A note on the existence of stationary vortex patches for the SQG equation in bounded domain

Cao, Daomin; Lai, Shanfa; Qin, Guolin

doi:10.1090/proc/16487

A note on the existence of stationary vortex patches for the SQG equation in bounded domain
HTML articles powered by AMS MathViewer

by Daomin Cao, Shanfa Lai and Guolin Qin;

Proc. Amer. Math. Soc. 151 (2023), 4881-4891

DOI: https://doi.org/10.1090/proc/16487

Published electronically: June 30, 2023

HTML | PDF | Request permission

Abstract:

By studying the contour dynamics equation and using the implicit function theorem, we prove the existence of stationary vortex patches with fixed vorticity and total flux for each patch for the surface quasi-geostrophic equation in a bounded domain near non-degenerate critical points of the Kirchhoff-Routh function.

References

Weiwei Ao, Juan Dávila, Manuel del Pino, Monica Musso, and Juncheng Wei, Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation, Trans. Amer. Math. Soc. 374 (2021), no. 9, 6665–6689. MR 4302173, DOI 10.1090/tran/8406
Daomin Cao, Guolin Qin, Weicheng Zhan, and Changjun Zou, Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation, J. Differential Equations 299 (2021), 429–462. MR 4295163, DOI 10.1016/j.jde.2021.07.026
Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J. 165 (2016), no. 5, 935–984. MR 3482335, DOI 10.1215/00127094-3449673
Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE 2 (2016), no. 1, Art. 1, 34. MR 3462104, DOI 10.1007/s40818-016-0007-3
Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, Mem. Amer. Math. Soc. 266 (2020), no. 1292, v+89. MR 4126257, DOI 10.1090/memo/1292
Dongho Chae, Peter Constantin, Diego Córdoba, Francisco Gancedo, and Jiahong Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math. 65 (2012), no. 8, 1037–1066. MR 2928091, DOI 10.1002/cpa.21390
Peter Constantin and Mihaela Ignatova, Critical SQG in bounded domains, Ann. PDE 2 (2016), no. 2, Art. 8, 42. MR 3595454, DOI 10.1007/s40818-016-0017-1
Peter Constantin and Mihaela Ignatova, Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications, Int. Math. Res. Not. IMRN 6 (2017), 1653–1673. MR 3658179, DOI 10.1093/imrn/rnw098
Peter Constantin and Mihaela Ignatova, Estimates near the boundary for critical SQG, Ann. PDE 6 (2020), no. 1, Paper No. 3, 30. MR 4093852, DOI 10.1007/s40818-020-00079-7
Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437, DOI 10.1088/0951-7715/7/6/001
Peter Constantin and Huy Quang Nguyen, Local and global strong solutions for SQG in bounded domains, Phys. D 376/377 (2018), 195–203. MR 3815216, DOI 10.1016/j.physd.2017.08.008
Peter Constantin and Huy Quang Nguyen, Global weak solutions for SQG in bounded domains, Comm. Pure Appl. Math. 71 (2018), no. 11, 2323–2333. MR 3862092, DOI 10.1002/cpa.21720
Coralie Renault, Relative equilibria with holes for the surface quasi-geostrophic equations, J. Differential Equations 263 (2017), no. 1, 567–614. MR 3631317, DOI 10.1016/j.jde.2017.02.050
Diego Córdoba, Javier Gómez-Serrano, and Alexandru D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal. 233 (2019), no. 3, 1211–1251. MR 3961297, DOI 10.1007/s00205-019-01377-6
Diego Córdoba, Marco A. Fontelos, Ana M. Mancho, and Jose L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005), no. 17, 5949–5952. MR 2141918, DOI 10.1073/pnas.0501977102
