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April  2021, 41(4): 1939-1969. doi: 10.3934/dcds.2020348

Steady asymmetric vortex pairs for Euler equations

Zineb Hassainia 1, and  Taoufik Hmidi 2,

1. 

NYU, Abu Dhabi, Saadiyat Marina District, P.O. Box 129188, Abu Dhabi, UAE
2. 

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

Received  January 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter-rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behavior for the angular velocity and the translating speed close to the point vortex pairs.

Keywords: Euler equations, vortex dynamics, steady vortex pairs, desingularization, implicit function theorem.
Mathematics Subject Classification: Primary: 35Q35, 35Q31; Secondary: 53C35, 76C05.
Citation: Zineb Hassainia, Taoufik Hmidi. Steady asymmetric vortex pairs for Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1939-1969. doi: 10.3934/dcds.2020348
References:
[1]

T. Bartsch and M. Sacchet, Periodic solutions with prescribed minimal period of vortex type problem in domains, Nonlinearity, 31 (2018), 2156-2172.  doi: 10.1088/1361-6544/aaaf2d.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating smooth solutions for the incompressible 2D Euler equations, Arch. Ration. Mech. Anal., 231 (2019), 719-785.  doi: 10.1007/s00205-018-1288-3.  Google Scholar

[6]

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

[7]

F. de la Hoz, Z. Hassainia and T. Hmidi, 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.  Google Scholar

[8]

F. de la Hoz, Z. Hassainia, T. Hmidi and J Mateu, An analytical and numerical study of steady patches in the disc, Anal. PDE, 9 (2016), 1609-1670.  doi: 10.2140/apde.2016.9.1609.  Google Scholar

[9]

D. G. Dritschel, A general theory for two-dimensional vortex interactions, J. Fluid Mech., 293 (1995), 269-303.  doi: 10.1017/S0022112095001716.  Google Scholar

[10]

D. G. Dritschel, T. Hmidi and C. Renault, Imperfect bifurcation for the quasi-geostrophic shallow-water equations, Arch. Ration. Mech. Anal., 231 (2019), 1853-1915.  doi: 10.1007/s00205-018-1312-7.  Google Scholar

[11]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, 128. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.  Google Scholar

[12]

C. Garcìa, T. Hmidi and J. Mateu, Time periodic solutions for 3D quasi-geostrophic model, preprint, arXiv: 2004.01644. Google Scholar

[13]

C. Garcìa, T. Hmidi and J. Soler, Non uniform rotating vortices and periodic orbits for the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 238 (2020), 929-1085.  doi: 10.1007/s00205-020-01561-z.  Google Scholar

[14]

J. Gómez-Serrano, On the existence of stationary patches, Adv. Math., 343 (2019), 110-140.  doi: 10.1016/j.aim.2018.11.012.  Google Scholar

[15]

J. Gómez-Serrano, J. Park, J. Shi and Y. Yao, Symmetry in stationary and uniformly-rotating solutions of active scalar equations, preprint, arXiv: 1908.01722. Google Scholar

[16]

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.  Google Scholar

[17]

Z. Hassainia, N. Masmoudi and M. H. Wheeler, Global bifurcation of rotating vortex patches, Comm. Pure Appl. Math., 73 (2020), 1933-1980.  doi: 10.1002/cpa.21855.  Google Scholar

[18]

T. Hmidi, On the trivial solutions for the rotating patch model, J. Evol. Equ., 15 (2015), 801-816.  doi: 10.1007/s00028-015-0281-7.  Google Scholar

[19]

T. Hmidi and C. Renault, Existence of small loops in a bifurcation diagram near degenerate eigenvalues, Nonlinearity, 30 (2017), 3821-3852.  doi: 10.1088/1361-6544/aa82ef.  Google Scholar

[20]

F. de la Hoz, T. Hmidi, J. Mateu and J. Verdera, Doubly connected $V$-states for the planar Euler equations, SIAM J. Math. Anal., 48 (2016), 1892-1928.  doi: 10.1137/140992801.  Google Scholar

[21]

