Abstract
In this paper, we consider patch solutions to the \(\alpha \)-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Strain Gancedo and Strain (2014), providing a condition on the growth of curvature of the patch necessary for the splash and an exponential in time lower bound on the distance between patches with bounded curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Castro, A., Cordoba, D., Fefferman, C., Gancedo, F., Gomez-Serrano, J.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. 178(3), 1061–1134 (2013)
Chae, D., Constantin, P., Córdoba, D., Gancedo, F., Wu, J.: Generalized surface quasi-geostrophic equations with singular velocities. Comm. Pure Appl. Math. 65(8), 1037–1066 (2012)
Chemin, J.-Y.: Persistance de structures geometriques dans les fluides incompressibles bidimensionnels. Ann. de l’École Norm. Supér 26, 1–26 (1993)
Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)
Constantin, P., Iyer, G., Wu, J.: Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 57(6), 2681–2692 (2008)
Cordoba, D.: Nonexistence of simple hyperbolic blow up for the quasi-geostrophic equation. Ann. Math. 148, 1135–1152 (1998)
Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15, 665–670 (2002)
Cordoba, D., Fefferman, C., de la Llave, R.: On squirt singularities in hydrodynamics. SIAM J. Math. Anal. 36(1), 204–213 (2004)
Cordoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949–5952 (2005)
Cordoba, A., Cordoba, D., Gancedo, F.: Uniqueness for SQG patch solutions. Trans. Amer. Math. Soc. Ser. B 5, 1–31 (2018)
Coutand, D., Shkoller, S.: On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Comm. Math. Phys. 325(1), 143–183 (2014)
Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Gancedo, F.: Existence for the \(\alpha \)-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217, 2569–2598 (2008)
Gancedo, F., Strain, R.: Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. USA 111, 635–639 (2014)
Gancedo, F., Patel, N.: On the local existence and blow-up for generalized SQG patches. Ann. PDE, 7(1):Paper No. 4, 63, (2021)
Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)
Jeon, J., Zlatoš, A.: An improved regularity criterion and absence of splash-like singularities for g-SQG patches. preprint, (2021)
Kiselev, A., Sverak, V.: Small scale creation for solutions of the incompressible two dimensional Euler equation. Ann. Math. 180, 1205–1220 (2014)
Kiselev, A., Li, C.: Global regularity and fast small scale formation for Euler patch equation in a smooth domain. Comm. Partial Differ. Equ 44(4), 279–308 (2019)
Kiselev, A., Ryzhik, L., Yao, Y., Zlatoš, A.: Finite time singularity for the modified SQG patch equation. Ann. Math. 184(3), 909–948 (2016)
Luo, G., Hou, T.: Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation. Multiscale Model. Simul. 12, 1722–1776 (2014)
Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York, Heidelberg (1994)
Rodrigo, J.L.: On the evolution of sharp fronts for the quasi-geostrophic equation. Comm. Pure Appl. Math. 58, 821–866 (2005)
Scott, R.K., Dritschel, D.G.: Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system. Phys. Rev. Lett. 112, 144505 (2014)
Scott, R.K., Dritschel, D.G.: Scale-invariant singularity of the surface quasigeostrophic patch. J. Fluid Mech. 863(12), 86–08 (2019)
Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Zh Vych Mat 3, 1032–1066 (1963)
Acknowledgements
AK acknowledges partial support of the NSF-DMS grant 2006372 and of the Simons Fellowship grant 667842. XL is partially supported by the NSF-DMS grant 1926686. We thank Andrej Zlatoš for noticing a miscalculation in the earlier version of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Robert Buckingham.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kiselev, A., Luo, X. On Nonexistence of Splash Singularities for the \(\alpha \)-SQG Patches. J Nonlinear Sci 33, 37 (2023). https://doi.org/10.1007/s00332-023-09893-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-023-09893-2