July  2022, 42(7): 3143-3168. doi: 10.3934/dcds.2022012

Vortex collapses for the Euler and Quasi-Geostrophic models

Laboratoire de Mathématiques Jean Leray, Nantes Université, 2, chemin de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France

Received  January 2021 Revised  January 2022 Published  July 2022 Early access  February 2022

Fund Project: The author is supported by grant ANR-18-CE40-0020-01

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

Citation: Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012
References:
[1]

W. AoJ. DavilaM. Del PinoM. Musso and W. Juncheng, Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation, Trans. Am. Math. Soc., 374 (2021), 6665-6689.  doi: 10.1090/tran/8406.

[2]

H. Aref, Motion of three vortices, Phys. of Fluids, 22 (1979), 393-400.  doi: 10.1063/1.862605.

[3]

H. Aref, Self-similar motion of three point vortices, Phys. of Fluids, 22 (2010), 057104.  doi: 10.1063/1.3425649.

[4]

V. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.

[5]

G. Badin and A. Barry, Collapse of generalized Euler and surface quasi-geostrophic point-vortices, Phys. Rev. E., 98.

[6]

T. BuckmasterS. Shkoller and V. Vicol, Nonuniqueness of weak solutions to the SQG equations, Comm. Pure Appl. Math., 72 (2016), 1809-1874.  doi: 10.1002/cpa.21851.

[7]

A. Castro, D. Córdoba and J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, Mem. Am. Math. Soc., 266 (2020), v+89 pp. doi: 10.1090/memo/1292.

[8]

G. Cavallaro, R. Garra and C. Marchioro, Long time localization of modified surface quasi-geostrophic equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5135–5148. arXiv: 2004.11183 [math-ph]. doi: 10.3934/dcdsb.2020336.

[9]

P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.  doi: 10.1088/0951-7715/7/6/001.

[10]

M. Donati, Two-dimensional point vortex dynamics in bounded domains: Global existence for almost every initial data, SIAM Journal on Mathematical Analysis, 54 (2022), 79-113. doi: 10.1137/21M1413213.

[11]

C. Garcia, Vortex patches choreography for active scalar equations, Jour. of Nonlinear Sci., 34 (2021), Paper No. 75, 31 pp. doi: 10.1007/s00332-021-09729-x.

[12]

C. Geldhauser and M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, Am. Ins. Math. Sci., 25 (2020), 2583-2606.  doi: 10.3934/dcdsb.2020023.

[13]

L. Godard-Cadillac, Smooth traveling-wave solutions to the generalized inviscid surface quasi-geostrophic equation, Comptes Rendus Math. Ac. Sci., 359 (2021), 85-98.  doi: 10.5802/crmath.159.

[14]

L. Godard-Cadillac, P. Gravejat and D. Smets, Co-rotating vortices with n fold symmetry for the inviscid surface quasi-geostrophic equation, Preprint, arXiv: 2010.08194.

[15]

P. Gravejat and D. Smets, Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation, Int. Math. Res. Not., 2019 (2019), 1744-1757.  doi: 10.1093/imrn/rnx177.

[16]

F. Grotto and U. Pappalettera, Burst of point-vortices and non-uniqueness of 2D euler equations., preprint, arXiv: 2011.13329 [math.DS].

[17]

T. Hmidi and J. Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys., 350 (2017), 699-747.  doi: 10.1007/s00220-016-2784-7.

[18]

A. Kiselev and F. Nazarov, A simple energy pump for the surface quasi-geostrophic equation, in Nonlinear Partial Differential Equations, vol. 7 of Abel Symposia, Holden, H. and Karlsen, K.H., 2012,175–179. doi: 10.1007/978-3-642-25361-4_9.

[19]

C. Marchioro and M. Pulvirenti, Vortex methods in two-dimensionnal fluid mechanics, Springer-Verlag, Berlin, 1984.

[20]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied mathematical sciences, Applied Mathematical Sciences, 96. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[21]

E. Noether, Invariante variationsprobleme, Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys, 235–255.

[22]

E. Novikov, Dynamics and statistics of a system of vortices, Zh. Eksp. Teor. Fiz., 41 (1975), 937-943. 

[23]

E. Novikov and Y. Sedov, Vortex collapse, Zh. Eksp. Teor. Fiz., 77 (1979), 588-597. 

[24] J. Pedlowsky, Geophysical Fluid Dynamics, Springer-Verlag, New-York, 1979. 
[25]

J. Reinaud, Self-similar collapse of three geophysical vortices, Geophysical & Astrophysical Fluid Dynamics, 115 (2021), 369-392.  doi: 10.1080/03091929.2020.1828402.

[26]

S. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, 1995.

[27]

M. Rosenzweig, Justification of the point vortex approximation for modified surface quasi-geostrophic equations, SIAM Journal on Mathematical Analysis, 52 (2020), 1690-1728.  doi: 10.1137/19M1262620.

[28]

D. Smets and J. Van Schaftingen, Desingularization of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.

[29] G. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006. 
[30]

V. Yudovich, Non-stationary flows of an ideal incompressible fluid, USSR Comput. Math. and Math. Phys., 3 (1963), 1407-1456.  doi: 10.1016/0041-5553(63)90247-7.

show all references

References:
[1]

W. AoJ. DavilaM. Del PinoM. Musso and W. Juncheng, Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation, Trans. Am. Math. Soc., 374 (2021), 6665-6689.  doi: 10.1090/tran/8406.

[2]

H. Aref, Motion of three vortices, Phys. of Fluids, 22 (1979), 393-400.  doi: 10.1063/1.862605.

[3]

H. Aref, Self-similar motion of three point vortices, Phys. of Fluids, 22 (2010), 057104.  doi: 10.1063/1.3425649.

[4]

V. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.

