Self-similar blowup for supercritical evolution equations

Nonlinear evolution equations, i.e., nonlinear time-dependent partial differential equations are central to mathematical description of natural phenomena. For example, fluid flow and the evolution of the universe in physics, population dynamics and cell division in biology, cancer growth and spreading of infectious diseases in medical sciences, are all modeled by means of nonlinear evolution equations. What is more, these equations play an important role in the treatment of pure mathematical problems as well. In this project we are concerned with two types of evolution equations, namely the wave and the parabolic ones. They arise in a number of areas from sciences and engineering, e.g., fluid dynamics, optics, general relativity, particle physics, population dynamics, image processing, heat conduction, and many more. However, in spite of their ubiquity in applications, the mathematical understanding of these equations is still unsatisfactory. A question that is particularly relevant yet underexplored is the one of finite time breakdown of solutions (also called singularity formation or blowup). Physically, singularity formation indicates a radical change in the modeled phenomenon (e.g., formation of a black hole), or the emergence of a new (singular) structure (e.g., drop formation in liquids), or that in fact some essential physics is missing from the model. Recently, there has been a significant research progress on the topic of blowup for wave and parabolic equations. However, most of the results concern the so-called critical and subcritical cases, i.e., the instances where conserved quantities of the equations can be used to control the evolution. If such control is not possible then the equations are referred to as supercritical, and their analysis is traditionally more difficult. What is more, numerical studies of various types of supercritical equations indicate that large data solutions generically break down, with blowup being governed by the so-called self-similar solutions. In view of these insights, the main aim of this project is the development of new, general and robust methods for the study of the existence and stability of self-similar blowup for supercritical wave and parabolic equations. We intend to do this through the analysis of concrete models: the focusing power nonlinearity wave and heat equations, wave maps, the hyperbolic Yang-Mills equations and the Keller- Segel model for chemotaxis. Our approach entails a combination of methods from the classical analysis of partial differential equations, ordinary differential equations theory, nonlinear functional analysis, operator theory and approximation theory. Our overarching aim is a systematic and rigorous study of stability of blowup in evolution equations in general, and development of techniques that can be efficiently applied to realistic physical models.

Nonlinear evolution equations are of central importance for the mathematical description of natural phenomena. A particularly relevant notion in this context is that of finite-time breakdown of solutions, also called singularity formation or blowup. The main aim of this project has been to explore, from a theoretical point of view, a specific type of blowup, namely the self-similar blowup, in the context of two classes of nonlinear evolution equations. For nonlinear wave equations, which arise in fields such as elasticity theory, optics, and general relativity, we developed a mathematical framework to study the stability of self-similar solutions. From a physical standpoint, understanding the stability of blowup solutions is essential, as only stable structures are likely to be observed in nature. We applied the developed framework to two nonlinear wave models: the hyperbolic Yang-Mills equations and the Wave Maps equation, both of which have applications in particle physics. In both cases, we established the stability of a particular self-similar solution, whose existence had been known for over a decade. The second part of the project focused on nonlinear parabolic equations, which appear in combustion theory, fluid dynamics, chemotaxis, image processing, and many other areas of applied science. We developed a corresponding mathematical framework for studying the stability of self-similar blowup solutions in this context as well. We then applied this framework to two important parabolic model equations. The first is the harmonic map heat flow, which arises in geometric analysis and concerns the geometry of high-dimensional objects. The second is the Keller-Segel system, which originates in mathematical biology and models aggregation of bacteria. For both equations, we demonstrated the stability of certain self-similar profiles, thereby describing the generic mechanism of singularity formation in these settings. The result for the Keller-Segel model is particularly significant, as it resolved a conjecture that had remained open for over two decades. The frameworks we developed have, on the one hand, brought many important problems within reach and, on the other hand, opened up a range of new and challenging mathematical questions to be addressed. We aim to pursue these directions in the forthcoming FWF project of the same title.