“It is as if there existed, for what seems like millennia [centuries], tracing back to the very origins of mathematics and of other arts and sciences, a sort of “conspiracy of silence” surrounding these “unspeakable labors” which precede the birth of each new idea, both big and small, and which thereby lead to a renewal of our understanding of a portion of this world in which we live, a world engaged in perpetual creation” (Recoltes et Semailles, Grothendieck, A., 3.6, English translation).
The fascination of thinkers and polymaths with “tides, winds, ocean flows, rise of sea level; fluids in general” dates back to the early periods of scientific revolution (16th and 17th century) and the age of enlightenment (18th century), when the phenomenon of attraction between big bulky objects in the heavens and their motion was given light to by Newton’s calculus (keeping aside the controversy around the priority disputes). Newton’s view on fluids were both mixed and unclear: he proposed both a molecular model where particles impact upon each other (leading to his famous law of resistance being proportional to the square of the velocity multiplied by sin-squared of angle of attack; which turned out to be later wrong) and a continuum model for the fluids without any voids in between (elaborated historical discussion in Truesdell 2018). Berlin Academy in 1748 went forward to improvise Newton’s footsteps and proposed the theory of the resistance of fluids as a subject for the prize competition, knowing that the Newton’s theory of the sine-square law of air resistance is wrong. Subsequently, d’Alembert in 1749 submitted his entry, concluding with the now-known physical paradox, the d’Alembert paradox, which leads to a strictly vanishing singularity in the theory of potential flow. The director of Berlin Academy, Leonhard Euler, who sent back all manuscripts and gave no prizes, including d’Alembert’s submission (being a non-resident member, d’Alembert withdrew and published the essay later on independently), was known to be highly preoccupied with the pure pleasure of mathematics, and considered the questions of mechanics and physics only secondary to his predominant passion. Such a culture divide would later become more explicit, as the field of hydraulics and theoretical fluid mechanics had their unfortunate split, famously explained by Sir Cyler Hinshelwood (cited here) as “..hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed“. Regardless of the split, the history of fluid mechanics in the enlightenment and pre-industrialisation age saw the arrival of many thinkers, D. Bernoulli, L. Euler, C.L.M.H. Navier, A. Cauchy, S.D. Poisson, and G.G. Stokes, and the year 1845 is often attributed as the when Stokes published in the Transactions of the Cambridge Philosophical Society his article on the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids (updated version here).
Outside the science community, the fascination of fluids spreaded throughout, to every continent, country, region, culture, and household – be it through common medieval dictum of separating storage vessels for drinking water from bathing water (a few names are: ibrik, olla, zeer, qirbah, and ewer), or through a bicycle tyre repair mechanic’s dictum of intuitively knowing how much open the fluid valve needs to be while filling up air in the punctured tyre of a cycle (which is loosely a function of the type of bike, the specific season of that specific country you are filling your tyre in, and numbers are communicated in the units of psi), using Bernoulli’s principle to siphon out the water from a liquid tank or pool placed at a certain height above the ground, by sucking in and building a relative negative pressure on the lower end of the tube, acting as an outlet; we all have some level of innate rigour and knowledge within us, when it comes down to dealing with fluids. In fact, within us, the vortex rings are formed in the left ventricle of the human heart during cardiac relaxation when the jet of valve enters the mitral valve (in fact, the recoil force generated during the vortex propulsion is found to be smaller for diseased cases of dilated cardiomyopathy and stenotic valves, in comparison to the healthy case); the immature ones are key to understanding if our hearts have a good health or not. It is not just humans but animals having evolved to use fluids for their advantage. Be it phalaropes and other shorebirds opening and closing their beak in tweezering motion and generating vortex to feed in small crustaceans and other invertebrates (Prakash et al. 2008), or the legs of Gerridae striders who support their weight on the surface of ponds, rivers, and the open ocean, by making use of the surface tension force generated by curvature of the free surface (Hu et al. 2003), or the carnivorous pitcher plants like N. Rafflesiana which imprint tiny portions of their digestive, viscoelastic fluids onto the surfaces of insects fallen inside them, and thus effectively wetting them, making it energetically difficult to come out from the stomach of the pitcher plant (Kang et. al 2021), it could be safe to say that the water being much older than the life on Earth, means that we have adapted to live peacefully with and around it.
