• Previous Article
    On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation
  • DCDS Home
  • This Issue
  • Next Article
    Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type
2016, 36(10): 5401-5422. doi: 10.3934/dcds.2016038

Bifurcation of rotating patches from Kirchhoff vortices

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.
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).

[2]

J. Burbea, Motions of vortex patches,, Lett. Math. Phys., 6 (1982), 1. 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. 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. 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). 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. 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. 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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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. doi: 10.1007/s00220-003-1017-z.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[25]

T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices,, Physics of Fluids, 20 (2008). 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. doi: 10.1137/0146049.

[27]

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

[28]

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

[29]

Y. H. Wan, The stability of rotating vortex patches,, Comm. Math. Phys., 107 (1986), 1. 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. 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.

show all references

References:
[1]

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

[2]

J. Burbea, Motions of vortex patches,, Lett. Math. Phys., 6 (1982), 1. 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. 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. 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). 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. 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. 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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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. doi: 10.1007/s00220-003-1017-z.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[25]

T. B. Mitchell and L. F. Rossi, The evolution of Kirchhoff elliptic vortices,, Physics of Fluids, 20 (2008). 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. doi: 10.1137/0146049.

[27]

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

[28]

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

[29]

Y. H. Wan, The stability of rotating vortex patches,, Comm. Math. Phys., 107 (1986), 1. 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. 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.

[1]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002

[2]

Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543

[3]

Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753

[4]

Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391

[5]

Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349

[6]

B. Emamizadeh, F. Bahrami, M. H. Mehrabi. Steiner symmetric vortices attached to seamounts. Communications on Pure & Applied Analysis, 2004, 3 (4) : 663-674. doi: 10.3934/cpaa.2004.3.663

[7]

Peter Constantin. Transport in rotating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 165-176. doi: 10.3934/dcds.2004.10.165

[8]

D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47

[9]

Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439

[10]

Takashi Suzuki. Brownian point vortices and dd-model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161

[11]

A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100

[12]

Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223

[13]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121

[14]

A. V. Borisov, I. S. Mamaev, S. M. Ramodanov. Dynamics of a circular cylinder interacting with point vortices. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 35-50. doi: 10.3934/dcdsb.2005.5.35

[15]

F. D. Araruna, F. O. Matias, M. P. Matos, S. M. S. Souza. Hidden regularity for the Kirchhoff equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1049-1056. doi: 10.3934/cpaa.2008.7.1049

[16]

Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin. Variational discretization for rotating stratified fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 477-509. doi: 10.3934/dcds.2014.34.477

[17]

Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365

[18]

Angel Castro, Diego Córdoba, Javier Gómez-Serrano, Alberto Martín Zamora. Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5045-5059. doi: 10.3934/dcds.2014.34.5045

[19]

Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

[20]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]