Sage Journals: Discover world-class research

Abstract

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Keywords

Incompressible Euler equations vorticity fronts asymptotic equations modified energy

Get full access to this article

View all access options for this article.

References

Bahouri,

J.-Y.

Chemin and

Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011.

Biello and

J.K.

Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities, Comm. Pure Appl. Math 63 (2010), 303–336.

Biello and

J.K.

Hunter, Contour dynamics for vorticity discontinuities. In preparation.

Bressan and

K.T.

Nguyen, Global existence of weak solutions for the Burgers–Hilbert equation, SIAM J. Math. Anal. 46(4) (2014), 2884–2904. doi:10.1137/140957536.

Bressan and

Zhang, Piecewise smooth solutions to the Burgers–Hilbert equation, Comm. Math. Sci. 15 (2017), 165–184. doi:10.4310/CMS.2017.v15.n1.a7.

Castro,

Córdoba and

Gancedo, Singularity formations for a surface wave model, Nonlinearity 23(11) (2010), 2835–2849. doi:10.1088/0951-7715/23/11/006.

Córdoba,

Gómez-Serrano and

A.D.

Ionescu, Global solutions for the generalized SQG patch equation, Arch. Rational Mech. Anal. 233(3) (2019), 1211–1251. doi:10.1007/s00205-019-01377-6.

Cummings and

C.E.

Wayne, Modified energy functionals and the NLS approximation, Discrete Contin. Dyn. Syst. 37 (2017), 1295–1321. doi:10.3934/dcds.2017054.

Dawson,

McGahagan and

Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc. 136(6) (2008), 2081–2090. doi:10.1090/S0002-9939-08-09355-6.

10.

D.G.

Dritschel, The repeated filamentation of two-dimensional vorticity interfaces, J. Fluid Mech. 194 (1988), 511–547. doi:10.1017/S0022112088003088.

11.

D.G.

Dritschel, Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows, Comput. Phys. Rep. 10 (1989), 77–146. doi:10.1016/0167-7977(89)90004-X.

12.

W.-P.

Düll, Justification of the nonlinear Schrödinger approximation for a quasilinear Klein–Gordon equation, Comm. Math. Phys. 355(3) (2017), 1189–1207. doi:10.1007/s00220-017-2966-y.

13.

W.-P.

Düll and

Heß, Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation, J. Differential Equations 264(4) (2018), 2598–2632. doi:10.1016/j.jde.2017.10.031.

14.

J.K.

Hunter, The Burgers–Hilbert equation. Theory, numerics and applications of hyperbolic problems II, Springer Proc. Math. Stat. 237 (2018), 41–57.

15.

J.K.

Hunter and

Ifrim, Enhanced life span of smooth solutions of a Burgers–Hilbert equation, SIAM J. Math. Anal. 44(3) (2012), 2039–2052. doi:10.1137/110849791.

16.

J.K.

Hunter,

Ifrim,

Tataru and

T.K.

Wong, Long time solutions for a Burgers–Hilbert equation via a modified energy method, Proc. Amer. Math. Soc. 143(8) (2015), 3407–3412. doi:10.1090/proc/12215.

17.

J.K.

Hunter and

Shu, Regularized and approximate equations for sharp fronts in the surface quasi-geostrophic equation and its generalizations, Nonlinearity 31(6) (2018), 2480–2517. doi:10.1088/1361-6544/aab1cc.

18.

J.K.

Hunter,

Shu and

Zhang, Contour dynamics for surface quasi-geostrophic fronts, Nonlinearity 33(9) (2020), 4699–4714. doi:10.1088/1361-6544/ab8d16.

19.

J.K.

Hunter,

Shu and

Zhang, Two-front SQG equation and its generalizations, Commun. Math. Sci. 18(6) (2020), 1685–1741. doi:10.4310/CMS.2020.v18.n6.a8.

20.

J.K.

Hunter,

Shu and

Zhang, Global solutions of a surface quasi-geostrophic front equation, Pure and Applied Analysis 3(3) (2021), 403–472. doi:10.2140/paa.2021.3.403.

21.

J.K.

Hunter,

Shu and

Zhang, Global solutions for a family of GSQG front equations, Preprint. arXiv:2005.09154.

22.

V.M.

Hur, Norm inflation for equations of KdV type with fractional dispersion, Differential Integral Equations 31(11–12) (2018), 833–850.

23.

Ifrim, Normal form transformations for quasilinear wave equations, Ph.D. thesis, University of California, Davis, 2012.

24.

Ifrim and

Tataru, The NLS approximation for two dimensional deep gravity waves, Sci. China Math. 62(6) (2019), 1101–1120. doi:10.1007/s11425-018-9501-y.

25.

Ifrim and

Tataru, Local well-posedness for quasilinear problems: A primer, Preprint. arXiv:2008.05684.

26.

C.E.

Kenig,

Pilod,

Ponce and

Vega, On the unique continuation of solutions to non-local non-linear dispersive equations, Comm. Partial Differential Equations 45(8) (2020), 872–886. doi:10.1080/03605302.2020.1739707.

27.

C.E.

Kenig,

Ponce and

Vega, Uniqueness properties of solutions to the Benjamin–Ono equation and related models, J. Funct. Anal. 278(5) (2020), 108396. doi:10.1016/j.jfa.2019.108396.

28.

S.G.

Krupa and

A.F.

Vasseur, Stability and uniqueness for piecewise smooth solutions to a nonlocal scalar conservation law with applications to Burgers–Hilbert equation, SIAM J. Math. Anal. 52(3) (2020), 2491–2530. doi:10.1137/19M1257883.

29.

Lord, On the stability or instability of certain fluid motions, Proc. Lond. Math. Soc. 11 (1880), 57.

30.

Lord, On the propagation of waves upon the plane surface separating two portions of fluid of different vorticities, Proc. Lond. Math. Soc. 27(1) (1895), 13–18.

31.

A.J.

Majda and

A.L.

Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.

32.

Marsden and

Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7(1–3) (1983), 305–323. doi:10.1016/0167-2789(83)90134-3.

33.

Pu and

Liu, Justification of the NLS approximation for the Euler–Poisson equation, Comm. Math. Phys. 371(2) (2019), 357–398. doi:10.1007/s00220-019-03576-4.

34.

J.-C.

Saut and

Wang, The wave breaking for Whitham-type equations revisited, Preprint. arXiv:2006.03803.

35.

Schneider and

Uecker, Nonlinear PDEs: A Dynamical Systems Approach, American Mathematical Society, Providence, RI, 2017.

36.

Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996.

37.

Yang, Shock formation for the Burgers–Hilbert equation, Preprint. arXiv:2006.05568.