Year: 2023
Type: preprint
Abstract: The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $ν_z=\mathcal{O}(\varepsilon^γ)$ where $γ=2$, one obtains the primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take two more limit equations into consideration: For $γ<2$ the $2D$-Navier-Stokes equations are obtained. For $γ>2$ the primitive equations with only horizontal viscosity $-Δ_H$ as $\varepsilon\to 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $ν_z=\varepsilon^2 δ$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $\varepsilon,δ\to 0$. The flexibility of our methods is also illustrated by the convergence for $δ\to \infty$ and $\varepsilon\to 0$ to the $2D$-Navier-Stokes equations.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Analysis on nonlinear diffusion and dynamic singular structure, Research on the well-posedness, regularity, and justification of numerical methods for fluid problems and related boundary-value problems, Development of the theory of diffusion equations for analysis on data separation, Analyse gekoppelter Systeme von Evolutionsgleichungen Kopplung durch Randbedingungen und Blockstrukturen, G6580465709
OpenAlex ID: W4389501470