November  2020, 40(11): 6473-6506. doi: 10.3934/dcds.2020287

On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces

1. 

LEDPA, Université de Batna 2, Faculté des Mathématiques et d'Informatique, Département de Mathématiques, 05000 Batna, Algérie

2. 

Université Côte d'Azur, CNRS, LJAD, Nice, France

* Corresponding author appreciates Taoufik Hmidi from Rennes 1 University for the fruitful discussions on the subject with the occasion of RAMA 11 organized by Sidi bel Abbess University, Algeria, November 2019. A particular greeting dedicates to the referee for the enormous efforts, careful reading and his appreciation of the work

Received  March 2020 Revised  April 2020 Published  July 2020

Fund Project: This work has been done while the second author is a PhD student at the University of Côte d’Azur-Nice-France, under the supervision of F. Planchon and P. Dreyfuss, in particular the second author would like to thank his supervisors, and the LJAD direction

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the initial data $ (v_0, \rho_0) $ is axisymmetric and $ (\omega_0, \rho_0) $ belongs to the critical space $ L^1(\Omega)\times L^1( \mathbb{R}^3) $, with $ \omega_0 $ is the initial vorticity associated to $ v_0 $ and $ \Omega = \{(r, z)\in \mathbb{R}^2:r>0\} $, then the viscous Boussinesq system has a unique global solution.

Citation: Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287
References:
[1]

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[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.  Google Scholar

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H. AbidiT. Hmidi and S. Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dyn. Syst., 29 (2011), 737-756.  doi: 10.3934/dcds.2011.29.737.  Google Scholar

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J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$D-Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

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M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal., 128 (1994), 329-358.  doi: 10.1007/BF00387712.  Google Scholar

[6]

H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal., 128 (1994), 359-360.  doi: 10.1007/BF00387713.  Google Scholar

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D. Chae, Global regularity for the $2$D-Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar

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G.-H. Cottet, Equations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105-108.   Google Scholar

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R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

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R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.  Google Scholar

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P. Dreyfuss and H. Houamed, Uniqueness result for Navier-Stokes-Boussinesq equations with horizontal dissipation, preprint, arXiv: 1904.00437. Google Scholar

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W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.  Google Scholar

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H. Feng and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal., 215 (2015), 89-123.  doi: 10.1007/s00205-014-0775-4.  Google Scholar

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I. Gallagher and T. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., 332 (2005), 287-327.  doi: 10.1007/s00208-004-0627-x.  Google Scholar

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T. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 1091-1131.  doi: 10.3934/dcdss.2012.5.1091.  Google Scholar

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T. Gallay and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Math., 7 (2015), 67-92.  doi: 10.5802/cml.25.  Google Scholar

[21]

T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.  doi: 10.1007/s00220-004-1254-9.  Google Scholar

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P. Germain, Équations de Navier-Stokes dans $ \mathbb{R}^2$: Existence et comportement asymptotique de solutions d'énergie infinie, Bull. Sci. Math., 130 (2006), 123-151.  doi: 10.1016/j.bulsci.2005.06.004.  Google Scholar

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Y. GigaT. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.  Google Scholar

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B. L. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica, 5 (1989), 208-218.  doi: 10.1007/BF02006004.  Google Scholar

[25]

T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.   Google Scholar

[26]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[27]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for an Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

[28]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Navier-Stokes-Boussinesq system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[29]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[30]

T. Hmidi and M. Zerguine, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239 (2010), 1387-1401.  doi: 10.1016/j.physd.2009.12.009.  Google Scholar

[31]

T. Hmidi and M. Zerguine, Inviscid limit for axisymmetric Navier-Stokes system, Differential Integral Equations, 22 (2009), 1223-1246.   Google Scholar

[32]

T. Hmidi and M. Zerguine, Vortex patch for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  doi: 10.4310/CMS.2014.v12.n8.a8.  Google Scholar

[33]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[34]

H. Houamed and M. Zerguine, On the global solvability of the axisymmetric Boussinesq system with critical regularity, Nonlinear Anal., 200 (2020), 26pp. doi: 10.1016/j.na.2020.112003.  Google Scholar

[35]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155-177.   Google Scholar

[36]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the $2$D-Boussinesq system with anisotropic viscosity without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[37]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[38]

X. LiuM. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[39]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.  Google Scholar

[40]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. (9), 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.  Google Scholar

[41]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67.  doi: 10.1007/s00220-013-1721-2.  Google Scholar

[42]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[43]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781139095143.  Google Scholar

