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Jean Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace.

Acta Math. 63 (1934), 193–248

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Notes

  1. The existence of the derivatives \(\frac{\partial u_{j}}{\partial t}\), \(\frac{\partial^{2} u _{j}}{\partial x_{k} \partial x_{l}}\), \(\frac{\partial p}{\partial x _{j}}\) is necessary neither for the basic equations [integral formulation of conservation laws] nor for the systems [integral representation of a solution by hydrodynamic potentials], but we had to require them only for the proof.

  2. Moreover, we can verify that the total kinetic energy of the fluid is bounded in time; but it seems to be impossible to deduce from this fact that the motion itself is regular.

  3. […] it is an incorrect terminology to speak about the velocity of the fluid at a particular point […] What is to be understood in hydrodynamics by the velocity of the fluid is the mean value of velocities of molecules within a certain domain, being small but nevertheless containing many molecules.

  4. […] if we want to continue this calculus we will obtain a new interval \((T_{0}, T_{0}+T_{1})\) where in general \(T_{1}\) differs from \(T_{0}\). If the numbers \(T_{0}\), \(T_{1}\) etc. define a convergent series, the calculus will stop at a finite time \(t\). […] it seems to be probable that irregularities will show up in the interior of a viscous incompresssible fluid even if […] and the initial value are completely regular.

  5. Unfortunately, I did not succeed in studying the system (6). Thus we let open the question whether irregularities show up or not.

  6. I could not prove a theorem of uniqueness stating that to any initial value there corresponds a unique turbulent solution.

  7. We tried to prove the existence of a solution of the Navier-Stokes system for any initial value: we succeeded only by dispensing with the regularity of the solution at certain instants, suitably chosen such that this set has vanishing measure; […]

  8. …simplifies a lot the presentation and throws more light on the main difficulties.

  9. The remarkable investigations Leray dedicates to the question of differentiability, suggest a curious difference between dimensions \(n=2\) and \(n=3\). Whereas in the first case, at least when \(G\) equals the whole plane, we succeed in proving that the solution is infinitely many times continuously differentiable, the methods fail when \(n \geq3\) […] Also the failure of proofs of uniqueness in three dimensions is strange […] It is hard to believe that the initial value problem of viscous fluids for \(n=3\) could have more than one solution […]

  10. For the definition and properties of Besov spaces see [81].

  11. The theoretical study of fluid flow with an initial value leads, in very different cases, to the same conclusion: the existence of at least one weak solution which is regular and unique near the initial time, and which exists for all time. That’s a theorem on “weak” existence; do there exist theorems about regularity and uniqueness to complete it? In other words, a fluid flow initially regular remains so over a certain interval of time; then it goes on indefinitely; but does it remain regular and well-determined? We do not know the answer to this double question. It was addressed sixty years ago in an extremely particular case [48]. At that time H. Lebesgue, when asked for, declared: “Do not spend too much time on such a refractory question. Do something different!”.

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Farwig, R. Jean Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace.. Jahresber. Dtsch. Math. Ver. 119, 249–272 (2017). https://doi.org/10.1365/s13291-017-0160-y

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