Self-similar blow up dynamics for nonlinear evolution equations

The description of time-dependent processes in terms of evolution equations, i.e., time-dependent partial differential equations (PDEs), plays a fundamental role in many areas of science and technology. An important property of nonlinear evolution equations is the formation of singularities in finite time from smooth initial data, which is referred to as blow up. In such a scenario, self-reinforcing processes dominate smoothing mechanisms such as dissipation or dispersion. Consequently, the amplitude of the solution blows up or the gradient diverges and discontinuities are formed. For PDEs which are invariant under certain scaling transformations, self-similar (scale invariant) solutions can provide explicit examples of finite time blow up. For many models it is proven that they admit countable families of self-similar solutions, however, they are usually not known in closed form and can only be constructed numerically. In addition to such explicit examples and conditions on the initial data that allow to predict the formation of a singularity, one is interested in features of generic blow up solutions, e.g. the divergence rate or the asymptotic profile. For several models - ranging from nonlinear wave equations, parabolic problems of second and higher order to more complicated coupled systems of equations - numerical experiments show convergence of singular solutions to a particular profile, which can be identified with the ground state of the above mentioned family of scale invariant solutions. Based on these observations it is suggested that blow up via the self-similar ground state provides a stable blow up mechanism for the respective equations. In this project we investigate these questions with functional analytical methods for different types of PDEs. First we study wave equations with focusing power nonlinearities. Here, the aim is to refine results on the stability of the so-called ODE blow up. Our approach is perturbative and involves highly non-selfadjoint operators. It relies on methods from the theory of strongly continuous one-parameter semigroups, operator theory as well as ODE theory. In the second part of the project our techniques will be generalized to non-hyperbolic problems, where we investigate the heat flows for harmonic maps and Yang-Mills connections in certain geometrical settings as well as the parabolic-elliptic Keller-Segel Model. The models considered in this project can be classified as supercritical. The investigation of such problems is a very active field of research at the present time and this project is supposed to contribute substantially to the rigorous understanding of stable blow up mechanisms for supercritical PDEs. Furthermore, we expect that the developed methods will allow the investigation of more involved problems for which at the present time no suitable mathematical techniques are available.

This project was concerned with the analytic investigation of blowup dynamics in non- linear time-dependent partial differential equations. In the course of the project robust methods have been developed to study the stability properties of self-similar blowup solutions for semilinear wave equations and heat flows with the focus on energy super- critical problems. The obtained results open up new possibilities and perspectives in the investigation of singular dynamics in supercritical time-evolution problems.The first part of the project was concerned with nonlinear hyperbolic problems. For wave equations with a focussing power nonlinearity previous results on the nonlinear asymptotic stability of the so-called ODE blowup solution were generalized to arbitrary odd space dimensions in the radial case and to non-radial perturbations in three space dimensions. The results are local in nature and valid in the backward lightcone of the blowup point. The existence and stability of global self-similar profiles was investigated for a different model known as the wave maps equation. In the project, we studied wave maps from Minkowski space into the 3?sphere under certain symmetry assumptions. For this model, we found an explicit solution which exists for all times and which is perfectly smooth on all of R3 except at time t = T , where the gradient blows up at the origin. By introducing novel adapted coordinates we showed that this solution provides a globally stable blowup profile. Furthermore, the new coordinates allowed us to track the time-evolution past the blowup time in a space-time region which is not influenced by the singularity at the origin because of finite speed of propagation.The main objective of the second part was to develop an analytic framework to study self-similar blowup solutions in parabolic problems. Here, prominent models from geometric analysis were considered: The heat flow of harmonic maps from R3 into the 3?sphere and the heat flow for Yang-Mills connections on R5 SO(5). Both problems were studied under particular symmetry assumptions. By following a similar approach as for wave equations, we established the nonlinear asymptotic stability of an explicitly known self-similar solution for the Yang-Mills heat flow. This was among the first results on stable blowup for a supercritical heat equation with a non-trivial blowup profile. For the harmonic map heat flow a similar result could be obtained. Here, the approach was further simplified by exploiting structural properties of the problem.