The method of concentration compactness and dispersive Hamiltonian evolution equations (Q2929275)

In this brief survey, the author presents recent progress on large data results for nonlinear wave equations such as \(\square u=F(u,Du)\), \(F(0)=DF(0)=0\), \((u(0),\dot{u}(0))=(f,g)\). The author distinguishes two basic scenarios: {\parindent= 0.5cm\begin{itemize}\item[1)] Small data theory: the local and global well-posedness, the existence of conserved quantities (energy), their relation to the basic symmetries of the equation (particularly the dilation symmetry) are presented and discussed. \item[2)] Large data theory: finite-time blow-up of solutions, the classification of possible blowup dynamics and structure are presented and investigated.NEWLINENEWLINE\end{itemize}} It is demonstrated that the concentration compactness procedure has turned out to much more versatile; for example, it has been a key ingredient in the classification of blow-up behavior of solutions.NEWLINENEWLINEFor the entire collection see [Zbl 1279.00048].