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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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