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

Abstract

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 $-\Delta$ is replaced by $-\Delta +\epsilon (-\Delta {)}^l$ or by $-\Delta +\epsilon (-{\Delta }_H{)}^l$, respectively, where ${\Delta }_H={\partial }_x^2+{\partial }_y^2$, $\Delta ={\Delta }_H+{\partial }_z^2$, $\epsilon >0$, $l>1$. 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 $-\Delta$ is replaced by $(-\Delta {)}^{5/4}$. Here, for the primitive equations the corresponding result is proven for $(-\Delta {)}^{8/5}$. 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.

arXiv comment18 pages, 1 figure

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Cited by in corpus (3)

• The primitive equations with stochastic wind driven boundary conditions
• Gaussian invariant measures and stationary solutions of 2D Primitive Equations
• Adaptive hyperviscosity stabilisation for the RBF-FD method in solving advection-dominated transport equations