The advancement of mathematics, science, technology, and society in totality, has evolved in tandem with our evolving understanding of fluids. On 16th July, 1945, prior to the Hiroshima and Nagasaki incident, the Trinity test in New Mexico saw a release of a mushroom cloud as high as 40,000 feet in about 7 minutes. A famed fluid dynamicist, G.I. Taylor, used the scaling analyis to guesstimate incumbent energy of the bomb in terms of density of the undisturbed atmosphere, radius of shock wave, and the time since the blast. This dimensional analysis helped him arrive at the number of 16.8 kilotons of TNT, close to the value 18.6 kilotons reported by DoE. Extending afar to the field of biofluids, E.M. Purcell in his famous paper, Life at low-Reynolds number, derived Scallop’s theorem which states that a highly viscous, low Re envionment is not conducive for a biological swimmer to achieve net displacement in its motion; for example, a scallop consists of one hinge which upon opening and closing in one period isn’t enough to cause its net migration in the fluid. Fluids, governed by the Navier-Stokes equations, also inspired and ignited a lot of interest of mathematicians in the theory of PDEs. Oseen, Leray, Hopf, and Ladyzhenskaya are often credited with working on the Navier-Stokes solutions by stepping away from the classical approach of guaraneeting uniqueness of a solution. The use of weak solutions was so unusual in those times that Ladyzhenskaya recalls: “I still remember years (the 1940s and 50s) when the majority of maitres (and first and foremost I.G. Petrovskii) regarded a problem as unsolved if on the chosen path of investigation the researcher did not guarantee the existence of a classical solution“. It is worth noting that Leray had also introduced the concept of generalised derivatives in the sense of distributions, which are Lebesgue measurable, square-integrable, with a generalised square-integrable gradient (later to be known as the Sobolev space) in his much celebrated weak solution to the Navier-Stokes equations, and he later turned away to jail and then in isolation to lay down the foundations of sheaf theory and what was to become foundations of modern algebraic geometry (miles away from the theory of PDEs, Navier-Stokes, and analysis, for the fear of being labelled as a mechanician by the German authorities in the camp and thus, delving it into his secondary interests).
The chaos theory in the mid 20th century helped in pushing forward the applied analysis of fluids, through the lens of computers, such as by the dynamical simulations, where the initial conditions were to be regarded as enough sensitively powerful to create reproducible structures on the screen; the phenomenon of the Lorenz system, chaotic advection, elastic turbulence, stirring of dye in water (low Re chaotic advection), strange attractors in the chaotic dynamical systems prevalent in metereology and hydrodynamics, and much more. Charles E. Doering and J.D. Gibbons are often regarded as abridging the gap between mathematical and applied fluid dynamics, and their attempt in this direction is apparent in the celebrated book Applied Analysis of the Navier-Stokes equations, where in Chapter 9, they derived estimates on the spectrum of the linearised evolution operator around solutions on the attractor, and thus using 2D Navier-Stokes equations to derive a global attractor; this approach fell short to be used for the 3D Navier-Stokes equations because of the lack of the regularity proof and thus that of the existence of a compact attractor (ref. Ch. 9). Towards the end of 20th century, Cristopher Moore’s program of undecidability suggested an alternative approach of considering motion with at least three degrees of freedom as equivalent to a Turing machine, and thus more notoriously difficult than a low-dimensional chaos for example.
Though the papers and examples cited above are picked by the author’s personal taste of interest and relationship with the field, what is undebatable is the ever-evolving science of PDEs which saw many schools of thoughts flourishing over the past many decades, and its percolation to many applied disciplines, such as, computational fluid dynamics (Basilisk CFD software based in Paris is known for promising solutions to PDEs on adaptive, Cartesian meshes), engineering fluid dynamics (GALCIT, Caltech, known for inventing numerous wind tunnels in mid 20th century), physics of fluid dynamics (DAMTP, Cambridge, UK, known for housing the community of Journal of Fluid Mechanics; started off by G.K. Batchelor), and the most fundamental and uniting of them – the mathematical fluid dynamics. The community of mathematical fluid dynamics had always acted as a shining and guiding light on the landscape of PDEs and fluid dynamics, and with the coming of results from the greats of O. Ladyzhenskaya (her paper on the relevance of millennium problem), P. Lax (most noted for the Lax pair), V. Sverak (Heinz Hopf prize winner), G. Seregin (most noted for his lecture notes), E. Titi (the course on the mathematical theory of the Navier-Stokes equations), J. Bourgain (norm inflation), P. Constantin (early years of the regularity theory of Navier-Stokes), T. Tao (for constructing artificial blowup using dyadic shell model), T. Buckmaster (Clay Research Award winner), and many thinkers, who produced results of important kinds. New tools, techniques, philosophies, and subschools of thoughts percolated. To list a few, the traditional school of analysis, divided into former school of uniqueness results, followed by a shift towards non-uniqueness results, the school of convex integration (due to Camillo de Lellis, Laszlo Szekelyhidi Jr.), the school of symplectic geometry (due to Eva Miranda, Daniel Peralta-Salas), the school of numerically stabilised PDEs solution certification (Thomas Hou in Caltech), the school of PINN blowups (Karniadakis), and slightly more abstractly, analysis-free analysis in Bonn (more recent and isolated from rest). This is not to forget the permeation of some of these ideas into applied disciplines; every mathematical fluid dynamical idea has had its application somewhere lying dormant in the physical world, it is just that someone would now and then work hard and appreciate it to bring it about. Collaborations like Constantin-Goldstein, Moffatt-Kimura, Charles R. Doering, Miranda–Peralta-Salas, Sharma-Wilson, and many results in turbulence are of those kind.