T. Hmidi and J. Mateu, Bifurcation of rotating patches from Kirchhoff vortices, Discrete Contin. Dyn. Syst., 36 (2016), 5401-5422.  doi: 10.3934/dcds.2016038.  Google Scholar

[22]

T. Hmidi and J. Mateu, Degenerate bifurcation of the rotating patches, Adv. Math., 302 (2016), 799-850.  doi: 10.1016/j.aim.2016.07.022.  Google Scholar

[23]

T. Hmidi and J. Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys., 350 (2017), 699-747.  doi: 10.1007/s00220-016-2784-7.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

G. Keady, Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, 26 (1985), 487-502.  doi: 10.1017/S0334270000004677.  Google Scholar

[27]

G. Kirchhoff, Vorlesungen uber Mathematische Physik, Leipzig, 1874. Google Scholar

[28]

P. K. Newton, The $N$-Vortex Problem, Analytical techniques. Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[29]

J. Norbury, Steady planar vortex pairs in an ideal fluid, Comm. Pure Appl. Math., 28, (1975), 679–700. doi: 10.1002/cpa.3160280602.  Google Scholar

[30]

E. A. Overman II, 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.  Google Scholar

[31]

R. T. Pierrehumbert, A family of steady, translating vortex pairs with distributed vorticity, Journal of Fluid Mechanics, 99 (1980), 129-144.  doi: 10.1017/S0022112080000559.  Google Scholar

[32]

C. Renault, Relative equilibria with holes for the surface quasi-geostrophic equations, J. Differential Equations, 263 (2017), 567-614.  doi: 10.1016/j.jde.2017.02.050.  Google Scholar

[33] P. G. Saffman, Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.   Google Scholar
[34]

P. G. Saffman and R. Szeto, Equilibrium shapes of a pair of equal uniform vortices, Phys. Fluids, 23 (1980), 2339-2342.  doi: 10.1063/1.862935.  Google Scholar

[35]

B. Turkington, Corotating steady vortex flows with $N$-fold symmety, Nonlinear Anal., 9 (1985), 351-369.  doi: 10.1016/0362-546X(85)90059-8.  Google Scholar

[36]

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. Numerical algorithms ans results, J. Comput. Phys., 53 (1984), 42-71.  doi: 10.1016/0021-9991(84)90051-2.  Google Scholar

[37]

V. I. Yudovič, Non-stationnary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz., 3 (1963), 1032–1066.  Google Scholar

show all references

References:
[1]

T. Bartsch and M. Sacchet, Periodic solutions with prescribed minimal period of vortex type problem in domains, Nonlinearity, 31 (2018), 2156-2172.  doi: 10.1088/1361-6544/aaaf2d.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

A. Castro, D. Córdoba and J. Gómez-Serrano, Uniformly rotating smooth solutions for the incompressible 2D Euler equations, Arch. Ration. Mech. Anal., 231 (2019), 719-785.  doi: 10.1007/s00205-018-1288-3.  Google Scholar

[6]

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

[7]

F. de la Hoz, Z. Hassainia and T. Hmidi, 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.  Google Scholar

[8]

F. de la Hoz, Z. Hassainia, T. Hmidi and J Mateu, An analytical and numerical study of steady patches in the disc, Anal. PDE, 9 (2016), 1609-1670.  doi: 10.2140/apde.2016.9.1609.  Google Scholar

[9]

D. G. Dritschel, A general theory for two-dimensional vortex interactions, J. Fluid Mech., 293 (1995), 269-303.  doi: 10.1017/S0022112095001716.  Google Scholar

[10]

D. G. Dritschel, T. Hmidi and C. Renault, Imperfect bifurcation for the quasi-geostrophic shallow-water equations, Arch. Ration. Mech. Anal., 231 (2019), 1853-1915.  doi: 10.1007/s00205-018-1312-7.  Google Scholar

[11]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, 128. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.  Google Scholar

[12]

C. Garcìa, T. Hmidi and J. Mateu, Time periodic solutions for 3D quasi-geostrophic model, preprint, arXiv: 2004.01644. Google Scholar

[13]

