Numerical verification of regularity for solutions of the 3D Navier-Stokes equations

Numerical verification of regularity for solutions of the 3D Navier-Stokes equations

Mon, 14/02/2011
17:00 		James Robinson (University of Warwick) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Numerical verification of regularity for solutions of the 3D Navier-Stokes equations
I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data. The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $ H^{1/2} $. I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $ u_0\in H^1 $ with $ \|u\|_{H^1}\le R $ gives rise to a solution of the unforced equation that remains regular for all $ t\ge 0 $. This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).