illustrate the connection between the theory of nonlinear operators and partial differential equations. These include primarily semilinear or quasilinear equations of elliptic or of parabolic type, degenerate cases with change of type, related systems and variational inequalities, and spatial boundary conditions of the usual Dirichlet, Neumann, Robin or dynamic type. The discussions of evolution equations include the usual initial-value problems as well as periodic or more general nonlocal constraints, history-value problems, those which may change type due to a possibly vanishing coefficient of the time derivative, and other implicit evolution equations or systems including hysteresis models. The scalar conservation law and semilinear wave equations are briefly mentioned, and hyperbolic systems arising from vibrations of elastic-plastic rods are developed. The origins of a representative sample of such problems are given in the appendix.

This work is based upon a Special Session on the Theory and Applications of Nonlinear Operators of Accretive and Monotone Type held during the recent meeting of the American Mathematical Society in San Francisco. It examines current developments in non-linear analysis, emphasizing accretive and monotone operator theory. The book presents a major survey/research article on partial functional differential equations with delay and an important survey/research article on approximation solvability.