This monograph presents a systematic theory of weak solutions in
Hilbert-Sobolev spaces of initial-boundary value problems for
parabolic systems of partial differential equations with general
essential and natural boundary conditions and minimal hypotheses on
coefficients. Applications to quasilinear systems are given,
including local existence for large data, global existence near an
attractor, the Leray and Hopf theorems for the Navier-Stokes
equations and results concerning invariant regions. Supplementary
material is provided, including a self-contained treatment of the
calculus of Sobolev functions on the boundaries of Lipschitz
domains and a thorough discussion of measurability considerations
for elements of Bochner-Sobolev spaces. This book will be
particularly useful both for researchers requiring accessible and
broadly applicable formulations of standard results as well as for
students preparing for research in applied analysis. Readers should
be familiar with the basic facts of measure theory and functional
analysis, including weak derivatives and Sobolev spaces. Prior work
in partial differential equations is helpful but not required.
General
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