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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 & Continuous Dynamical Systems - A, 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, (2002). Google Scholar

[2]

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

[3]

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

[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. doi: 10.1215/00127094-3449673. Google Scholar

[5]

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

[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. doi: 10.1017/S0022112003005536. Google Scholar

[7]

J.-Y. Chemin, Perfect Incompressible Fluids,, Oxford University Press, (1998). Google Scholar

[8]

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

[9]

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

[10]

J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1960). Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1007/s00205-015-0953-z. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions,, PhD thesis, (1987). Google Scholar

[20]

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

[21]

H. Lamb, Hydrodynamics,, Dover Publications, (1945). Google Scholar

[22]

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

[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. Google Scholar

[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. doi: 10.1016/j.matpur.2009.01.010. Google Scholar

[25]

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

[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. doi: 10.1137/0146049. Google Scholar

[27]

P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, (1992). Google Scholar

[28]

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

[29]

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

[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. doi: 10.1016/0021-9991(84)90051-2. Google Scholar

[31]

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

show all references

References:
[1]

A. Bertozzi and A. Majda, Vorticity and Incompressible Flow,, Cambridge texts in applied Mathematics, (2002). Google Scholar

[2]

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

[3]

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

[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. doi: 10.1215/00127094-3449673. Google Scholar

[5]

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

[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. doi: 10.1017/S0022112003005536. Google Scholar

[7]

J.-Y. Chemin, Perfect Incompressible Fluids,, Oxford University Press, (1998). Google Scholar

[8]

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

[9]

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

[10]

J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1960). Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1007/s00205-015-0953-z. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

J. R. Kamm, Shape and Stability of Two-Dimensional Uniform Vorticity Regions,, PhD thesis, (1987). Google Scholar

[20]

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

[21]

H. Lamb, Hydrodynamics,, Dover Publications, (1945). Google Scholar

[22]

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

[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. Google Scholar

[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. doi: 10.1016/j.matpur.2009.01.010. Google Scholar

[25]

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

[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. doi: 10.1137/0146049. Google Scholar

[27]

P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, (1992). Google Scholar

[28]

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

[29]

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

[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. doi: 10.1016/0021-9991(84)90051-2. Google Scholar

[31]

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

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