Point vortices for inviscid generalized surface quasi-geostrophic models

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July 2020, 25(7): 2583-2606. doi: 10.3934/dcdsb.2020023

Point vortices for inviscid generalized surface quasi-geostrophic models

Carina Geldhauser ^1, and Marco Romito ^2,

1.	School of Mathematics and Statistics, The University of Sheffield, Hounsfield Rd, Sheffield S3 7RH, United Kingdom
2.	Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I–56127 Pisa, Italia

Carina Geldhauser, http://www.cgeldhauser.de
Marco Romito, http://people.dm.unipi.it/romito

Received January 2019 Published July 2020 Early access April 2020

Fund Project: The first author was supported by Deutsche Forschungsgemeinschaft in the context of TU Dresden's Institutional Strategy "The Synergetic University". The second author acknowledges the partial support of the University of Pisa, through project PRA 2018_49

We give a rigorous proof of the validity of the point vortex description for a class of inviscid generalized surface quasi-geostrophic models on the whole plane.

Keywords: Inviscid generalized surface quasi-geostrophic, weak solutions, point vortex motion, vortex approximation, localization, stability.

Mathematics Subject Classification: Primary: 76B47, 76M23; Secondary: 76E20, 86A99.

Citation: Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023

References:

[1]	G. Badin and A. M. Barry, Collapse of generalized Euler and surface quasi-geostrophic point vortices, Phys. Rev. E, 98 (2018), 023110. doi: 10.1103/PhysRevE.98.023110.
[2]	D. Bernard, G. Boffetta, A. Celani and G. Falkovich, Conformal invariance in two-dimensional turbulence, Nature Physics, 2 (2006), 124-128. doi: 10.1038/nphys217.
[3]	D. Bernard, G. Boffetta, A. Celani and G. Falkovich, Inverse turbulent cascades and conformally invariant curves, Physical Review Letters, 98 (2007), 024501. doi: 10.1103/PhysRevLett.98.024501.
[4]	T. Bodineau and A. Guionnet, About the stationary states of vortex systems, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 205-237. doi: 10.1016/S0246-0203(99)80011-9.
[5]	E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.
[6]	E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. Ⅱ, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602.
[7]	G. Cavallaro, R. Garra and C. Marchioro, Localization and stability of active scalar flows, Riv. Math. Univ. Parma (N.S.), 4 (2013), 175-196.
[8]	D. Chae, P. Constantin, D. Córdoba, F. Gancedo and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math., 65 (2012), 1037-1066. doi: 10.1002/cpa.21390.
[9]	D. Chae, P. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62. doi: 10.1007/s00205-011-0411-5.
[10]	D. Chae, P. Constantin and J. Wu, Dissipative models generalizing the 2D Navier-Stokes and surface quasi-geostrophic equations, Indiana Univ. Math. J., 61 (2012), 1997-2018. doi: 10.1512/iumj.2012.61.4756.
[11]	P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97–107, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). doi: 10.1512/iumj.2001.50.2153.
[12]	P. Constantin, A. J. 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.
[13]	A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.
[14]	D. Córdoba, C. Fefferman and J. L. Rodrigo, Almost sharp fronts for the surface quasi-geostrophic equation, Proc. Natl. Acad. Sci. USA, 101 (2004), 2687-2691. doi: 10.1073/pnas.0308154101.
[15]	D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA, 102 (2005), 5949-5952. doi: 10.1073/pnas.0501977102.
[16]	D. Córdoba, J. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251. doi: 10.1007/s00205-019-01377-6.
[17]	J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6.
[18]	G. Falkovich, Symmetries of the turbulent state, J. Phys. A, 42 (2009), 123001, 18pp. doi: 10.1088/1751-8113/42/12/123001.
[19]	F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, J. Evol. Equat., 19 (2019), 1071-1090. doi: 10.1007/s00028-019-00506-8.
[20]	C. Geldhauser and M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, 2018, arXiv: 1810.12706.
[21]	M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384. doi: 10.1142/S0218202509003814.
[22]	I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, Journal of Fluid Mechanics, 282 (1995), 1-20. doi: 10.1017/S0022112095000012.
[23]	I. M. Held, R. T. Pierrehumbert and S. K. L., Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons & Fractals, 4 (1994), 1111–1116, Special Issue: Chaos Applied to Fluid Mixing.
[24]	A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. of Math, 184 (2016), 909-948. doi: 10.4007/annals.2016.184.3.7.
[25]	P.-L. Lions, On Euler Equations and Statistical Physics, Cattedra Galileiana. [Galileo Chair], Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.
[26]	F. Marchand, Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces $L^p$ or $\dot H^{-1/2}$, Comm. Math. Phys., 277 (2008), 45-67. doi: 10.1007/s00220-007-0356-6.
[27]	C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503. doi: 10.1007/BF01209630.
[28]	C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, vol. 203 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1984.
[29]	C. Marchioro and M. Pulvirenti, Vortices and localization in Euler flows, Comm. Math. Phys., 154 (1993), 49-61. doi: 10.1007/BF02096831.
[30]	C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.
[31]	S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–The University of Chicago.
[32]	J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059.
[33]	S. Schochet, The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations, 20 (1995), 1077-1104. doi: 10.1080/03605309508821124.
[34]	S. Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math., 49 (1996), 911-965. doi: 10.1002/(SICI)1097-0312(199609)49:9<911::AID-CPA2>3.0.CO;2-A.
[35]	N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E, 61 (2000), 6572-6577. doi: 10.1103/PhysRevE.61.6572.
[36]	C. V. Tran, Nonlinear transfer and spectral distribution of energy in $\alpha$ turbulence, Phys. D, 191 (2004), 137-155. doi: 10.1016/j.physd.2003.11.005.
[37]	C. V. Tran, D. G. Dritschel and R. K. Scott, Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence, Phys. Rev. E, 81 (2010), 016301. doi: 10.1103/PhysRevE.81.016301.
[38]	A. Venaille, T. Dauxois and S. Ruffo, Violent relaxation in two-dimensional flows with varying interaction range, Phys. Rev. E, 92 (2015), 011001. doi: 10.1103/PhysRevE.92.011001.

