September  2021, 26(9): 5135-5148. doi: 10.3934/dcdsb.2020336

Long time localization of modified surface quasi-geostrophic equations

1. 

Sapienza Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 2, 00185 Roma, Italy

2. 

Sapienza Università di Roma, Dipartimento di Scienze Statistiche, Piazzale Aldo Moro 2, 00185 Roma, Italy

3. 

International Research Center M&MOCS, Università di L'Aquila, Italy

* Corresponding author: Guido Cavallaro

Received  April 2020 Published  September 2021 Early access  November 2020

We discuss the time evolution of a two-dimensional active scalar flow, which extends some properties valid for a two-dimensional incompressible nonviscous fluid. In particular we study some characteristics of the dynamics when the field is initially concentrated in $ N $ small disjoint regions, and we discuss the conservation in time of this localization property. We discuss also how long this localization persists, showing that in some cases this happens for quite long times.

Citation: Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336
References:
[1]

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

[2]

T. L. AshbeeJ. G. Esler and N. R. McDonald, Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions, Journal of Computational Physics, 246 (2013), 289-303.  doi: 10.1016/j.jcp.2013.03.044.

[3]

G. Badin and A. M. Barry, Collapse of generalized Euler and surface quasigeostrophic point vortices, Phys. Rev. E, 98 (2018), 023110. doi: 10.1103/PhysRevE.98.023110.

[4]

P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50 (2018), 735-760.  doi: 10.1137/16M1103725.

[5]

G. CavallaroR. Garra and C. Marchioro, Localization and stability of active scalar flows, Riv. Mat. Univ. Parma, 4 (2013), 175-196. 

[6]

D. Cetrone and G. Serafini, Long time evolution of fluids with concentrated vorticity and convergence to the point-vortex model, Rend. Mat. Appl., 39 (2018), 29-78. 

[7]

D. ChaeP. 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.

[8]

P. ConstantinA. 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.

[9]

P. ConstantinG. Iyer and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 57 (2001), 2681-2692.  doi: 10.1512/iumj.2008.57.3629.

[10]

F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, Journal of Evolution Equations, 19 (2019), 1071-1090.  doi: 10.1007/s00028-019-00506-8.

[11]

T. Gallay, Interaction of vortices in weakly viscous planar flows, Arch. Ration. Mech. Anal., 200 (2011), 445-490.  doi: 10.1007/s00205-010-0362-2.

[12]

F. Gancedo, Existence for the alpha-patch model and the QG sharp front in Sobolev spaces, Advances in Mathematics, 217 (2008), 2569-2598.  doi: 10.1016/j.aim.2007.10.010.

[13]

R. Garra, Confinement of a hot temperature patch in the modified SQG model, Discrete & Continuous Dynamical Systems-B, 24 (2019), 2407-2416.  doi: 10.3934/dcdsb.2018258.

[14]

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

[15]

C. Geldhauser and M. Romito, The point vortex model for the Euler equation, AIMS Mathematics, 4 (2019), 534-575.  doi: 10.3934/math.2019.3.534.

[16]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.

[17]

D. Luo and M. Saal, Regularization by noise for the point vortex model of mSQG equations, Acta. Math. Sin. - English Ser., (2020) doi: 10.1007/s10114-020-9256-x.

[18]

C. Marchioro, On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data, Math. Meth. Appl. Sci., 12 (1990), 463-470.  doi: 10.1002/mma.1670120602.

[19]

C. Marchioro, On the inviscid limit for a fluid with a concentrated vorticity, Commun. Math. Phys., 196 (1998), 53-65.  doi: 10.1007/s002200050413.

[20]

C. Marchioro, Vanishing viscosity limit for an incompressible fluid with concentrated vorticity,, J. Math. Phys., 48 (2007), 065302, 16 pp. doi: 10.1063/1.2347901.

[21]

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

[22]

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

[23]

R. May, Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space $H^1$, J. Differential Equations, 250 (2011), 320-339.  doi: 10.1016/j.jde.2010.09.021.

[24]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[25]

R. T. PierrehumbertI. M. Held and K. L. Swanson, Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons and Fractals, 4 (1994), 1111-1116.  doi: 10.1016/0960-0779(94)90140-6.

[26]

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

[27]

W. TanB. Q. Dong and Z. M. Chen, Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces, Discrete & Continuous Dynamical Systems-A, 39 (2019), 3749-3765.  doi: 10.3934/dcds.2019152.

