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October  2016, 36(10): 5401-5422. doi: 10.3934/dcds.2016038

Bifurcation of rotating patches from Kirchhoff vortices

Taoufik Hmidi 1, and  Joan Mateu 2,

1. 

IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France
2. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Received  September 2015 Revised  April 2016 Published  July 2016

In this paper we investigate the existence of a new family of rotating patches for the planar Euler equations. We shall prove the existence of countable branches bifurcating from the ellipses at some implicit angular velocities. The proof uses bifurcation tools combined with the explicit parametrization of the ellipse through the exterior conformal mappings. The boundary is shown to belong to Hölderian class.
Keywords: Kirchhoff vortices, bifurcation., rotating patches.
Mathematics Subject Classification: Primary: 35Q31; Secondary: 35Q35, 76B0.
Citation: Taoufik Hmidi, Joan Mateu. Bifurcation of rotating patches from Kirchhoff vortices. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5401-5422. doi: 10.3934/dcds.2016038
References:
[1]

A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge texts in applied Mathematics, Cambridge University Press, Cambridge, 2002.

[2]

J. Burbea, Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16. doi: 10.1007/BF02281165.

[3]

J. Burbea and M. Landau, The Kelvin waves in vortex dynamics and their stability, Journal of Computational Physics, 45 (1982), 127-156. doi: 10.1016/0021-9991(82)90106-1.

[4]

A. Castro, D. Córdoba and J. Gómez-Serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J., 165 (2016), 935-984. doi: 10.1215/00127094-3449673.

[5]

A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, 2 (2016), Art. 1, 34 pp. doi: 10.1007/s40818-016-0007-3.

[6]

C. Cerretelli and C. H. K. Williamson, A new family of uniform vortices related to vortex configurations before fluid merger, J. Fluid Mech., 493 (2003), 219-229. doi: 10.1017/S0022112003005536.

[7]

J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. of Func. Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[9]

G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Phys. Rev. Lett., 40 (1978), 859-862.

[10]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.

[11]

D. G. Dritschel, The nonlinear evolution of rotating configurations of uniform vorticity, J. Fluid Mech., 172 (1986), 157-182. doi: 10.1017/S0022112086001696.

[12]

G. R. Flierl and L. M. Polvani, Generalized Kirchhoff vortices, Phys. Fluids, 29 (1986), 2376-2379.

[13]

Z. Hassainia and T. Hmidi, On the V-States for the generalized quasi-geostrophic equations, Comm. Math. Phys., 337 (2015), 321-377. doi: 10.1007/s00220-015-2300-5.

[14]

Z. Hassainia, T. Hmidi and F. de la Hoz, Doubly connected V-states for the generalized surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 220 (2016), 1209-1281. doi: 10.1007/s00205-015-0953-z.

[15]

T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., 209 (2013), 171-208. doi: 10.1007/s00205-013-0618-8.

[16]

T. Hmidi, J. Mateu and J. Verdera, On rotating doubly connected vortices, J. Differential Equations, 258 (2015), 1395-1429. doi: 10.1016/j.jde.2014.10.021.

[17]

T. Hmidi, F. de la Hoz, J. Mateu and J. Verdera, Doubly connected V-states for the planar Euler equations,, Preprint, (). 

[18]

Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354. doi: 10.1007/s00220-003-1017-z.

[19]

J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions, PhD thesis, California Institute of Technology, 1987.

[20]

G. Kirchhoff, Vorlesungen Uber Mathematische Physik, (Leipzig, 1874).

[21]

H. Lamb, Hydrodynamics, Dover Publications, New York, 1945.

[22]

A. E. H. Love, On the Stability of certain Vortex Motions, Proc. London Math. Soc., 25 (1893), 18-43. doi: 10.1112/plms/s1-25.1.18.

[23]

P. Luzzatto-Fegiz and C. H. K. Williamson, Stability of elliptical vortices from "Imperfect-Velocity-Impulse" diagrams, Theor. Comput. Fluid Dyn., 24 (2010), 181-188.

[24]

J. Mateu, J. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431. doi: 10.1016/j.matpur.2009.01.010.

[25]

T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices, Physics of Fluids, 20 (2008), 054103. doi: 10.1063/1.2912991.

[26]

E. A. II Overman, Steady-state solutions of the Euler equations in two dimensions. II. Local analysis of limiting V-states, SIAM J. Appl. Math., 46 (1986), 765-800. doi: 10.1137/0146049.

[27]

P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992.

[28]

Y. Tang, Nonlinear stability of vortex patches, Trans. Amer. Math. Soc., 304 (1987), 617-638. doi: 10.1090/S0002-9947-1987-0911087-X.