The current state of affairs are very unsettling and tense. The source of joy of the unknown around the Navier-Stokes millennium question and the source of silence that permeated us all has vanished in a shocking state of affair and in a matter of few days and with vast amount of financial and computing resources deployed by OpenAI. Terry Tao has likened the OpenAI’s approach to treading one’s way to the end goal like a fast-paced automobile in a highway, without caring at all of proof digestion, proof exposition, proof canonicalisation, and proof publication. The community usually would have been happy and excited and positively surprised and happy about it – and there is indeed a first-order emotion of happiness in all of us, for the verified, machine-generated solution would let us investigate the theory of Navier-Stokes equations in a more goal-directed manner. But there are much higher-order, nuanced emotions in the mathematical community of a loss of mathematical culture, child’s play, and muddy-sluggish-inconsistent way to live one’s life around mathematics and mathematical questions; it is this spirit that had previously let us all live and stay close to the “silence” that had weirdly conspired to “force” us to work hard and give birth to each new idea for past many millennia, both big and small, and renewal of our understanding of a portion of this fluid-world in which we lived for centuries, a world engaged in perpetual creation (quoting Grothendieck verbatim here, for the lack of better words).
What lies ahead in the world of fluid dynamics, in the world of PDEs, and in the world of Navier-Stokes is unclear, unsure, and unknown to me, now that the shining and guiding light on the landscape is gone, and the most recent status around the proof is that is in “incomprehensible” to the community (though that should change soon to a better end). Everything seems bearish from here point onwards, to choose the language that the ones who caused this prefer to use, given that the Anthropic IPO is on the horizon. The mysterious forces that exist in the Bay area and the likes, do not care to snatch more of such stories of centuries, if not millennia, of human endeavours in a fraction of a second, and use the winds of derision to leave us with soul-less new foundations and abandoned construction sites (quoting Grothendieck again from Recoltes et Semailles, 3.6); for us, the chisel-and-hammer workers to sit, beautify, paint, colour, and fill with purpose, warmth, meaning, judgement, understanding, and a collective-consciousness-fabric. In the words of Terry Tao and his talk at SAIR, Caltech on 11th Sept, 2026, the steps following proof verification, of that of proof digestion, proof exposition, and ruminating on the impact of the proof on the adjacent fields seem to be all arid lands and empty at the moment. Broadly, there are a few suggestions to move forward: the mathematical community is suggesting to come together and design our own LLM, directed to our academic interests, as outlined in this declaration; SAIR has proposed to hold new LLM-free mathematical challenges; mathematical discourse video journal to give a talk on the problem, and inviting or asking the first person to give a mathematical insight on a machine-generated proof. In addition, call for slow down and not bringing the time factor (influenced from market) into the rest open problems is at rise (c.f., statements like “What has RH anything to do with 10 years” are at risk of losing their purity). Alain Connes has nicely suggested to not let AI take the agency of brain to be able to create mental images for them. A few decades ago Grothendieck had felt something similar when he outlined his experiences in the following paragraph
While a lot of suggestions have been in the forefront, such as ignoring the fear-mongering tactics of the AI industry, building a community-owned academic LLM, not caring much about the machine solution and doing mathematics at one’s own pace like before, adjusting the economy of doing mathematics, the issues that lie ahead in the context of Navier-Stokes to resolve are, how the new proof – once out with a decent digestion and explanation – helps one to think about arriving at an updated theory of PDEs, aids us in obtaining better predictive models in turbulence, tease out any underlying algebraic structure in the Navier-Stokes equations that might have been lying waiting to be discovered underneath these hundreds of pages of calculations, and last but not the least, investigate if a curiosity-inspired question like Navier-Stokes regularity has a real-world application (which I believe it to be the case; c.f., unreasonable effectiveness of mathematics, E. Wigner).
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