## Analysis & PDE

• Anal. PDE
• Volume 11, Number 7 (2018), 1587-1624.

### On the global stability of a beta-plane equation

#### Abstract

We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as the $β$-plane approximation. In vorticity formulation, the model we consider is then given by the Euler equation with the addition of a linear anisotropic, nondegenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate.

Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a “double null form” that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.

#### Article information

Source
Anal. PDE, Volume 11, Number 7 (2018), 1587-1624.

Dates
Received: 21 October 2016
Accepted: 4 March 2018
First available in Project Euclid: 15 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.apde/1547521406

Digital Object Identifier
doi:10.2140/apde.2018.11.1587

Mathematical Reviews number (MathSciNet)
MR3810467

Zentralblatt MATH identifier
1393.35180

#### Citation

Pusateri, Fabio; Widmayer, Klaus. On the global stability of a beta-plane equation. Anal. PDE 11 (2018), no. 7, 1587--1624. doi:10.2140/apde.2018.11.1587. https://projecteuclid.org/euclid.apde/1547521406

#### References

• S. Alinhac, “An example of blowup at infinity for a quasilinear wave equation”, pp. 1–91 in Autour de l'analyse microlocale, edited by G. Lebeau, Astérisque 284, Société Mathématique de France, Paris, 2003.
• Y. Deng, A. D. Ionescu, B. Pausader, and F. Pusateri, “Global solutions of the gravity-capillary water wave system in 3 dimensions”, preprint, 2016. To appear in Acta Math.
• Y. Deng, A. D. Ionescu, and B. Pausader, “The Euler–Maxwell system for electrons: global solutions in 2D”, Arch. Ration. Mech. Anal. 225:2 (2017), 771–871.
• S. A. Denisov, “Double exponential growth of the vorticity gradient for the two-dimensional Euler equation”, Proc. Amer. Math. Soc. 143:3 (2015), 1199–1210.
• P. G. Drazin, Introduction to hydrodynamic stability, Cambridge University Press, Cambridge, 2002.
• T. M. Elgindi and K. Widmayer, “Long time stability for solutions of a $\beta$-plane equation”, Comm. Pure Appl. Math. 70:8 (2017), 1425–1471.
• P. Germain, N. Masmoudi, and J. Shatah, “Global solutions for 2D quadratic Schrödinger equations”, J. Math. Pures Appl. $(9)$ 97:5 (2012), 505–543.
• P. Germain, N. Masmoudi, and J. Shatah, “Global solutions for the gravity water waves equation in dimension 3”, Ann. of Math. $(2)$ 175:2 (2012), 691–754.
• Z. Hani, F. Pusateri, and J. Shatah, “Scattering for the Zakharov system in 3 dimensions”, Comm. Math. Phys. 322:3 (2013), 731–753.
• N. Hayashi and P. I. Naumkin, “Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations”, Amer. J. Math. 120:2 (1998), 369–389.
• A. D. Ionescu and B. Pausader, “Global solutions of quasilinear systems of Klein–Gordon equations in 3D”, J. Eur. Math. Soc. $($JEMS$)$ 16:11 (2014), 2355–2431.
• J. Kato and F. Pusateri, “A new proof of long-range scattering for critical nonlinear Schrödinger equations”, Differential Integral Equations 24:9-10 (2011), 923–940.
• A. Kiselev and V. Šverák, “Small scale creation for solutions of the incompressible two-dimensional Euler equation”, Ann. of Math. $(2)$ 180:3 (2014), 1205–1220.
• S. Klainerman, “The null condition and global existence to nonlinear wave equations”, pp. 293–326 in Nonlinear systems of partial differential equations in applied mathematics, I (Santa Fe, NM, 1984), edited by B. Nicolaenko et al., Lectures in Appl. Math. 23, American Mathematical Society, Providence, RI, 1986.
• A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics 9, Courant Institute of Mathematical Sciences, New York, 2003.
• J. C. McWilliams, Fundamentals of geophysical fluid dynamics, Cambridge University Press, 2006.
• S.-J. Oh and F. Pusateri, “Decay and scattering for the Chern–Simons–Schrödinger equations”, Int. Math. Res. Not. 2015:24 (2015), 13122–13147.
• J. Pedlosky, Geophysical fluid dynamics, Springer, 1987.
• F. Pusateri and J. Shatah, “Space-time resonances and the null condition for first-order systems of wave equations”, Comm. Pure Appl. Math. 66:10 (2013), 1495–1540.
• T. C. Sideris, “Formation of singularities in three-dimensional compressible fluids”, Comm. Math. Phys. 101:4 (1985), 475–485.
• J. Sukhatme and L. M. Smith, “Local and nonlocal dispersive turbulence”, Physics of Fluids 21:5 (2009), art. id. 056603.
• A. Zlatoš, “Exponential growth of the vorticity gradient for the Euler equation on the torus”, Adv. Math. 268 (2015), 396–403.