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

The $3$-D primitive equations and incompressible Navier-Stokes equations with full hyper-viscosity and only horizontal hyper-viscosity are considered on the torus, i.e., the diffusion term $-\mathrm{\Delta }$ is replaced by $-\Delta+ \varepsilon(-\Delta)^{l}$orby$-\Delta + \varepsilon(-\Delta_H)^{l}$, respectively, where ${\mathrm{\Delta }}_H={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2$, $\Delta= \Delta_H

• \partial_z^2$,$\varepsilon> 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 solutions for initial data in $L^2$ holds if $-\mathrm{\Delta }$ is replaced by $(-\mathrm{\Delta }{)}^{5\mathrm{/}4}$. Here, for the primitive equations the corresponding result is proven for $(-\mathrm{\Delta }{)}^{8\mathrm{/}5}$. For the case of horizontal hyper-viscosity $l=2$ is sufficient in both cases. Strong convergence for $\epsilon \to 0$ 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 $L^2$ and weak-strong uniqueness.