Stability and dynamics of self-similarity in evolution equations

Abstract

Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

Similarity and singularity in adhesive elastohydrodynamic touchdown

“…In Fig. 2(a) and Table I, a comparison of these solutions of ODE (6) and (7) with the solution of the time-dependent partial differential equation (3) shows that only the fundamental solution associated with m = 1 agrees with the solution of complete PDE (3), a scenario also seen for capillary film rupture 9,11,12,14 and justified using a stability analysis in similarity coordinates. A qualitative explanation of why this is so follows by noting that solutions of (6) and (7) with m 1 has multiple minima in the height h; any perturbation will cause one of them to be lower, and run away dynamically from the others, and thus eventually end up locally like the fundamental discrete solution with m = 1.…”

“…[3][4][5] Indeed, analogs of these flows have been well studied in the context of the rupture of capillary thin fluid films on substrates [6][7][8][9][10][11][12][13][14][15] that reveal a finite-time singularity. 10,11 Our analysis focuses on small scale flows wherein inertial effects in the fluid and the sheet may be neglected.…”

Similarity: generalizations, applications and open problems

“…A comprehensive survey of rigorous, formal, and numerical approaches to local self-similarity in describing singularity behaviour in a wide class of physically based PDE models, relating primarily to fluid systems, is given in [27]. An extensive bibliography of further problems is given in the bibliography of [28].…”

“…Remarkably, such solutions arise often in applications (cf. [27,28]) and their analysis requires a synthesis of similarity, dynamical systems, perturbation, and numerical methods; -rigorous analysis of the above issues (cf. [12][13][14][15][16][17]29]).…”

“…In [28], a methodology is presented for studying linear stability for self-similar solutions. It is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems.…”