Steady asymmetric vortex pairs for Euler equations

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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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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