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On the Hydrostatic Approximation of Compressible Anisotropic Navier–Stokes Equations–Rigorous Justification

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Abstract

In this work, we derive the hydrostatic approximation by taking the small aspect ratio limit to the Navier–Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in the general large scale geophysical motions meaning that the vertical scale is significantly smaller than horizontal. We derive the versatile relative entropy inequality. Applying the versatile relative entropy inequality we gave the rigorous derivation of the limit from the compressible Navier–Stokes equations to the compressible Primitive Equations. This is the first work where the relative entropy inequality was used for proving hydrostatic approximation - the compressible Primitive Equations.

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Notes

  1. For completeness we will write the system (PE):

    $$\begin{aligned} \left\{ \begin{array}{l} \partial _t {\mathbf {v}}+\text {div}_x({\mathbf {v}}\otimes {\mathbf {v}})+\partial _z (\mathbf{vw})+\nabla _x p=\Delta _x{\mathbf {v}}+\partial _{zz}{\mathbf {v}},\\ \partial _zp=0\\ \text {div}_x {\mathbf {v}}+\partial _z w=0, \end{array}\right. \end{aligned}$$
    (1.3)

  2. Let us mention that explanation of the periodicity boundary conditions for compressible PE can be found in paper by Liu and Titi–arxiv:1905.09367, Page 4, line 14.

  3. By (PE) we mean the incompressible Primitive Equation.

  4. Let us emphasize the result of Bresch and Jabin is valid only for small coefficients of viscosities.

  5. The calcuation of \(\rho w\) can be seen in details see Liu and Titi [46] page 1920 and see [54], page 4.

  6. The idea of relative entropy inequality was introduced by Dafermos [17] for hyperbolic equations and by Germain ([32]) to fluid dynamics. After that Feireisl and his coauthors proved that the relative entropy inequality is valid for all smooth test functions with appropriate boundary conditions, see [23]. Precisely, the test functions \(r, {\varvec{U}}\) in the relative entropy inequality are arbitrary smooth, r strictly positive, and \({\varvec{U}}\) satisfy the corresponding boundary conditions. Moreover, it is easy to check that the relative entropy inequality is satisfied as an equality as soon as the solution \(\rho , {{\varvec{u}}}\) is smooth enough. Moreover there is shown so-called the weak-strong uniqueness principle, which means that if the initial data of weak solutions and strong solutions coincide, weak solutions coincide with the unique strong solution on the time interval where the strong solution lives. This principle gives us the possibility to use relative entropy inequality and weak-strong principle to show e.g. singular limits from weak solutions to strong solution with corresponding scaling if the corresponding reminder can be estimated in appropriate way. For details see e.g. book of Feireisl, Novotný, [24].

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Acknowledgements

The authors are very much indebted to the anonymous referees for many helpful suggestions. Moreover, we would like to thank to Prof. A. Novotný for his remarks and suggestions during his visit in Prague in the spring 2021. We thank to Prof. Edriss Titi for his wonderful and great talks on the course“compact course Mathematical Analysis of Geophysical Models and Data Assimilation” held from 26 June to 10 July 2020, which offers insightful and constructive suggestions. The research of H. G is partially supported by the NSFC Grant No. 12171084 and the fundamental Research Funds for the Central Universities No. 2242022R10013. The research of Š.N. was supported by the Czech Sciences Foundation (GAČR), GA19-04243S, 22-01591S (final version), Premium Academia of Š. Nečasová and RVO 67985840. The research of T.T. was supported by the NSFC Grant No. 11801138.

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Authors and Affiliations

  1. School of Mathematical Sciences, Southeast University, Nanjing, 211189, People’s Republic of China

    Hongjun Gao

  2. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 11567, Praha 1, Czech Republic

    Šárka Nečasová

  3. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, People’s Republic of China

    Tong Tang

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  1. Hongjun Gao
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Correspondence to Šárka Nečasová.

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Communicated by Tohru Ozawa.

Dedicated to Professor Yoshihiro Shibata on the occasion of his 70th birthday.

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Gao, H., Nečasová, Š. & Tang, T. On the Hydrostatic Approximation of Compressible Anisotropic Navier–Stokes Equations–Rigorous Justification. J. Math. Fluid Mech. 24, 86 (2022). https://doi.org/10.1007/s00021-022-00717-z

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