Mathematicians Find Wrinkle in Famed Fluid Equations

Post by [deleted] » Sat, Dec 23, 2017, 10:17 AM UTC

fspeech wrote: post #119

geezerjay wrote: post #112

Earlier quoted context omitted.

> but a case where multiple different outcomes can arise from identical inputs. If they are identical but not equal and these small differences in the input lead to large deviations in the output then that's pretty much the definition of a chaotic system. Edit: I've just browsed through the paper in question and it actually demonstrates thar an approximate (weak) solution is not unique, which means that the exact sam…

"Weak" here does not mean numerical approximations. These are mathematicians so any quantitative approximation would be bounded o(1) otherwise the work would be meaningless. One should think of "weak" as in constraints, for example constraints at lower spatial resolutions (but precise).

Post by xaedes » Sun, Dec 24, 2017, 11:46 PM UTC

CamperBob2 wrote: post #10

Using this approach, Buckmaster and Vicol prove that these very weak solutions to the Navier-Stokes equations are nonunique. They demonstrate, for example, that if you start with a completely calm fluid, like a glass of water sitting still by your bedside, two scenarios are possible. The first scenario is the obvious one: The water starts still and remains still forever. The second is fantastical but mathematically p…

   The second is 
   fantastical but mathematically permissible: 
   The water starts still, erupts in the middle
   of the night, then returns to stillness.