# American Institute of Mathematical Sciences

December  2019, 39(12): 6801-6824. doi: 10.3934/dcds.2019232

## Superfluids passing an obstacle and vortex nucleation

 1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY. 10012, USA 2 Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada

Dedicated to Professor Luis Caffarelli on the occasion of his 70th birthday, with deep admiration

Received  June 2018 Revised  July 2018 Published  June 2019

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle
 $\epsilon^2 \Delta u+ u(1-|u|^2) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu} = 0 \ \mbox{on}\ \partial \Omega$
where
 $\Omega$
is a smooth bounded domain in
 ${\mathbb R}^d$
(
 $d\geq 2$
), which is referred as the obstacle and
 $\epsilon>0$
is sufficiently small. We first construct a vortex free solution of the form
 $u = \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}}$
with
 $\rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x)$
where
 $\Phi^\delta (x)$
is the unique solution for the subsonic irrotational flow equation
 $\nabla ( (1-|\nabla \Phi|^2)\nabla \Phi ) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} = 0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty$
and
 $|\delta | <\delta_{*}$
(the sound speed).
In dimension
 $d = 2$
, on the background of this vortex free solution we also construct solutions with single vortex close to the maximum or minimum points of the function
 $|\nabla \Phi^\delta (x)|^2$
(which are on the boundary of the obstacle). The latter verifies the vortex nucleation phenomena (for the steady states) in superfluids described by the Gross-Pitaevskii equations. Moreover, after some proper scalings, the limits of these vortex solutions are traveling wave solution of the Gross-Pitaevskii equation. These results also show rigorously the conclusions drawn from the numerical computations in [26,27].
Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see [1] and references therein) for the trapped Bose-Einstein condensates, are also discussed.
Citation: Fanghua Lin, Juncheng Wei. Superfluids passing an obstacle and vortex nucleation. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6801-6824. doi: 10.3934/dcds.2019232
##### References:
 [1] A. Aftalion, Q. Du and Y. Pomeau, Dissipative flow and vortex shedding in the Painleve boundary layer of a Bose-Einstein condensate, Phys. Rev. Lett., 91 (2003), 090407-1-4. [2] F. Bethuel and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅰ, Ann. Inst. H. Poincare' Phys. The'or., 70 (1999), 147-238. [3] F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94. [4] F. Bethuel, P. Gravejat and J.-G. Saut, Travelling waves for the Gross-Pitaevskii equation, Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2. [5] F. Bethuel, H. Brezis and F. He'lein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. and PDE., 1 (1993), 123-148.  doi: 10.1007/BF01191614. [6] F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994 doi: 10.1007/978-1-4612-0287-5. [7] F. Bethuel, P. Gravejat and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2. [8] L. Bers, Ezistence and uniqueness of a subsonic pow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303. [9] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958. [10] S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. Lond. Math. Soc., 90 (2005), 245-272.  doi: 10.1112/S0024611504014960. [11] R. Carles, R. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843. [12] D. Chiron and M. Maris, Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit, Comm. Math. Phys., 326 (2014), 329-392.  doi: 10.1007/s00220-013-1879-7. [13] C. Coste, Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.  doi: 10.1007/s100510050178. [14] M. del Pino, M. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature, Journal of Differential Geometry, 83 (2013), 67-131.  doi: 10.4310/jdg/1357141507. [15] M. del Pino, M. Kowalczyk and M. Musso, Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497-541.  doi: 10.1016/j.jfa.2006.07.006. [16] G.