[5]

G. Badin and A. Barry, Collapse of generalized Euler and surface quasi-geostrophic point-vortices, Phys. Rev. E., 98.

[6]

T. BuckmasterS. Shkoller and V. Vicol, Nonuniqueness of weak solutions to the SQG equations, Comm. Pure Appl. Math., 72 (2016), 1809-1874.  doi: 10.1002/cpa.21851.

[7]

A. Castro, D. Córdoba and J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, Mem. Am. Math. Soc., 266 (2020), v+89 pp. doi: 10.1090/memo/1292.

[8]

G. Cavallaro, R. Garra and C. Marchioro, Long time localization of modified surface quasi-geostrophic equations, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5135–5148. arXiv: 2004.11183 [math-ph]. doi: 10.3934/dcdsb.2020336.

[9]

P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.  doi: 10.1088/0951-7715/7/6/001.

[10]

M. Donati, Two-dimensional point vortex dynamics in bounded domains: Global existence for almost every initial data, SIAM Journal on Mathematical Analysis, 54 (2022), 79-113. doi: 10.1137/21M1413213.

[11]

C. Garcia, Vortex patches choreography for active scalar equations, Jour. of Nonlinear Sci., 34 (2021), Paper No. 75, 31 pp. doi: 10.1007/s00332-021-09729-x.

[12]

C. Geldhauser and M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, Am. Ins. Math. Sci., 25 (2020), 2583-2606.  doi: 10.3934/dcdsb.2020023.

[13]

L. Godard-Cadillac, Smooth traveling-wave solutions to the generalized inviscid surface quasi-geostrophic equation, Comptes Rendus Math. Ac. Sci., 359 (2021), 85-98.  doi: 10.5802/crmath.159.

[14]

L. Godard-Cadillac, P. Gravejat and D. Smets, Co-rotating vortices with n fold symmetry for the inviscid surface quasi-geostrophic equation, Preprint, arXiv: 2010.08194.

[15]

P. Gravejat and D. Smets, Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation, Int. Math. Res. Not., 2019 (2019), 1744-1757.  doi: 10.1093/imrn/rnx177.

[16]

F. Grotto and U. Pappalettera, Burst of point-vortices and non-uniqueness of 2D euler equations., preprint, arXiv: 2011.13329 [math.DS].

[17]

T. Hmidi and J. Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys., 350 (2017), 699-747.  doi: 10.1007/s00220-016-2784-7.

[18]

A. Kiselev and F. Nazarov, A simple energy pump for the surface quasi-geostrophic equation, in Nonlinear Partial Differential Equations, vol. 7 of Abel Symposia, Holden, H. and Karlsen, K.H., 2012,175–179. doi: 10.1007/978-3-642-25361-4_9.

[19]

C. Marchioro and M. Pulvirenti, Vortex methods in two-dimensionnal fluid mechanics, Springer-Verlag, Berlin, 1984.

[20]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied mathematical sciences, Applied Mathematical Sciences, 96. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[21]

E. Noether, Invariante variationsprobleme, Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys, 235–255.

[22]

E. Novikov, Dynamics and statistics of a system of vortices, Zh. Eksp. Teor. Fiz., 41 (1975), 937-943. 

[23]

E. Novikov and Y. Sedov, Vortex collapse, Zh. Eksp. Teor. Fiz., 77 (1979), 588-597. 

[24] J. Pedlowsky, Geophysical Fluid Dynamics, Springer-Verlag, New-York, 1979. 
[25]

J. Reinaud, Self-similar collapse of three geophysical vortices, Geophysical & Astrophysical Fluid Dynamics, 115 (2021), 369-392.  doi: 10.1080/03091929.2020.1828402.

[26]

S. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, PhD thesis, University of Chicago, 1995.

[27]

M. Rosenzweig, Justification of the point vortex approximation for modified surface quasi-geostrophic equations, SIAM Journal on Mathematical Analysis, 52 (2020), 1690-1728.  doi: 10.1137/19M1262620.

[28]

D. Smets and J. Van Schaftingen, Desingularization of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.

[29] G. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006. 
[30]

V. Yudovich, Non-stationary flows of an ideal incompressible fluid, USSR Comput. Math. and Math. Phys., 3 (1963), 1407-1456.  doi: 10.1016/0041-5553(63)90247-7.

[1]

Dominic Breit, Eduard Feireisl, Martina Hofmanová. Generalized solutions to models of inviscid fluids. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3831-3842. doi: 10.3934/dcdsb.2020079

[2]

Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233

[3]

K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133

[4]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41

[5]

Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497

[6]

Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033

[7]

Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1

[8]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic and Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[9]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001

[10]

Tomáš Roubíček. From quasi-incompressible to semi-compressible fluids. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4069-4092. doi: 10.3934/dcdss.2020414

[11]

Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, 2022, 11 (4) : 1251-1283. doi: 10.3934/eect.2021043

[12]

Paolo Secchi. An alpha model for compressible fluids. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351

[13]

Peter Constantin. Transport in rotating fluids. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 165-176. doi: 10.3934/dcds.2004.10.165

[14]

Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383

[15]

D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47

[16]

Robert Cardona. The topology of Bott integrable fluids. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022054

[17]

Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072

[18]

Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic and Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373

[19]

Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin. Variational discretization for rotating stratified fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 477-509. doi: 10.3934/dcds.2014.34.477

[20]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks and Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (218)
  • HTML views (123)
  • Cited by (0)

Other articles
by authors

[Back to Top]