Self-similar singularities in PDE and computer assisted proof

This talk will present recent work on constructing self-similar imploding solutions to compressible fluid equations and their connection to blow-up phenomena in the defocusing nonlinear Schrödinger equation and the defocusing complex nonlinear wave equation. I will showcase the application of interval arithmetic as a tool for rigorous construction of these solutions through a computer-assisted proof. Additionally, I hope to briefly discuss ongoing work that illustrates a general framework for extending these techniques to more complex problems, such as nonlocal equations.