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Steady asymmetric vortex pairs for Euler equations
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 |
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.
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. |
[2] |
J. Burbea,
Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16.
doi: 10.1007/BF02281165. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
D. G. Dritschel,
A general theory for two-dimensional vortex interactions, J. Fluid Mech., 293 (1995), 269-303.
doi: 10.1017/S0022112095001716. |
[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. |
[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. |
[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. |
[14] |
J. Gómez-Serrano,
On the existence of stationary patches, Adv. Math., 343 (2019), 110-140.
doi: 10.1016/j.aim.2018.11.012. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
G. Keady,
Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, 26 (1985), 487-502.
doi: 10.1017/S0334270000004677. |
[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. |
[29] |
J. Norbury, Steady planar vortex pairs in an ideal fluid, Comm. Pure Appl. Math., 28, (1975), 679–700.
doi: 10.1002/cpa.3160280602. |
[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. |
[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. |
[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. |
[33] |
P. G. Saffman, Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.
![]() |
[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. |
[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. |
[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. |
[37] |
V. I. Yudovič, Non-stationnary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz., 3 (1963), 1032–1066. |
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. |
[2] |
J. Burbea,
Motions of vortex patches, Lett. Math. Phys., 6 (1982), 1-16.
doi: 10.1007/BF02281165. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
D. G. Dritschel,
A general theory for two-dimensional vortex interactions, J. Fluid Mech., 293 (1995), 269-303.
doi: 10.1017/S0022112095001716. |
[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. |
[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. |
[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. |
[14] |
J. Gómez-Serrano,
On the existence of stationary patches, Adv. Math., 343 (2019), 110-140.
doi: 10.1016/j.aim.2018.11.012. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
G. Keady,
Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, 26 (1985), 487-502.
doi: 10.1017/S0334270000004677. |
[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. |
[29] |
J. Norbury, Steady planar vortex pairs in an ideal fluid, Comm. Pure Appl. Math., 28, (1975), 679–700.
doi: 10.1002/cpa.3160280602. |
[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. |
[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. |
[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. |
[33] |
P. G. Saffman, Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.
![]() |
[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. |
[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. |
[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. |
[37] |
V. I. Yudovič, Non-stationnary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz., 3 (1963), 1032–1066. |
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