Partial and full hyper-viscosity for Navier-Stokes and primitive equations

Abstract

The 3-D primitive equations and incompressible Navier-Stokes equations with either full hyper-viscosity or only horizontal hyper-viscosity are considered on the torus, i.e., the diffusion term −Δ is replaced by −Δ+ε(−Δ)l or by −Δ+ε(−ΔH)l, respectively, where ΔH=∂x2+∂y2, Δ=ΔH+∂z2, ε>0, l>1. Hyper-viscosity is applied in many numerical schemes, and in particular horizontal hyper-viscosity appears in meteorological models. A classical result by Lions states that for the Navier-Stokes equations uniqueness of global weak Leray-Hopf solutions for initial data in L2 holds if −Δ is replaced by (−Δ)5/4. Here, for the primitive equations the corresponding result is proven for (−Δ)8/5. For the case of horizontal hyper-viscosity l=2 is sufficient in both cases. Strong convergence for ε→0 of hyper-viscous solutions to a weak solution of the Navier-Stokes and primitive equations, respectively, is proven as well. The approach presented here is based on the construction of strong solutions via an evolution equation approach for initial data in L2 and weak-strong uniqueness.