July  2022, 21(7): 2479-2493. doi: 10.3934/cpaa.2022075

Geophysics and Stuart vortices on a sphere meet differential geometry

International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

Received  December 2021 Revised  March 2022 Published  July 2022 Early access  April 2022

Fund Project: Łukasz Rudnicki is supported by the Foundation for Polish Science (IRAP project, ICTQT, Contract No. 2018/MAB/5, cofinanced by the EU within the Smart Growth Operational Programme)

We prove new existence criteria relevant for the non-linear elliptic PDE of the form $ \Delta_{S^2} u = C-he^{u} $, considered on a two dimensional sphere $ S^2 $, in the parameter regime $ 2\leq C<4 $. We apply this result, as well as several previously known results valid when $ C<2 $, to discuss existence of solutions of a particular PDE modelling ocean surface currents.

Citation: Łukasz Rudnicki. Geophysics and Stuart vortices on a sphere meet differential geometry. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2479-2493. doi: 10.3934/cpaa.2022075
References:
[1]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. A., 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.

[2]

A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 869 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109.

[3]

A. ConstantinD. G. CrowdyV. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discret. Contin. Dynam. Syst., 41 (2021), 201-215.  doi: 10.3934/dcds.2020263.

[4]

J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440. 

[5]

D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 398 (2004), 381-402.  doi: 10.1017/S0022112003007043.

[6]

J. L. Kazdan and F. W. Warner, Curvature Functions for Compact 2-Manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.

[7]

T. Aubin, Meilleures constantes dans le théorème d'nclusion de Sobolev et un théeorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal., 32 (1979), 148-174.  doi: 10.1016/0022-1236(79)90052-1.

[8]

J. DolbeaultM. J. Esteban and and G. Jankowiak, Onofri inequalities and rigidity results, Discret. Contin. Dynam. Syst., 37 (2017), 3059-3078.  doi: 10.3934/dcds.2017131.

[9]

J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differ. Geom., 10 (1975), 113-134. 

[10]

J. L. Kazdan and F. W. Warner, Existence and Conformal Deformation of Metrics With Prescribed Gaussian and Scalar Curvatures, Ann. Math., 101 (1975), 317-331.  doi: 10.2307/1970993.

[11]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[12]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer monographs in mathematics, Springer, 1998. doi: 10.1007/978-3-662-13006-3.

[13]

J. DolbeaultM. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri Inequality, Chin. Ann. Math., 36 (2015), 777-802.  doi: 10.1007/s11401-015-0976-7.

[14] R. A. Horn and R. Johnson, Matrix Analysis, 2nd Edition, Cambridge University Press, Cambridge, 2013. 
[15]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $R^2$, Duke Math. J., 71 (1993), 427-439.  doi: 10.1215/S0012-7094-93-07117-7.

[16]

S. Y. A. Chang and F. Hang, Improved Moser-Trudinger-Onofri inequality under constraints, Commun. Pure Appl. Math., 75 (2022), 197-220. 

[17]

P. DelsarteJ. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388.  doi: 10.1007/bf03187604.

[18]

T. Sakajo, Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus, Proc. R. Soc. A, 475 (2019), 20180666. 

[19]

J. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc., 301 (1987), 723-736.  doi: 10.2307/2000667.

show all references

References:
[1]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. A., 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.

[2]

A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 869 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109.

[3]

A. ConstantinD. G. CrowdyV. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discret. Contin. Dynam. Syst., 41 (2021), 201-215.  doi: 10.3934/dcds.2020263.

[4]

J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440. 

[5]

D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 398 (2004), 381-402.  doi: 10.1017/S0022112003007043.

[6]

J. L. Kazdan and F. W. Warner, Curvature Functions for Compact 2-Manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.

[7]

T. Aubin, Meilleures constantes dans le théorème d'nclusion de Sobolev et un théeorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal., 32 (1979), 148-174.  doi: 10.1016/0022-1236(79)90052-1.

[8]

J. DolbeaultM. J. Esteban and and G. Jankowiak, Onofri inequalities and rigidity results, Discret. Contin. Dynam. Syst., 37 (2017), 3059-3078.  doi: 10.3934/dcds.2017131.

[9]

J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differ. Geom., 10 (1975), 113-134. 

[10]

J. L. Kazdan and F. W. Warner, Existence and Conformal Deformation of Metrics With Prescribed Gaussian and Scalar Curvatures, Ann. Math., 101 (1975), 317-331.  doi: 10.2307/1970993.

[11]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[12]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer monographs in mathematics, Springer, 1998. doi: 10.1007/978-3-662-13006-3.

[13]

J. DolbeaultM. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri Inequality, Chin. Ann. Math., 36 (2015), 777-802.  doi: 10.1007/s11401-015-0976-7.

[14] R. A. Horn and R. Johnson, Matrix Analysis, 2nd Edition, Cambridge University Press, Cambridge, 2013. 
[15]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $R^2$, Duke Math. J., 71 (1993), 427-439.  doi: 10.1215/S0012-7094-93-07117-7.

[16]

S. Y. A. Chang and F. Hang, Improved Moser-Trudinger-Onofri inequality under constraints, Commun. Pure Appl. Math., 75 (2022), 197-220. 

[17]

P. DelsarteJ. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388.  doi: 10.1007/bf03187604.

[18]

T. Sakajo, Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus, Proc. R. Soc. A, 475 (2019), 20180666. 

[19]

J. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc., 301 (1987), 723-736.  doi: 10.2307/2000667.

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