-C. Dong and B. Ou, Subsonic flows around a body in space, Comm. Partial Differential Equations, 18 (1993), 355-379.  doi: 10.1080/03605309308820933. [17] M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., (2004), 1511-1527.  doi: 10.1155/S1073792804133588. [18] Q. Du, J. Wei and C. Zhao, Vortex solutions of the high-$\kappa$ high-field Ginzburg-Landau model with an applied current, SIAM J. Math. Anal., 42 (2010), 2368-2401.  doi: 10.1137/090769983. [19] R. Finn and D. Gilbarg, Three dimensional subsonicflows and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476. [20] T. Frisch, Y. Pomeau and S. Rica, Transition to dissipation in a model of superflow, Phys. Rev. Lett., 69 (1992), 1644-1647.  doi: 10.1103/PhysRevLett.69.1644. [21] P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299. [22] P. Gravejat, Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation, Differential Integral Equations, 17 (2004), 1213-1232. [23] P. Gravejat, Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 591-637.  doi: 10.1016/j.anihpc.2003.09.001. [24] P. Gravejat, A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation, Comm. Math. Phys., 243 (2003), 93-103.  doi: 10.1007/s00220-003-0961-y. [25] J. Grant and P. H. Roberts, Motions in a Bose condensate. Ⅲ. The structure and effective masses of charged and uncharged impurities, J. Phys. A: Math., Nucl. Gen., 7 (1974), 260-279.  doi: 10.1088/0305-4470/34/1/306. [26] C. Huepe and M. E. Brachet, Scaling laws for vortical nucleation solutions in a model of superflow, Phys. D, 140 (2000), 126-140.  doi: 10.1016/S0167-2789(99)00229-8. [27] M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman and M. E. Brachet, Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynam. Res., 33 (2003), 509-544.  doi: 10.1016/j.fluiddyn.2003.09.001. [28] C. A. Jones, S. J. Putterman and P. H. Roberts, Stability of wave solutions of nonlinear Schrodinger equations in two and three dimensions, J. Phys A: Math. Gen., 19 (1986), 2991-3011. [29] C. A. Jones and P. H. Roberts, Motion in a Bose condensate Ⅳ, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619.  doi: 10.1088/0305-4470/15/8/036. [30] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.  doi: 10.1088/0951-7715/14/5/201. [31] C. Josserand, Y. Pomeau and S. Rica, Vortex shedding in a model of superflow, Phys. D, 134 (1999), 111-125.  doi: 10.1016/S0167-2789(99)00066-4. [32] L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153. [33] F.-H. Lin and J. Wei, Traveling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621.  doi: 10.1002/cpa.20338. [34] F.-H. Lin and P. Zhang, Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Ration. Mech. Anal., 179 (2006), 79-107.  doi: 10.1007/s00205-005-0383-4. [35] Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. [36] P. I. Lizorkin, Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290. [37] M. Maris, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl., 81 (2002), 1207-1239.  doi: 10.1016/S0021-7824(02)01274-6. [38] M. Maris, Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40 (2008), 1076-1103.  doi: 10.1137/070711189. [39] M. Maris, Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity, Ann. Math., 178 (2013), 107-182.  doi: 10.4007/annals.2013.178.1.2. [40] C.-T. Pham, C. Nore and M. E. Brachet, Boundary layers and emitted excitations in nonlinear Schröinger superflow past a disk, Phys. D, 210 (2005), 203-226.  doi: 10.1016/j.physd.2005.07.013. [41] S. Rica, A remark on the critical speed of vortex nucleation in the nonlinear Schrodinger equation, Phys. D, 148 (2001), 221-226.  doi: 10.1016/S0167-2789(00)00168-8. [42] O. Rey and J. Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part Ⅱ: $N \geq 4$, Ann. Non linearie, Annoles de l'Institut H. Poincaré, 22 (2005), 459-484.  doi: 10.1016/j.anihpc.2004.07.004. [43] M. Shiffman, On the ezistence of subsonic flows of a compressible fluid, Arch. Rational Mech. Anal., 2 (1952), 605-652.  doi: 10.1512/iumj.1952.1.51020. [44] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, NJ, 1971. [45] J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.  doi: 10.1017/S0308210500001487.