[29]

Y. H. Wan, The stability of rotating vortex patches, Comm. Math. Phys., 107 (1986), 1-20. doi: 10.1007/BF01206950.

[30]

H. M. Wu, E. A. Overman II and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. Algorithms ans results, J. Comput. Phys., 53 (1984), 42-71. doi: 10.1016/0021-9991(84)90051-2.

[31]

V. I. Yudovich, Non-stationnary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3 (1963), 1032-1066.

show all references

References:
[1]

A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge texts in applied Mathematics, Cambridge University Press, Cambridge, 2002.

[2]

J. Burbea, Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16. doi: 10.1007/BF02281165.

[3]

J. Burbea and M. Landau, The Kelvin waves in vortex dynamics and their stability, Journal of Computational Physics, 45 (1982), 127-156. doi: 10.1016/0021-9991(82)90106-1.

[4]

A. Castro, D. Córdoba and J. Gómez-Serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J., 165 (2016), 935-984. doi: 10.1215/00127094-3449673.

[5]

A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, 2 (2016), Art. 1, 34 pp. doi: 10.1007/s40818-016-0007-3.

[6]

C. Cerretelli and C. H. K. Williamson, A new family of uniform vortices related to vortex configurations before fluid merger, J. Fluid Mech., 493 (2003), 219-229. doi: 10.1017/S0022112003005536.

[7]

J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. of Func. Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[9]

G. S. Deem and N. J. Zabusky, Vortex waves: Stationary "V-states", interactions, recurrence, and breaking, Phys. Rev. Lett., 40 (1978), 859-862.

[10]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.

[11]

D. G. Dritschel, The nonlinear evolution of rotating configurations of uniform vorticity, J. Fluid Mech., 172 (1986), 157-182. doi: 10.1017/S0022112086001696.

[12]

G. R. Flierl and L. M. Polvani, Generalized Kirchhoff vortices, Phys. Fluids, 29 (1986), 2376-2379.

[13]

Z. Hassainia and T. Hmidi, On the V-States for the generalized quasi-geostrophic equations, Comm. Math. Phys., 337 (2015), 321-377. doi: 10.1007/s00220-015-2300-5.

[14]

Z. Hassainia, T. Hmidi and F. de la Hoz, Doubly connected V-states for the generalized surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 220 (2016), 1209-1281. doi: 10.1007/s00205-015-0953-z.

[15]

T. Hmidi, J. Mateu and J. Verdera, Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., 209 (2013), 171-208. doi: 10.1007/s00205-013-0618-8.

[16]

T. Hmidi, J. Mateu and J. Verdera, On rotating doubly connected vortices, J. Differential Equations, 258 (2015), 1395-1429. doi: 10.1016/j.jde.2014.10.021.

[17]

T. Hmidi, F. de la Hoz, J. Mateu and J. Verdera, Doubly connected V-states for the planar Euler equations,, Preprint, (). 

[18]

Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354. doi: 10.1007/s00220-003-1017-z.

[19]

J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions, PhD thesis, California Institute of Technology, 1987.

[20]

G. Kirchhoff, Vorlesungen Uber Mathematische Physik, (Leipzig, 1874).

[21]

H. Lamb, Hydrodynamics, Dover Publications, New York, 1945.

[22]

A. E. H. Love, On the Stability of certain Vortex Motions, Proc. London Math. Soc., 25 (1893), 18-43. doi: 10.1112/plms/s1-25.1.18.

[23]

P. Luzzatto-Fegiz and C. H. K. Williamson, Stability of elliptical vortices from "Imperfect-Velocity-Impulse" diagrams, Theor. Comput. Fluid Dyn., 24 (2010), 181-188.

[24]

J. Mateu, J. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431. doi: 10.1016/j.matpur.2009.01.010.

[25]

T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices, Physics of Fluids, 20 (2008), 054103. doi: 10.1063/1.2912991.

[26]

E. A. II Overman, Steady-state solutions of the Euler equations in two dimensions. II. Local analysis of limiting V-states, SIAM J. Appl. Math., 46 (1986), 765-800. doi: 10.1137/0146049.

[27]

P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992.

[28]

Y. Tang, Nonlinear stability of vortex patches, Trans. Amer. Math. Soc., 304 (1987), 617-638. doi: 10.1090/S0002-9947-1987-0911087-X.

[29]

Y. H. Wan, The stability of rotating vortex patches, Comm. Math. Phys., 107 (1986), 1-20. doi: 10.1007/BF01206950.

[30]

H. M. Wu, E. A. Overman II and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. Algorithms ans results, J. Comput. Phys., 53 (1984), 42-71. doi: 10.1016/0021-9991(84)90051-2.

[31]

V. I. Yudovich, Non-stationnary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3 (1963), 1032-1066.

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