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Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms
Point vortices for inviscid generalized surface quasi-geostrophic models
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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