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Roger Temam writes in SOME DEVELOPMENTS ON NAVIER-STOKES EQUATIONS IN THE SECOND HALF OF THE 20th CENTURY:

A remarkable property of the Navier-Stokes equations is that they are one of the very few (if not the only) nonlinear equations in mathematical physics for which the nonlinearity is derived from mathematical argument (just chain rule differentiation) and not from physical modelling.

What are the other "very few" PDE possessing this property?

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The "very few" include at least

  • Einstein's equation in general relativity, derivable from a Lagrangian formulation by looking for the critical point of the Einstein-Hilbert functional.

  • Any place where minimal surfaces come up (cosmic strings and D-branes, soap film, fluids with surface tension).

  • The eikonal equation.

Furthermore, the line between "mathematical argument" and "physical modelling" is really blurry in PDEs. Sure, there are situations like the approximation of optical propagation by the nonlinear Schrodinger equation which are pretty clearly "physical models", but what about equations of kinetic theory? I find it hard to justify regarding the collision kernel as "physical modelling" while assigning to the $v\cdot\nabla v$ transport term of Navier Stokes as "pure mathematics".

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