Re:When Math Gets Impossibly Hard
by btrettel on Friday September 18, 2020 @07:51AM
(#24515551)
> Mathematicians do not yet know how to prove that they have a solution
Probably just a typo, but it's not hard to find particular solutions. The question is more whether solutions exist for all reasonable initial conditions. Here's a famous particular solution:
I've used a variant of this one to test CFD software.
The Millenium prize problem has a succinct statement of what's under question:
> In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
Keep in mind that this is for incompressible Navier-Stokes with a constant viscosity. For inviscid flows you could get discontinuities as you've indicated with the inviscid Burgers equations, and I know that existence and uniqueness problem for the incompressible Navier-Stokes equations has been proved for certain uncommon viscosity laws (which should be reasonably realistic as a constant viscosity is not quite right).
(And of course there are other complications not considered in this problem like boundary conditions.)
Probably just a typo, but it's not hard to find particular solutions. The question is more whether solutions exist for all reasonable initial conditions. Here's a famous particular solution:
https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex
I've used a variant of this one to test CFD software.
The Millenium prize problem has a succinct statement of what's under question:
> In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
Keep in mind that this is for incompressible Navier-Stokes with a constant viscosity. For inviscid flows you could get discontinuities as you've indicated with the inviscid Burgers equations, and I know that existence and uniqueness problem for the incompressible Navier-Stokes equations has been proved for certain uncommon viscosity laws (which should be reasonably realistic as a constant viscosity is not quite right).
(And of course there are other complications not considered in this problem like boundary conditions.)