show all references

References:

[1]	G. Badin and A. M. Barry, Collapse of generalized Euler and surface quasi-geostrophic point vortices, Phys. Rev. E, 98 (2018), 023110. doi: 10.1103/PhysRevE.98.023110.
[2]	D. Bernard, G. Boffetta, A. Celani and G. Falkovich, Conformal invariance in two-dimensional turbulence, Nature Physics, 2 (2006), 124-128. doi: 10.1038/nphys217.
[3]	D. Bernard, G. Boffetta, A. Celani and G. Falkovich, Inverse turbulent cascades and conformally invariant curves, Physical Review Letters, 98 (2007), 024501. doi: 10.1103/PhysRevLett.98.024501.
[4]	T. Bodineau and A. Guionnet, About the stationary states of vortex systems, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 205-237. doi: 10.1016/S0246-0203(99)80011-9.
[5]	E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.
[6]	E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. Ⅱ, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602.
[7]	G. Cavallaro, R. Garra and C. Marchioro, Localization and stability of active scalar flows, Riv. Math. Univ. Parma (N.S.), 4 (2013), 175-196.
[8]	D. Chae, P. Constantin, D. Córdoba, F. Gancedo and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math., 65 (2012), 1037-1066. doi: 10.1002/cpa.21390.
[9]	D. Chae, P. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62. doi: 10.1007/s00205-011-0411-5.
[10]	D. Chae, P. Constantin and J. Wu, Dissipative models generalizing the 2D Navier-Stokes and surface quasi-geostrophic equations, Indiana Univ. Math. J., 61 (2012), 1997-2018. doi: 10.1512/iumj.2012.61.4756.
[11]	P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97–107, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). doi: 10.1512/iumj.2001.50.2153.
[12]	P. Constantin, A. J. 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.
[13]	A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.
[14]	D. Córdoba, C. Fefferman and J. L. Rodrigo, Almost sharp fronts for the surface quasi-geostrophic equation, Proc. Natl. Acad. Sci. USA, 101 (2004), 2687-2691. doi: 10.1073/pnas.0308154101.
[15]	D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA, 102 (2005), 5949-5952. doi: 10.1073/pnas.0501977102.
[16]	D. Córdoba, J. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251. doi: 10.1007/s00205-019-01377-6.
[17]	J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6.
[18]	G. Falkovich, Symmetries of the turbulent state, J. Phys. A, 42 (2009), 123001, 18pp. doi: 10.1088/1751-8113/42/12/123001.
[19]	F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, J. Evol. Equat., 19 (2019), 1071-1090. doi: 10.1007/s00028-019-00506-8.
[20]	C. Geldhauser and M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, 2018, arXiv: 1810.12706.
[21]	M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384. doi: 10.1142/S0218202509003814.
[22]	I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, Journal of Fluid Mechanics, 282 (1995), 1-20. doi: 10.1017/S0022112095000012.
[23]	I. M. Held, R. T. Pierrehumbert and S. K. L., Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons & Fractals, 4 (1994), 1111–1116, Special Issue: Chaos Applied to Fluid Mixing.
[24]	A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. of Math, 184 (2016), 909-948. doi: 10.4007/annals.2016.184.3.7.
[25]	P.-L. Lions, On Euler Equations and Statistical Physics, Cattedra Galileiana. [Galileo Chair], Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.
[26]	F. Marchand, Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces $L^p$ or $\dot H^{-1/2}$, Comm. Math. Phys., 277 (2008), 45-67. doi: 10.1007/s00220-007-0356-6.
[27]	C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503. doi: 10.1007/BF01209630.
[28]	C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, vol. 203 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1984.
[29]	C. Marchioro and M. Pulvirenti, Vortices and localization in Euler flows, Comm. Math. Phys., 154 (1993), 49-61. doi: 10.1007/BF02096831.
[30]	C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.
[31]	S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–The University of Chicago.
[32]	J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059.
[33]	S. Schochet, The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations, 20 (1995), 1077-1104. doi: 10.1080/03605309508821124.
[34]	S. Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math., 49 (1996), 911-965. doi: 10.1002/(SICI)1097-0312(199609)49:9<911::AID-CPA2>3.0.CO;2-A.
[35]	N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E, 61 (2000), 6572-6577. doi: 10.1103/PhysRevE.61.6572.
[36]	C. V. Tran, Nonlinear transfer and spectral distribution of energy in $\alpha$ turbulence, Phys. D, 191 (2004), 137-155. doi: 10.1016/j.physd.2003.11.005.
[37]	C. V. Tran, D. G. Dritschel and R. K. Scott, Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence, Phys. Rev. E, 81 (2010), 016301. doi: 10.1103/PhysRevE.81.016301.
[38]	A. Venaille, T. Dauxois and S. Ruffo, Violent relaxation in two-dimensional flows with varying interaction range, Phys. Rev. E, 92 (2015), 011001. doi: 10.1103/PhysRevE.92.011001.

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