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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Falkovich, Symmetries of the turbulent state, J. Phys. A, 42 (2009), 123001, 18pp. doi: 10.1088/1751-8113/42/12/123001.  Google Scholar

[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.  Google Scholar

[20]

C. Geldhauser and M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, 2018, arXiv: 1810.12706. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

P.-L. Lions, On Euler Equations and Statistical Physics, Cattedra Galileiana. [Galileo Chair], Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.  Google Scholar

[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.  Google Scholar

[27]

C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503.  doi: 10.1007/BF01209630.  Google Scholar

[28]

C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, vol. 203 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1984.  Google Scholar

[29]

C. Marchioro and M. Pulvirenti, Vortices and localization in Euler flows, Comm. Math. Phys., 154 (1993), 49-61.  doi: 10.1007/BF02096831.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E, 61 (2000), 6572-6577.  doi: 10.1103/PhysRevE.61.6572.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Falkovich, Symmetries of the turbulent state, J. Phys. A, 42 (2009), 123001, 18pp. doi: 10.1088/1751-8113/42/12/123001.  Google Scholar

[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.  Google Scholar

[20]

C. Geldhauser and M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, 2018, arXiv: 1810.12706. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

P.-L. Lions, On Euler Equations and Statistical Physics, Cattedra Galileiana. [Galileo Chair], Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.  Google Scholar

[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.  Google Scholar

[27]

C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503.  doi: 10.1007/BF01209630.  Google Scholar

[28]

C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, vol. 203 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1984.  Google Scholar

[29]

C. Marchioro and M. Pulvirenti, Vortices and localization in Euler flows, Comm. Math. Phys., 154 (1993), 49-61.  doi: 10.1007/BF02096831.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E, 61 (2000), 6572-6577.  doi: 10.1103/PhysRevE.61.6572.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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