Partial Differential Equations with Singularities

From January 2025 the project is funded and administered by the Higher Education and Science Committee of Armenia.

Principal Investigator: Prof. Michael Ruzhansky

University: University of Ghent

Research team: Zhirayr Avetisyan, Zahra Keyshams, Monire Mikaeili Nia, Aram Nazaryan, Gor Chalyan, Mery Galstyan

Contributing Researchers: Alesya Ghandilyan

Duration: 2023-2027

Project Importance

The analysis of partial differential equations (PDEs) occupies a central place among a wide range of sciences. Whenever there are processes of evolution or static models, starting from the ground-breaking works of Isaac Newton, they are often described by partial differential equations. The mathematical insights and understanding of solutions of such equations often help resolving fundamental problems in many sciences. The concept of a solution to a PDE has been evolving over the centuries: pointwise, weak, distributional, viscosity and other notions of solutions have been used in order to handle the well-posedness of equations and to analyse fundamental properties of their solutions.

An inseparable part of the analysis of PDEs are the singularities of their solutions. However, when the coefficients of equations also become singular, or an equation contains nonlinear terms, it is often no longer possible to work with singular data due to the famous Schwartz’54 impossibility result on the multiplication of distributions.

The present project proposes a new concept of very weak solutions for dealing with PDEs with strong singularities, by adapting, in a specific way, the notion of solution to properties of solutions of regularising families of equations. We will work on the development of this notion in a number of concrete physical problems, followed by the development of a general theory for PDEs of elliptic, hyperbolic, parabolic, and general evolution types with strong singularities. Having the rigorous very weak well-posedness, we will study the limiting properties of solutions to regularised problems and apply it to the description of qualitative and quantitative properties of their limit(s). The theory developed will provide a fundamental basis for major advances in a range of methods for dealing with many previously unapproachable problems with strong singularities.

Expected Results

In sub-project (A) we expect to achieve a deep and very detailed understanding of very weak solutions and very weakwell-posedness in many important particular systems, e.g., acoustics, Schrödinger equations, elliptic equations, waves in shallow water etc. We expect that in each of these model cases the method of very weak solutions will capture the fine physical phenomena associated with singularities and fit well with the numerical simulations. Insights gained and patterns observed on these special cases will also inform the other sub-projects. In sub-project (B) we expect to establish precise formulations for very weak solutions and well-posedness for entire classes of hyperbolic, parabolic, general evolution-type, elliptic singular equations. In sub-project (C) we expect to establish the nature of the limit procedure used in the construction of very weak solutions in sub-projects (A) and (B), including the dependence on the chosen mollification, the singular structure (wavefront set) of the limit functions etc.