Continuity equations, also known as conservations laws, are differential equations relevant in a variety of contexts. The Navier Stokes equations are examples of continuity equations that describe how fluids move. They rest on assumptions such as Newton's second law, applied to liquids and gases. The equations involve quantities like velocity, pressure, density, viscosity, and external forces. In 2000, the Clay Mathematics Institute established seven \$1 million Millennium Prize Problems, one of them on Navier-Stokes equations. This Millenium problem asks for a proof deciding whether smooth, physically reasonable solutions always exist for Navier-Stokes equations in three dimensions. At issue is whether or not physically reasonable assumptions can nevertheless result in "blowups" where solutions become infinite.

In a recent article in Scientific American titled Humans and AI race to "blow up" math's toughest equations, Steve Shkoller describes a vision inspired by his observations of waves while surfing. Shkoller's vision goes beyond the formal constraints used by AI led searches for this type of phenomenon. It has been turned into mathematics in a lengthy paper posted on the arXiv. While the paper may not provide any examples relevant to the precise framework of the Navier-Stokes Millenium Prize, the key insight may yet turn out to be relevant. The article is an interesting glimpse into the different areas of this emerging research.