Blow-up of a class of solutions with free boundaries for the Navier-Stokes equations (Q1575807)

A particular type of Navier-Stokes equations is considered, given in terms of two functions \(f\) and \(g\). The mechanical interpretation is a class of free-boundary problems, describing a plane jet. For \(f\), a nonlinear heat equation with a quadratic reaction term and a nonlocal term appears. This equation is ``simplified'' into a first-order Hamilton-Jacobi equation. Positive functions \(f\) are considered. The main point is to prove that the solution blows up in a finite time. An asymptotical method is used, developed in a previous paper of the authors. Some types of unstable blow-up and global patterns (with a non-generic behaviour) are considered in the last part.