Linear and quasi-linear evolution equations in Hilbert spaces (Q2904605)

The objective of this monograph is to present a number of well-posedness and global existence results for equations of the form NEWLINE\[NEWLINE \varepsilon u_{tt}+\sigma u_t-\sum_{i,j}a_{ij}(t,x,u,u_t,\nabla u,\partial_{ij}u)=f(t,x), NEWLINE\]NEWLINE in \(N\)-space, where \((a_{ij})\) is elliptic for all values of its argument, where \(\varepsilon\) and \(\sigma\) are nonnegative constants. For initial-boundary value problems for these equations, the authors refer to two of their papers on this issue. The authors are keen to treat the hyperbolic and parabolic cases in parallel settings, so that the limit when \(\epsilon\to 0\), or when \(t\to\infty\) may be conveniently studied.NEWLINENEWLINENEWLINENEWLINEThe volume is also meant to give an introduction to some of the general-purpose techniques to solve nonlinear evolution equations, and to contribute to bridging the gap between a first-course in PDEs and the many research-level monographs quoted in the introduction: ``these notes are not meant to serve as an advanced PDEs textbook; rather, their didactical scope and subject range is restricted to the effort of explaining [\dots] one possible way to study two simple and fundamental examples of equations (hyperbolic, both dissipative or not, and parabolic) on the whole space \({\mathbb R}^N\).''NEWLINENEWLINENEWLINENEWLINEThe book contains seven chapters, of which the first contains useful elementary tools, such as the action of nonlinear functions on Sobolev spaces, or commutator estimates, proved here in a simple setting: characterization results and recent refinements are not mentioned. The book closes with a four-page list of function spaces, 171 references and a four-page index. There are no exercises. The seventh and last chapter deals with two problems similar to those treated in the other chapters, and motivated by applications. The first is Maxwell's equations with constitutive relations taken to be of the form \(D=\varepsilon E\) and \(H=\zeta(B)\). The second deals with equations of von Kármán type; here, the authors refer to an expanded version of this section on the book web page (\texttt{http://www.ams.org/bookpages/gsm-135}, about fifty pages; note that the preface, table of contents and the second chapter may also be downloaded from the AMS bookstore web page \texttt{http://www.ams.org/bookstore}). This supplement states: ``While their physical significance may not be evident, their interest resides in a number of specific analytical features, which makes their study a rich subject of investigation.''NEWLINENEWLINENEWLINENEWLINEThis volume could be used as supplemental reading in a graduate course on nonlinear PDEs.