C. Garcìa, T. Hmidi and J. Soler, Non uniform rotating vortices and periodic orbits for the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 238 (2020), 929-1085.  doi: 10.1007/s00205-020-01561-z.  Google Scholar

[14]

J. Gómez-Serrano, On the existence of stationary patches, Adv. Math., 343 (2019), 110-140.  doi: 10.1016/j.aim.2018.11.012.  Google Scholar

[15]

J. Gómez-Serrano, J. Park, J. Shi and Y. Yao, Symmetry in stationary and uniformly-rotating solutions of active scalar equations, preprint, arXiv: 1908.01722. Google Scholar

[16]

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.  Google Scholar

[17]

Z. Hassainia, N. Masmoudi and M. H. Wheeler, Global bifurcation of rotating vortex patches, Comm. Pure Appl. Math., 73 (2020), 1933-1980.  doi: 10.1002/cpa.21855.  Google Scholar

[18]

T. Hmidi, On the trivial solutions for the rotating patch model, J. Evol. Equ., 15 (2015), 801-816.  doi: 10.1007/s00028-015-0281-7.  Google Scholar

[19]

T. Hmidi and C. Renault, Existence of small loops in a bifurcation diagram near degenerate eigenvalues, Nonlinearity, 30 (2017), 3821-3852.  doi: 10.1088/1361-6544/aa82ef.  Google Scholar

[20]

F. de la Hoz, T. Hmidi, J. Mateu and J. Verdera, Doubly connected $V$-states for the planar Euler equations, SIAM J. Math. Anal., 48 (2016), 1892-1928.  doi: 10.1137/140992801.  Google Scholar

[21]

T. Hmidi and J. Mateu, Bifurcation of rotating patches from Kirchhoff vortices, Discrete Contin. Dyn. Syst., 36 (2016), 5401-5422.  doi: 10.3934/dcds.2016038.  Google Scholar

[22]

T. Hmidi and J. Mateu, Degenerate bifurcation of the rotating patches, Adv. Math., 302 (2016), 799-850.  doi: 10.1016/j.aim.2016.07.022.  Google Scholar

[23]

T. Hmidi and J. Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys., 350 (2017), 699-747.  doi: 10.1007/s00220-016-2784-7.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

G. Keady, Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, 26 (1985), 487-502.  doi: 10.1017/S0334270000004677.  Google Scholar

[27]

G. Kirchhoff, Vorlesungen uber Mathematische Physik, Leipzig, 1874. Google Scholar

[28]

P. K. Newton, The $N$-Vortex Problem, Analytical techniques. Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[29]

J. Norbury, Steady planar vortex pairs in an ideal fluid, Comm. Pure Appl. Math., 28, (1975), 679–700. doi: 10.1002/cpa.3160280602.  Google Scholar

[30]

E. A. Overman II, 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.  Google Scholar

[31]

R. T. Pierrehumbert, A family of steady, translating vortex pairs with distributed vorticity, Journal of Fluid Mechanics, 99 (1980), 129-144.  doi: 10.1017/S0022112080000559.  Google Scholar

[32]

C. Renault, Relative equilibria with holes for the surface quasi-geostrophic equations, J. Differential Equations, 263 (2017), 567-614.  doi: 10.1016/j.jde.2017.02.050.  Google Scholar

[33] P. G. Saffman, Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.   Google Scholar
[34]

P. G. Saffman and R. Szeto, Equilibrium shapes of a pair of equal uniform vortices, Phys. Fluids, 23 (1980), 2339-2342.  doi: 10.1063/1.862935.  Google Scholar

[35]

B. Turkington, Corotating steady vortex flows with $N$-fold symmety, Nonlinear Anal., 9 (1985), 351-369.  doi: 10.1016/0362-546X(85)90059-8.  Google Scholar

[36]

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. Numerical algorithms ans results, J. Comput. Phys., 53 (1984), 42-71.  doi: 10.1016/0021-9991(84)90051-2.  Google Scholar

[37]

V. I. Yudovič, Non-stationnary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz., 3 (1963), 1032–1066.  Google Scholar

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