Stability of postulated, self-similar, hydrodynamic blowup solutions
A solution with real time singularity is assumed to exist that is steady under a Leray-type normalization. This solution is further assumed to be reached asymptotically as t-->t(0) in the renormalized plane, and thus can be thought of as the leading behavior of an inner solution. Constraints due to conserved quantities like energy are shown to be weakened in this scenario. In the wake region that trails the collapsing structure, it is shown that eigenfunctions associated with initial conditions are stable and decay, allowing the attracting singular solution to be shielded from details of the initial conditions. The parameters of the normalization are t(0), r(0), v(0), lambda, and alpha, which are the critical time, the location of the singularity, the velocity of the singular point, a scaling factor, and the scaling exponent of the velocity (t(0)-t)(alpha). The stability of the eigenfunctions of this solution obtained from the perturbation of these parameters is also examined in this work. Perturbations in the critical time and location are shown to be unstable whereas perturbations in velocity and scaling are not. The condition that the amplitude of the unstable eigenfunctions vanishes determines the time and location of the singularity.
Greene
JM
General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA.
Pelz
RB