Cauchy convergence schemes for some nonlinear partial differential equations

Abstract

Motivated by ongoing work in the theory of stochastic partial differential equations we develop direct methods to infer that the Galerkin approximations of certain nonlinear partial differential equations are Cauchy (and therefore convergent). We develop such a result for the Navier–Stokes equations in space dimensions two and three, and for the primitive equations in space dimension two. The analysis requires novel estimates for the nonlinear portion of these equations and delicate interpolation results concerning subspaces.

Keywords:

• nonlinear PDE
• primitive equations
• Navier–Stokes
• Galerkin approximations
• compactness methods
• fractional spaces
• interpolation
• comparison estimates
• stochastic PDE

AMS Subject Classifications::

• 35Q86
• 35Q30
• 46B70

Acknowledgements

This work was partially supported by the National Science Foundation under the grants NSF-DMS-0604235 and NSF-DMS-0906440, and by the Research Fund of Indiana University.

Notes

Notes

1. One sometimes also finds the more general definition (U, U ^♯) ≔ ∫_ℳ v · v ^♯ dℳ + κ∫_ℳ TT ^♯ dℳ with κ > 0 fixed. This κ is useful for the coherence of physical dimensions and for (mathematical) coercivity. Since this is not needed here we take κ = 1.

2. B is also continuous from V × V to V′ and satisfies important cancellation properties and estimates. Since we are considering strong solutions in C([0, t], V) ∩ L ²([0, t], D(A)) these properties will not be needed here.