[28]

J. Wu, Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 46 (1997), 1113-1124.  doi: 10.1512/iumj.1997.46.1275.

show all references

References:
[1]

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

[2]

T. L. AshbeeJ. G. Esler and N. R. McDonald, Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions, Journal of Computational Physics, 246 (2013), 289-303.  doi: 10.1016/j.jcp.2013.03.044.

[3]

G. Badin and A. M. Barry, Collapse of generalized Euler and surface quasigeostrophic point vortices, Phys. Rev. E, 98 (2018), 023110. doi: 10.1103/PhysRevE.98.023110.

[4]

P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50 (2018), 735-760.  doi: 10.1137/16M1103725.

[5]

G. CavallaroR. Garra and C. Marchioro, Localization and stability of active scalar flows, Riv. Mat. Univ. Parma, 4 (2013), 175-196. 

[6]

D. Cetrone and G. Serafini, Long time evolution of fluids with concentrated vorticity and convergence to the point-vortex model, Rend. Mat. Appl., 39 (2018), 29-78. 

[7]

D. ChaeP. 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.

[8]

P. ConstantinA. 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.

[9]

P. ConstantinG. Iyer and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 57 (2001), 2681-2692.  doi: 10.1512/iumj.2008.57.3629.

[10]

F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, Journal of Evolution Equations, 19 (2019), 1071-1090.  doi: 10.1007/s00028-019-00506-8.

[11]

T. Gallay, Interaction of vortices in weakly viscous planar flows, Arch. Ration. Mech. Anal., 200 (2011), 445-490.  doi: 10.1007/s00205-010-0362-2.

[12]

F. Gancedo, Existence for the alpha-patch model and the QG sharp front in Sobolev spaces, Advances in Mathematics, 217 (2008), 2569-2598.  doi: 10.1016/j.aim.2007.10.010.

[13]

R. Garra, Confinement of a hot temperature patch in the modified SQG model, Discrete & Continuous Dynamical Systems-B, 24 (2019), 2407-2416.  doi: 10.3934/dcdsb.2018258.

[14]

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

[15]

C. Geldhauser and M. Romito, The point vortex model for the Euler equation, AIMS Mathematics, 4 (2019), 534-575.  doi: 10.3934/math.2019.3.534.

[16]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.

[17]

D. Luo and M. Saal, Regularization by noise for the point vortex model of mSQG equations, Acta. Math. Sin. - English Ser., (2020) doi: 10.1007/s10114-020-9256-x.

[18]

C. Marchioro, On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data, Math. Meth. Appl. Sci., 12 (1990), 463-470.  doi: 10.1002/mma.1670120602.

[19]

C. Marchioro, On the inviscid limit for a fluid with a concentrated vorticity, Commun. Math. Phys., 196 (1998), 53-65.  doi: 10.1007/s002200050413.

[20]

C. Marchioro, Vanishing viscosity limit for an incompressible fluid with concentrated vorticity,, J. Math. Phys., 48 (2007), 065302, 16 pp. doi: 10.1063/1.2347901.

[21]

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

[22]

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

[23]

R. May, Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space $H^1$, J. Differential Equations, 250 (2011), 320-339.  doi: 10.1016/j.jde.2010.09.021.

[24]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[25]

R. T. PierrehumbertI. M. Held and K. L. Swanson, Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons and Fractals, 4 (1994), 1111-1116.  doi: 10.1016/0960-0779(94)90140-6.

[26]

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

[27]

W. TanB. Q. Dong and Z. M. Chen, Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces, Discrete & Continuous Dynamical Systems-A, 39 (2019), 3749-3765.  doi: 10.3934/dcds.2019152.

[28]

J. Wu, Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 46 (1997), 1113-1124.  doi: 10.1512/iumj.1997.46.1275.

[1]

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

[2]

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

[3]

T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171

[4]

May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179

[5]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025

[6]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023

[7]

Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016

[8]

Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525

[9]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[10]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152

[11]

Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010

[12]

Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277

[13]

T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119

[14]

Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133

[15]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[16]

Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293

[17]

Tongtong Liang, Yejuan Wang. Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4697-4726. doi: 10.3934/dcdsb.2020309

[18]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093

[19]

Samira Amraoui, Didier Auroux, Jacques Blum, Emmanuel Cosme. Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022058

[20]

W. Patrick Hooper. An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs. Journal of Modern Dynamics, 2019, 14: 243-276. doi: 10.3934/jmd.2019009

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (163)
  • HTML views (256)
  • Cited by (0)

Other articles
by authors

[Back to Top]