Analysis of Fluids and Related Topics: Brief survey of computer assisted proofs for partial differential equations

I will present a brief survey of computer assisted methods of studying partial differential equations that I have worked on. The methods I am going to discuss allow for obtaining proofs of the existence of particular solutions of a certain class of PDEs in a prescribed range of parameters.  I will discuss opportunities  and limitations of the presented approach. In particular most of the presented results have not been obtained using known techniques of 'classical analysis'.  I will focus on two particular examples from my research, namely 1) a proof of the existence of globally attracting solutions for the 1d viscous Burgers equation (with non-autonomous forcing) https://arxiv.org/abs/1403.7170, and 2) recent proof of the heteroclinic connections in the 1d Ohta-Kawasaki (diblock copolymers) model https://arxiv.org/abs/1703.01022