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

Long time localization of modified surface quasi-geostrophic equations

Guido Cavallaro 1,, , Roberto Garra 2, and  Carlo Marchioro 3,

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.

Keywords: Surface quasi-geostrophic, localization, pseudo-vortices.
Mathematics Subject Classification: Primary: 76B47, 76M23; Secondary: 37C10, 86A99.
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. Ashbee, J. 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. Cavallaro, R. 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. 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.

[8]

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.

[9]

P. Constantin, G. 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. Pierrehumbert, I. 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. Tan, B. 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. Ashbee, J. 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. Cavallaro, R. 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. 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.

[8]

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.

[9]

P. Constantin, G. 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. Pierrehumbert, I. 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. Tan, B. 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.

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