I just came across an interesting publication in the Quanta magazine called “Mathematicians Coax Fluid Equations Into Nonphysical Solutions” by Leila Sloman. The article states that a new paper by D. Albritton, E. Brué, and M. Colombo “set to appear in the Annals of Mathematics” and it will show non-unique solutions of the Navier-Stokes equations of hydrodynamics. The author further says that “The advance is another step toward understanding the discrepancy between Navier-Stokes and the physical world — a mystery that underlies one of math’s most famous open problems.”
There are many areas of mechanics which lead to paradoxical (non-unique or non-existent) solutions. Some of them involve friction, for example, the ill-posed (unstable) problems of frictional sliding of elastic bodies or problems involving the Painlevé paradoxes of friction.
Ill-posedness and instability are deeply related to each other. There may be different ways to regularize the ill-posedness in mechanical problems. In the past we have suggested an unusual approach: use three-valued logic to resolve the paradoxes. Please see our paper “Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes” by Michael Nosonovsky and Alexander D. Breki is published in Entropy 2019, 21(6), 620; https://doi.org/10.3390/e21060620
Related blog entries
Three-valued logic deals with frictional paradoxes (June 23, 2019)