Mathematicians Find Wrinkle in Famed Fluid Equations

Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#6

Post by BucketSort » Fri, Dec 22, 2017, 5:01 AM UTC

Btw this is work relating to one of the famed Millennial Problems - http://www.claymath.org/millennium-problems/navier%E2%80%93s...

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#7

Post by westurner » Fri, Dec 22, 2017, 5:10 AM UTC

Navier-Stokes equations: https://en.wikipedia.org/wiki/Navier–Stokes_equations

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#8

Post by boxfire » Fri, Dec 22, 2017, 5:16 AM UTC

foota wrote: post #5

Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?

That's correct, while taking away the entropy inequality as well. The thing is that very inequality is one of these laws of thermodynamics we so love, so I kind of see this as hot air.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#9

Post by ianai » Fri, Dec 22, 2017, 5:20 AM UTC

boxfire wrote: post #8

foota wrote: post #5

Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?

That's correct, while taking away the entropy inequality as well. The thing is that very inequality is one of these laws of thermodynamics we so love, so I kind of see this as hot air.

Aye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#10

Post by CamperBob2 » Fri, Dec 22, 2017, 5:24 AM UTC

   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 permissible: 
   The water starts still, erupts in the middle
   of the night, then returns to stillness.

Also permissible, and also vanishingly unlikely, under quantum theory.

It's a stretch, but I wonder if it's reasonable to think of data points in the vector field describing fluid motion as probabilities rather than definite measurements. That might allow their behavior to be modeled with different mathematical tools.