[44]

V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$veràk, Selected Topics in Fluid Mechanics. Course notes, (2011). Google Scholar

[45]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-69.  doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

[46]

M. Zerguine, The regular vortex patch problem for stratified Euler equations with critical fractional dissipation, J. Evol. Equ., 15 (2015), 667-698.  doi: 10.1007/s00028-015-0277-3.  Google Scholar

show all references

References:
[1]

H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. Math., 132 (2008), 592-624.  doi: 10.1016/j.bulsci.2007.10.001.  Google Scholar

[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.  Google Scholar

[3]

H. AbidiT. Hmidi and S. Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dyn. Syst., 29 (2011), 737-756.  doi: 10.3934/dcds.2011.29.737.  Google Scholar

[4]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$D-Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[5]

M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal., 128 (1994), 329-358.  doi: 10.1007/BF00387712.  Google Scholar

[6]

H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal., 128 (1994), 359-360.  doi: 10.1007/BF00387713.  Google Scholar

[7]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, in Approximation Methods for Navier-Stokes Problems, Lecture Notes in Math., 771, Springer, Berlin, 1980,129–144. doi: 10.1007/BFb0086903.  Google Scholar

[8]

D. Chae, Global regularity for the $2$D-Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar

[9]

J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations, 21 (1996), 1771-1779.  doi: 10.1080/03605309608821245.  Google Scholar

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[11]

G.-H. Cottet, Equations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105-108.   Google Scholar

[12]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

[13]

R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.  Google Scholar

[14]

P. Dreyfuss and H. Houamed, Uniqueness result for Navier-Stokes-Boussinesq equations with horizontal dissipation, preprint, arXiv: 1904.00437. Google Scholar

[15]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.  Google Scholar

[16]

H. Feng and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal., 215 (2015), 89-123.  doi: 10.1007/s00205-014-0775-4.  Google Scholar

[17]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[18]

I. Gallagher and T. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., 332 (2005), 287-327.  doi: 10.1007/s00208-004-0627-x.  Google Scholar

[19]

T. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 1091-1131.  doi: 10.3934/dcdss.2012.5.1091.  Google Scholar

[20]

T. Gallay and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Math., 7 (2015), 67-92.  doi: 10.5802/cml.25.  Google Scholar

[21]

T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.  doi: 10.1007/s00220-004-1254-9.  Google Scholar

[22]

P. Germain, Équations de Navier-Stokes dans $ \mathbb{R}^2$: Existence et comportement asymptotique de solutions d'énergie infinie, Bull. Sci. Math., 130 (2006), 123-151.  doi: 10.1016/j.bulsci.2005.06.004.  Google Scholar

[23]

Y. GigaT. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.  Google Scholar

[24]

B. L. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica, 5 (1989), 208-218.  doi: 10.1007/BF02006004.  Google Scholar

[25]

T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.   Google Scholar

[26]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[27]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for an Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

[28]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Navier-Stokes-Boussinesq system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[29]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[30]

T. Hmidi and M. Zerguine, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239 (2010), 1387-1401.  doi: 10.1016/j.physd.2009.12.009.  Google Scholar

[31]

T. Hmidi and M. Zerguine, Inviscid limit for axisymmetric Navier-Stokes system, Differential Integral Equations, 22 (2009), 1223-1246.   Google Scholar

[32]

T. Hmidi and M. Zerguine, Vortex patch for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  doi: 10.4310/CMS.2014.v12.n8.a8.  Google Scholar

[33]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[34]

H. Houamed and M. Zerguine, On the global solvability of the axisymmetric Boussinesq system with critical regularity, Nonlinear Anal., 200 (2020), 26pp. doi: 10.1016/j.na.2020.112003.  Google Scholar

[35]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155-177.   Google Scholar

[36]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the $2$D-Boussinesq system with anisotropic viscosity without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[37]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[38]

X. LiuM. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[39]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.  Google Scholar

[40]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. (9), 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.  Google Scholar

[41]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67.  doi: 10.1007/s00220-013-1721-2.  Google Scholar

[42]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[43]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781139095143.  Google Scholar

[44]

V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$veràk, Selected Topics in Fluid Mechanics. Course notes, (2011). Google Scholar

[45]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-69.  doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

[46]

M. Zerguine, The regular vortex patch problem for stratified Euler equations with critical fractional dissipation, J. Evol. Equ., 15 (2015), 667-698.  doi: 10.1007/s00028-015-0277-3.  Google Scholar

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