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

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