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##### References:
 [1] A. Aftalion, Q. Du and Y. Pomeau, Dissipative flow and vortex shedding in the Painleve boundary layer of a Bose-Einstein condensate, Phys. Rev. Lett., 91 (2003), 090407-1-4. [2] F. Bethuel and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅰ, Ann. Inst. H. Poincare' Phys. The'or., 70 (1999), 147-238. [3] F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94. [4] F. Bethuel, P. Gravejat and J.-G. Saut, Travelling waves for the Gross-Pitaevskii equation, Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2. [5] F. Bethuel, H. Brezis and F. He'lein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. and PDE., 1 (1993), 123-148.  doi: 10.1007/BF01191614. [6] F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994 doi: 10.1007/978-1-4612-0287-5. [7] F. Bethuel, P. Gravejat and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2. [8] L. Bers, Ezistence and uniqueness of a subsonic pow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303. [9] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958. [10] S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. Lond. Math. Soc., 90 (2005), 245-272.  doi: 10.1112/S0024611504014960. [11] R. Carles, R. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843. [12] D. Chiron and M. Maris, Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit, Comm. Math. Phys., 326 (2014), 329-392.  doi: 10.1007/s00220-013-1879-7. [13] C. Coste, Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.  doi: 10.1007/s100510050178. [14] M. del Pino, M. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature, Journal of Differential Geometry, 83 (2013), 67-131.  doi: 10.4310/jdg/1357141507. [15] M. del Pino, M. Kowalczyk and M. Musso, Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497-541.  doi: 10.1016/j.jfa.2006.07.006. [16] G.-C. Dong and B. Ou, Subsonic flows around a body in space, Comm. Partial Differential Equations, 18 (1993), 355-379.  doi: 10.1080/03605309308820933. [17] M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., (2004), 1511-1527.  doi: 10.1155/S1073792804133588. [18] Q. Du, J. Wei and C. Zhao, Vortex solutions of the high-$\kappa$ high-field Ginzburg-Landau model with an applied current, SIAM J. Math. Anal., 42 (2010), 2368-2401.  doi: 10.1137/090769983. [19] R. Finn and D. Gilbarg, Three dimensional subsonicflows and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476. [20] T. Frisch, Y. Pomeau and S. Rica, Transition to dissipation in a model of superflow, Phys. Rev. Lett., 69 (1992), 1644-1647.  doi: 10.1103/PhysRevLett.69.1644. [21] P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299. [22] P. Gravejat, Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation, Differential Integral Equations, 17 (2004), 1213-1232. [23] P. Gravejat, Decay for travelling waves in the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 591-637.  doi: 10.1016/j.anihpc.2003.09.001. [24] P. Gravejat, A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation, Comm. Math. Phys., 243 (2003), 93-103.  doi: 10.1007/s00220-003-0961-y. [25] J. Grant and P. H. Roberts, Motions in a Bose condensate. Ⅲ. The structure and effective masses of charged and uncharged impurities, J. Phys. A: Math., Nucl. Gen., 7 (1974), 260-279.  doi: 10.1088/0305-4470/34/1/306. [26] C. Huepe and M. E. Brachet, Scaling laws for vortical nucleation solutions in a model of superflow, Phys. D, 140 (2000), 126-140.  doi: 10.1016/S0167-2789(99)00229-8. [27] M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman and M. E. Brachet, Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynam. Res., 33 (2003), 509-544.  doi: 10.1016/j.fluiddyn.2003.09.001. [28] C. A. Jones, S. J. Putterman and P. H. Roberts, Stability of wave solutions of nonlinear Schrodinger equations in two and three dimensions, J. Phys A: Math. Gen., 19 (1986), 2991-3011. [29] C. A. Jones and P. H. Roberts, Motion in a Bose condensate Ⅳ, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619.  doi: 10.1088/0305-4470/15/8/036. [30] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25-R62.  doi: 10.1088/0951-7715/14/5/201. [31] C. Josserand, Y. Pomeau and S. Rica, Vortex shedding in a model of superflow, Phys. D, 134 (1999), 111-125.  doi: 10.1016/S0167-2789(99)00066-4. [32] L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153. [33] F.-H. Lin and J. Wei, Traveling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621.  doi: 10.1002/cpa.20338. [34] F.-H. Lin and P. Zhang, Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Ration. Mech. Anal., 179 (2006), 79-107.  doi: 10.1007/s00205-005-0383-4. [35] Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. [36] P. I. Lizorkin, Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290. [37] M. Maris, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl., 81 (2002), 1207-1239.  doi: 10.1016/S0021-7824(02)01274-6. [38] M. Maris, Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40 (2008), 1076-1103.  doi: 10.1137/070711189. [39] M. Maris, Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity, Ann. Math., 178 (2013), 107-182.  doi: 10.4007/annals.2013.178.1.2. [40] C.-T. Pham, C. Nore and M. E. Brachet, Boundary layers and emitted excitations in nonlinear Schröinger superflow past a disk, Phys. D, 210 (2005), 203-226.  doi: 10.1016/j.physd.2005.07.013. [41] S. Rica, A remark on the critical speed of vortex nucleation in the nonlinear Schrodinger equation, Phys. D, 148 (2001), 221-226.  doi: 10.1016/S0167-2789(00)00168-8. [42] O. Rey and J. Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part Ⅱ: $N \geq 4$, Ann. Non linearie, Annoles de l'Institut H. Poincaré, 22 (2005), 459-484.  doi: 10.1016/j.anihpc.2004.07.004. [43] M. Shiffman, On the ezistence of subsonic flows of a compressible fluid, Arch. Rational Mech. Anal., 2 (1952), 605-652.  doi: 10.1512/iumj.1952.1.51020. [44] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, NJ, 1971. [45] J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.  doi: 10.1017/S0308210500001487.
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