June  2019, 24(6): 2407-2416. doi: 10.3934/dcdsb.2018258

Confinement of a hot temperature patch in the modified SQG model

Department of Statistical Sciences, Sapienza University of Rome, P.le Aldo Moro, 5, 00185, Rome, Ital

 

Received  July 2017 Revised  February 2018 Published  October 2018

In this paper we study the time evolution of a temperature patch in $\mathbb{R}^2$ according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

Citation: Roberto Garra. Confinement of a hot temperature patch in the modified SQG model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2407-2416. doi: 10.3934/dcdsb.2018258
References:
[1]

C. Bucur and A. L. Karakhanyan, Potential theoretic approach to Schauder estimates for the fractional Laplacian, Proceedings of the American Mathematical Society, 145 (2017), 637-651. doi: 10.1090/proc/13227. Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[3]

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

[4]

D. ChaeP. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the Surface Quasi-geostrophic equations, Arch. Rational Mech. Anal., 202 (2011), 35-62. doi: 10.1007/s00205-011-0411-5. Google Scholar

[5]

P. ConstantinD. Cordoba and J. Wu, On the Critical Dissipative Quasi-geostrophic Equation, Indiana University mathematics journal, 50 (2001), 97-107. doi: 10.1512/iumj.2008.57.3629. Google Scholar

[6]

P. ConstantinA. J. Majda and E. Tabak, Formation of strong fronts in the 2-d quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. Google Scholar

[7]

A. CordobaD. Cordoba and F. Gancedo, Uniqueness for SQG patch solutions, Transactions of the American Mathematical Society, Series B, 5 (2018), 1-31. doi: 10.1090/btran/20. Google Scholar

[8]

D. CordobaM. A. FontelosA. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proceedings of the National Academy of Sciences, 102 (2005), 5949-5952. doi: 10.1073/pnas.0501977102. Google Scholar

[9]

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

[10]

I. M. HeldR. T. PierrehumbertS. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012. Google Scholar

[11]

D. IftimieT. C. Sideris and P. Gamblin, On the evolution of compactly supported planar vorticity, Comm. Partial Differential Equations, 24 (1999), 1709-1730. Google Scholar

[12]

D. Iftimie, Evolution de Tourbillon à Support Compact, Actes du Colloque de Saint-Jean-de-Monts, 1999. Google Scholar

[13]

T. Iwayama and T. Watanabe, Green's function for a generalized two-dimensional fluid, Physical Review E, 82 (2010), 036307.Google Scholar

[14]

A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatos, Finite time singularity formation for the modified SQG patch equation, Ann. of Math. (2), 184 (2016), 909–948, arXiv:1508.07613. doi: 10.4007/annals.2016.184.3.7. Google Scholar

[15]

D. Li, Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions, Nonlinearity, 22 (2009), 1639-1651. doi: 10.1088/0951-7715/22/7/008. Google Scholar

[16]

A. J. Majda and E. Tabak, A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow, Physica D, 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5. Google Scholar

[17]

A. Mancho, Numerical studies on the self-similar collapse of the alpha-patches problem, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 152-166. doi: 10.1016/j.cnsns.2015.02.009. Google Scholar

[18]

C. Marchioro, Bounds on the growth of the support of a vortex patch, Comm. Math. Phys., 164 (1994), 507-524. Google Scholar

[19]

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

[20]

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. Google Scholar

[21]

J. L. Rodrigo, On the evolution of sharp Fronts for the Quasi-Geostrophic equation, Commun. Pure and Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059. Google Scholar

[22]

P. Serfati, Borne en temps des caractéristiques de l'équation d'Euler 2D á tourbillon positif et localisation pour le modéle point-vortex, Preprint.Google Scholar

show all references

References:
[1]

C. Bucur and A. L. Karakhanyan, Potential theoretic approach to Schauder estimates for the fractional Laplacian, Proceedings of the American Mathematical Society, 145 (2017), 637-651. doi: 10.1090/proc/13227. Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[3]

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

[4]

D. ChaeP. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the Surface Quasi-geostrophic equations, Arch. Rational Mech. Anal., 202 (2011), 35-62. doi: 10.1007/s00205-011-0411-5. Google Scholar

[5]

P. ConstantinD. Cordoba and J. Wu, On the Critical Dissipative Quasi-geostrophic Equation, Indiana University mathematics journal, 50 (2001), 97-107. doi: 10.1512/iumj.2008.57.3629. Google Scholar

[6]

P. ConstantinA. J. Majda and E. Tabak, Formation of strong fronts in the 2-d quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. Google Scholar

[7]

A. CordobaD. Cordoba and F. Gancedo, Uniqueness for SQG patch solutions, Transactions of the American Mathematical Society, Series B, 5 (2018), 1-31. doi: 10.1090/btran/20. Google Scholar

[8]

D. CordobaM. A. FontelosA. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proceedings of the National Academy of Sciences, 102 (2005), 5949-5952. doi: 10.1073/pnas.0501977102. Google Scholar

[9]

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

[10]

I. M. HeldR. T. PierrehumbertS. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012. Google Scholar

[11]

D. IftimieT. C. Sideris and P. Gamblin, On the evolution of compactly supported planar vorticity, Comm. Partial Differential Equations, 24 (1999), 1709-1730. Google Scholar

[12]

D. Iftimie, Evolution de Tourbillon à Support Compact, Actes du Colloque de Saint-Jean-de-Monts, 1999. Google Scholar

[13]

T. Iwayama and T. Watanabe, Green's function for a generalized two-dimensional fluid, Physical Review E, 82 (2010), 036307.Google Scholar

[14]

A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatos, Finite time singularity formation for the modified SQG patch equation, Ann. of Math. (2), 184 (2016), 909–948, arXiv:1508.07613. doi: 10.4007/annals.2016.184.3.7. Google Scholar

[15]

D. Li, Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions, Nonlinearity, 22 (2009), 1639-1651. doi: 10.1088/0951-7715/22/7/008. Google Scholar

[16]

A. J. Majda and E. Tabak, A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow, Physica D, 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5. Google Scholar

[17]

A. Mancho, Numerical studies on the self-similar collapse of the alpha-patches problem, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 152-166. doi: 10.1016/j.cnsns.2015.02.009. Google Scholar

[18]

C. Marchioro, Bounds on the growth of the support of a vortex patch, Comm. Math. Phys., 164 (1994), 507-524. Google Scholar

[19]

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

[20]

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. Google Scholar

[21]

J. L. Rodrigo, On the evolution of sharp Fronts for the Quasi-Geostrophic equation, Commun. Pure and Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059. Google Scholar

[22]

P. Serfati, Borne en temps des caractéristiques de l'équation d'Euler 2D á tourbillon positif et localisation pour le modéle point-vortex, Preprint.Google Scholar

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