This work is licensed under the Creative Commons Attribution 4.0 International License.
D. Lannes, Modeling shallow water waves, Nonlinearity, vol. 33, no. 5, pp. R1–R57, 2020.Search in Google Scholar
S. M. Griffies, Fundamentals of ocean climate models. Princeton University Press, Princeton, NJ, 2004. With a foreword by Trevor J. McDougall.Search in Google Scholar
D. J. Benney, Some properties of long nonlinear waves, Stud. Appl. Math, no. 52, pp. 45–50, 1973.Search in Google Scholar
V. Zakharov, Benney equations and quasiclassical approximation in the method of the inverse problem, Funct. Anal. Appl., vol. 14, pp. 89–98, 1980.Search in Google Scholar
V. Zakharov, On the Benney equations, Physica D: Nonlinear Phenomena, vol. 3, pp. 193–202, 1981.Search in Google Scholar
D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci.Éc. Norm. Supér. (4), vol. 49, no. 6, pp. 1445–1495, 2016.Search in Google Scholar
B. Di Martino, C. El Hassanieh, E. Godlewski, J. Guillod, and J. Sainte-Marie, Hyperbolicity of a semi-lagrangian formulation of the hydrostatic free-surface Euler system, arXiv preprint arXiv:2308.15083, 2023.Search in Google Scholar
V. Teshukov, On the hyperbolicity of long wave equations, Internat. Ser. Numer. Math., pp. 413–421, 1991.Search in Google Scholar
V. Teshukov, On Cauchy problem for long wave equations, Internat. Ser. Numer. Math., pp. 331–338, 1992.Search in Google Scholar
E. Knyazeva and A. A. Chesnokov, Stability criterion of shear fluid flow and the hyperbolicity of the long-wave equations, J. Appl. Mech. Tech. Phys., vol. 53, no. 5, pp. 657–663, 2012.Search in Google Scholar
A. A. Chesnokov, G. A. El, S. L. Gavrilyuk, and M. V. Pavlov, Stability of shear shallow water flows with free surface, SIAM J. Appl. Math., vol. 77, no. 3, pp. 1068–1087, 2017.Search in Google Scholar
Y. Brenier, Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math., vol. 127, no. 7, pp. 585–595, 2003.Search in Google Scholar
E. Grenier, On the derivation of homogeneous hydrostatic equations, M2AN Math. Model. Numer. Anal., vol. 33, no. 5, pp. 965–970, 1999.Search in Google Scholar
N. Masmoudi and T. Wong, On the Hs theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., vol. 204, no. 1, pp. 231–271, 2012.Search in Google Scholar
T. Gallay, Stability of vortices in ideal fluids: the legacy of Kelvin and Rayleigh, in Hyperbolic problems: theory, numerics, applications, vol. 10 of AIMS Ser. Appl. Math., pp. 42–59, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2020.Search in Google Scholar
D. Gérard-Varet and T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., vol. 77, no. 1-2, pp. 71–88, 2012.Search in Google Scholar
C. Collot, S. Ibrahim, and Q. Lin, Stable singularity formation for the inviscid primitive equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, vol. 41, no. 2, pp. 317–356, 2023.Search in Google Scholar
Y. Brenier, Comparaison du régime hydrostatique des fluides incompressibles non-visqueux et du régime quasineutre des plasmas, in Trends in applications of mathematics to mechanics (Nice, 1998), vol. 106 of Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., pp. 285–292, 2000.Search in Google Scholar
Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., vol. 52, no. 4, pp. 411–452, 1999.Search in Google Scholar
P. R. Gent and J.-C. McWilliams, Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., no. 1, pp. 591–609, 1990.Search in Google Scholar
B. Desjardins, D. Lannes, and J.-C. Saut, Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids, Water Waves, vol. 3, no. 1, pp. 153–192, 2021.Search in Google Scholar
T. Fradin, Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates, arXiv preprint arXiv:2406.13263, 2024.Search in Google Scholar
R. Bianchini and V. Duchêne, On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity, Comm. Partial Differential Equations in press. Available at arXiv:2206.01058., 2024.Search in Google Scholar
J. W. Miles, On the stability of heterogeneous shear flows, Journal of Fluid Mechanics, vol. 10, no. 4, pp. 496–508, 1961.Search in Google Scholar
L. N. Howard, Note on a paper of John W. Miles, J. Fluid Mech., no. 4, pp. 509–512, 1961.Search in Google Scholar
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., vol. 76, no. 12, pp. 3685–3768, 2023.Search in Google Scholar
G. K. Vallis, Atmospheric and oceanic fluid dynamics: Fundamentals and large-scale circulation, Cambridge University Press, 2017.Search in Google Scholar
T. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid boussinesq systems, SIAM J. Math. Anal, vol. 47, no. 6, pp. 4672–4684, 2015.Search in Google Scholar
M. Coti Zelati and M. Nualart, Limiting absorption principles and linear inviscid damping in the Euler-Boussinesq system in the periodic channel, arXiv preprint arXiv:2309.08445, 2023.Search in Google Scholar
R. Bianchini, M. Coti Zelati, and L. Ertzbischoff, Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit, arXiv preprint arXiv:2403.17857, 2024.Search in Google Scholar
D. Han-Kwan and M. Hauray, Stability issues in the quasineutral limit of the one-dimensional vlasov-poisson equation, Comm. Math. Phys, vol. 334, no. 2, pp. 1101–1152, 2015.Search in Google Scholar
D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Ration. Mech. Anal., vol. 221, no. 3, pp. 1317–1344, 2016.Search in Google Scholar
E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., vol. 53, no. 9, pp. 1067–1091, 2000.Search in Google Scholar
S. Friedlander, On nonlinear instability and stability for stratified shear flow, J. Math. Fluid Mech., vol. 3, no. 1, pp. 82–97, 2001.Search in Google Scholar
M. N. Rosenbluth, Necessary and sufficient condition for the stability of plane parallel inviscid flow, Internat. Ser. Numer. Math., vol. 7, no. 4, pp. 413–421, 1964.Search in Google Scholar
M. Renardy, Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., vol. 194, no. 3, pp. 877–886, 2009.Search in Google Scholar
A.-L. Dalibard, H. Dietert, D. Gérard-Varet, and F. Marbach, High frequency analysis of the unsteady interactive boundary layer model, SIAM J. Math. Anal., vol. 50, no. 4, pp. 4203–4245, 2018.Search in Google Scholar
P. Korn and E. S. Titi, Global well-posedness of the primitive equations of large-scale ocean dynamics with the gent- mcwilliams-redi eddy parametrization model, arXiv preprint arXiv:2304.03242, 2023.Search in Google Scholar
M. Adim, R. Bianchini, and V. Duchêne, Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems, Compt. Rend. Math. (in press), 2024.Search in Google Scholar
P. R. Gent, The energetically consistent shallow-water equations, Journal of the atmospheric sciences, vol. 50, no. 9, pp. 1323–1325, 1993.Search in Google Scholar
D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., vol. 238, no. 1-2, pp. 211–223, 2003.Search in Google Scholar
