From Navier-Stokes millennium-prize problem to soft matter computing

Abstract

Clay Mathematical Institute, in the year 2000, formulated a list of seven unsolved mathematical problems which might influence and direct the course of the research in the 21st century. Navier-Stokes regularity problem is one of the seven problems and has not been solved till date. These equations govern the motion of the viscous fluids, such as liquids, gases, gels, polymers. The present article is a spotlight on the NSE problem: a review of the past and current efforts to solve it, and our changing perspectives and understanding of these equations. The major debate is to either prove that the solutions to Navier-Stokes equations (NSE) are smooth or to prove that the solutions reach singularity at a finite-time (blowup); given some initial conditions. While in the past, the focus was on finding bounds for the smooth solutions, recently, there has been growing interest in finding solutions that blowup. The current article places the Turing-completeness of Euler equations (a subclass of NSE) into the centre-stage, by discussing the recent developments that are pointing towards that the idea that the fluid equations should be viewed as a 'computer program'. Given this is the case, the article then argues to see 'continuum fluid' from a complexity notion, and thus, as a Turing machine. This notion connects the discipline of fluid mechanics to disciplines like, 'neuromorphic computing, reservoir computing' and potentially leading to emerging new discipline, "soft matter complexity, computing, and learning", each of which has been discussed in brief.