Navier-Stokes equations documented in four Wikipedia languages; existence problem has own entries

The material available on this subject is encyclopedia text rather than dated reporting. Six Wikipedia entries cover the Navier-Stokes equations across four languages: French (s1), German (s2, s3), Spanish (s4) and English (s5, s6). None of the six carries a date tied to an event, a result, or an announcement. The one structural fact they do establish is that the existence-and-smoothness question is catalogued apart from the equations themselves, in both English (s5) and German (s3).

The French lead article (s1) organizes the equations around conservation laws. Its contents list notation and the relations used, the law of conservation itself, an Eulerian formulation, variations on the equation system, conservation of entropy, and Lagrangian and Lagrange-Euler formulations (s1). It extends to coordinate-specific expressions in Cartesian, cylindrical and spherical systems, and to reference frames in translation and rotation (s1).

Physical modelling sits further down the same French page: a section on the Newtonian fluid and the Stokes hypothesis, then thermodynamic properties and transport properties split between gases and liquids (s1). The German article (s2) follows a comparable order, opening with a history section (Geschichte) before formulation (Formulierung) and the momentum equation (Impulsgleichung).

The open question is documented as a subject in its own right. The English entry Navier-Stokes existence and smoothness (s5) states the problem in the whole space, separates two settings, unbounded and periodic space, and adds a hypotheses section. The German counterpart (s3) carries the same topic and includes a progress section (Fortschritte) alongside its own formulation of the equation.

Spanish coverage (s4) reaches the equations through prior concepts. Its opening sections list the substantial, or material, derivative and the Reynolds transport theorem, the tools a reader is expected to hold before the formulation begins.

English carries two separate entries. The general equations page (s6) is organized around flow velocity, general continuum equations with convective acceleration, and distinct treatments of compressible and incompressible flow. The existence-and-smoothness page (s5) stands apart from it, so a reader looking for the general system and a reader looking for the unsolved problem land in different articles.

Nothing in the supplied material supports a claim that something changed. The six sources are reference entries without dates, and none reports a proof, a counterexample, a dispute, or a new numerical result. Reading across the language editions shows how the encyclopedias divide the topic, not that the field has moved. A reversal or breakout framing would not be defensible on this sourcing, and the category hint of a reversal is not matched by anything in the text.

What a reader can watch is the structure the sources themselves expose. The separate existence-and-smoothness entries (s5, s3) each carry a progress section, which is where a documented advance would surface first. Until a dated source appears, the only defensible statement is that the equations and the problem named after them are catalogued separately across at least four language editions, and that the provided material establishes nothing further.