Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows
Roberta Bianchini1; Michele Coti Zelati2; Michele Dolce3
1 IAC, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy
2 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
3 Institute of Mathematics, EPFL, Station 8, 1015 Lausanne, Switzerland
Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 5, 17 p.

Significant advancements have emerged in the theory of asymptotic stability of shear flows in stably stratified fluids. In this comprehensive review, we spotlight these recent developments, with particular emphasis on novel approaches that exhibit robustness and applicability across various contexts.

Published online:
DOI: 10.5802/slsedp.160
Roberta Bianchini 1; Michele Coti Zelati 2; Michele Dolce 3

1 IAC, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy
2 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
3 Institute of Mathematics, EPFL, Station 8, 1015 Lausanne, Switzerland 
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Roberta Bianchini; Michele Coti Zelati; Michele Dolce. Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows. Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 5, 17 p. doi : 10.5802/slsedp.160. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.160/

[1] P. Antonelli, M. Dolce, and P. Marcati, Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid, Ann. PDE, 7 (2021), pp. Paper No. 24, 53. | DOI | MR | Zbl

[2] J. Bedrossian, R. Bianchini, M. Coti Zelati, and M. Dolce, Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations, Comm. Pure Appl. Math., to appear. 2021. | arXiv

[3] J. Bedrossian and N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations, Publ. Math. Inst. Hautes Études Sci., 122 (2015), pp. 195–300. | DOI | MR | Zbl

[4] R. Bianchini, M. Coti Zelati, and M. Dolce, Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime, Indiana Univ. Math. J., 71 (2022), pp. 1467–1504. | DOI | MR | Zbl

[5] O. Bühler, Waves and Mean Flows, Cambridge University Press, 2009. | DOI | Zbl

[6] Q. Chen, D. Wei, P. Zhang, and Z. Zhang, Nonlinear inviscid damping for 2-D inhomogeneous incompressible Euler equations, arXiv e-prints. 2023. | arXiv

[7] P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2), 168 (2008), pp. 643–674. | DOI | MR | Zbl

[8] M. Coti Zelati and A. Del Zotto, Suppression of lift-up effect in the 3D Boussinesq equations around a stably stratified Couette flow. 2023. | arXiv

[9] M. Coti Zelati and M. Nualart, Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip. 2023. | arXiv

[10] —, Limiting absorption principles and linear inviscid damping in the Euler-Boussinesq system in the periodic channel. 2023. | arXiv

[11] T. Dauxois, S. Joubaud, P. Odier, and A. Venaille, Instabilities of internal gravity wave beams, in Annual review of fluid mechanics. Vol. 50, vol. 50 of Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 2018, pp. 131–156. | DOI | MR | Zbl

[12] Y. Deng and N. Masmoudi, Long-time instability of the Couette flow in low Gevrey spaces, Communications on Pure and Applied Mathematics, 76 (2023), pp. 2804–2887. | DOI | MR | Zbl

[13] M. Dolce, Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables. 2023. | arXiv

[14] P. Drazin and L. Howard, Hydrodynamic stability of parallel flow of inviscid fluid, Advances in Applied Mechanics, 9 (1966), pp. 1–89. | DOI

[15] P. G. Drazin and W. H. Reid, Hydrodynamic stability, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second ed., 2004. With a foreword by John Miles.

[16] T. Gallay, Stability of vortices in ideal fluids: the legacy of Kelvin and Rayleigh, Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the XVII International Conference on Hyperbolic Problems (HYP2018), (2020), p. 690.

[17] R. Hartman, Wave propagation in a stratified shear flow, J. Fluid Mech., 71 (1975), pp. 89–104. | DOI | Zbl

[18] L. N. Howard, Note on a paper of John W. Miles, Journal of Fluid Mechanics, 10 (1961), pp. 509–512. | DOI | MR | Zbl

[19] N. Masmoudi, B. Said-Houari, and W. Zhao, Stability of the Couette flow for a 2D Boussinesq system without thermal diffusivity, Arch. Ration. Mech. Anal., 245 (2022), pp. 645–752. | DOI | MR | Zbl

[20] N. Masmoudi and W. Zhao, Stability threshold of two-dimensional Couette flow in Sobolev spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39 (2022), pp. 245–325. | DOI | MR | Zbl

[21] J. W. Miles, On the stability of heterogeneous shear flows, Journal of Fluid Mechanics, 10 (1961), pp. 496–508. | DOI | MR | Zbl

[22] M. Rieutord, Fluid dynamics: an introduction, Springer, 2014.

[23] J. Yang and Z. Lin, Linear inviscid damping for Couette flow in stratified fluid, J. Math. Fluid Mech., 20 (2018), pp. 445–472. | DOI | MR | Zbl

[24] C. Zhai and W. Zhao, Stability threshold of the Couette flow for Navier-Stokes Boussinesq system with large Richardson Number γ 2 >1 4, SIAM J. Math. Anal., 55 (2023), pp. 1284–1318. | DOI | MR | Zbl

[25] W. Zhao, Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel. 2023. | arXiv

[26] C. Zillinger, On enhanced dissipation for the Boussinesq equations, J. Differential Equations, 282 (2021), pp. 407–445. | DOI | MR | Zbl

[27] —, On enhanced dissipation for the Boussinesq equations, J. Differential Equations, 282 (2021), pp. 407–445. | DOI | MR | Zbl

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