Francisco de la Hoz, Zineb Hassainia, and Taoufik Hmidi, Doubly connected V-states for the generalized surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 220 (2016), no. 3, 1209–1281. MR 3466846, DOI 10.1007/s00205-015-0953-z
Tarek M. Elgindi and In-Jee Jeong, Symmetries and critical phenomena in fluids, Comm. Pure Appl. Math. 73 (2020), no. 2, 257–316. MR 4054357, DOI 10.1002/cpa.21829
Claudia García, Kármán vortex street in incompressible fluid models, Nonlinearity 33 (2020), no. 4, 1625–1676. MR 4072388, DOI 10.1088/1361-6544/ab6309
Claudia García, Vortex patches choreography for active scalar equations, J. Nonlinear Sci. 31 (2021), no. 5, Paper No. 75, 31. MR 4284365, DOI 10.1007/s00332-021-09729-x
C. García and S. V. Haziot, Global bifurcation for corotating and counter-rotating vortex pairs, Comm. Math. Phys., published online, 2023, DOI 10.1007/s00220-023-04741-6.
Ludovic Godard-Cadillac, Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations, C. R. Math. Acad. Sci. Paris 359 (2021), 85–98. MR 4229041, DOI 10.5802/crmath.159
Ludovic Godard-Cadillac, Philippe Gravejat, and Didier Smets, Co-rotating vortices with $N$ fold symmetry for the inviscid surface quasi-geostrophic equation, Indiana Univ. Math. J. 72 (2023), no. 2, 603–650. MR 4596187, DOI 10.1512/iumj.2023.72.9206
Javier Gómez-Serrano, On the existence of stationary patches, Adv. Math. 343 (2019), 110–140. MR 3880855, DOI 10.1016/j.aim.2018.11.012
Philippe Gravejat and Didier Smets, Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation, Int. Math. Res. Not. IMRN 6 (2019), 1744–1757. MR 3932594, DOI 10.1093/imrn/rnx177
Zineb Hassainia and Taoufik Hmidi, On the V-states for the generalized quasi-geostrophic equations, Comm. Math. Phys. 337 (2015), no. 1, 321–377. MR 3324164, DOI 10.1007/s00220-015-2300-5
Zineb Hassainia and Taoufik Hmidi, Steady asymmetric vortex pairs for Euler equations, Discrete Contin. Dyn. Syst. 41 (2021), no. 4, 1939–1969. MR 4211209, DOI 10.3934/dcds.2020348
Zineb Hassainia and Miles H. Wheeler, Multipole vortex patch equilibria for active scalar equations, SIAM J. Math. Anal. 54 (2022), no. 6, 6054–6095. MR 4510650, DOI 10.1137/21M1415339
Siming He and Alexander Kiselev, Small-scale creation for solutions of the SQG equation, Duke Math. J. 170 (2021), no. 5, 1027–1041. MR 4255049, DOI 10.1215/00127094-2020-0064
Taoufik Hmidi and Joan Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys. 350 (2017), no. 2, 699–747. MR 3607460, DOI 10.1007/s00220-016-2784-7
T. Hmidi, L. Xue and Z. Xue, Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc, Preprint, arXiv:2210.08760, 2022.
Alexander Kiselev, Lenya Ryzhik, Yao Yao, and Andrej Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. of Math. (2) 184 (2016), no. 3, 909–948. MR 3549626, DOI 10.4007/annals.2016.184.3.7
Alexander Kiselev, Yao Yao, and Andrej Zlatoš, Local regularity for the modified SQG patch equation, Comm. Pure Appl. Math. 70 (2017), no. 7, 1253–1315. MR 3666567, DOI 10.1002/cpa.21677
G. Lapeyre, Surface quasi-geostrophy, Fluids 2 (2017), no. 1.
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
Huy Quang Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE 11 (2018), no. 4, 1029–1047. MR 3749375, DOI 10.2140/apde.2018.11.1029
Logan F. Stokols and Alexis F. Vasseur, Hölder regularity up to the boundary for critical SQG on bounded domains, Arch. Ration. Mech. Anal. 236 (2020), no. 3, 1543–1591. MR 4076071, DOI 10.1007/s00205-020-01498-3
V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189