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 & Continuous Dynamical Systems - A, 2019, 39 (12) : 6801-6824. doi: 10.3934/dcds.2019232
References:
[1]

A. AftalionQ. 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.   Google Scholar

[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.   Google Scholar

[3]

F. BethuelG. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.   Google Scholar

[4]

F. BethuelP. 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.  Google Scholar

[5]

F. BethuelH. 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.  Google Scholar

[6]

F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994 doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[7]

F. BethuelP. 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.  Google Scholar

[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.  Google Scholar

[9]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958.  Google Scholar

[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.  Google Scholar

[11]

R. CarlesR. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843.  Google Scholar

[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.  Google Scholar

[13]

C. Coste, Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.  doi: 10.1007/s100510050178.  Google Scholar

[14]

M. del PinoM. 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.  Google Scholar

[15]

M. del PinoM. 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.  Google Scholar

[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.  Google Scholar

[17]

M. del PinoP. 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.  Google Scholar

[18]

Q. DuJ. 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.  Google Scholar

[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.  Google Scholar

[20]

T. FrischY. 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.  Google Scholar

[21]

P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. AbidC. HuepeS. MetensC. NoreC. T. PhamL. 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.  Google Scholar

[28]

C. A. JonesS. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. JosserandY. 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.  Google Scholar

[32]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. Google Scholar

[36]

P. I. Lizorkin, Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

C.-T. PhamC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.  doi: 10.1017/S0308210500001487.  Google Scholar

show all references

References:
[1]

A. AftalionQ. 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.   Google Scholar

[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.   Google Scholar

[3]

F. BethuelG. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.   Google Scholar

[4]

F. BethuelP. 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.  Google Scholar

[5]

F. BethuelH. 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.  Google Scholar

[6]

F. Bethuel, H. Brezis and F. He'lein, Ginzburg-Landau Vortices, Birkha"user, Boston, 1994 doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[7]

F. BethuelP. 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.  Google Scholar

[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.  Google Scholar

[9]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, New York, 1958.  Google Scholar

[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.  Google Scholar

[11]

R. CarlesR. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843.  Google Scholar

[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.  Google Scholar

[13]

C. Coste, Nonlinear Schrodinger equation and superfluid hydrodynamics, Eur. Phys. J. B Condens. Matter Phys., 1 (1998), 245-253.  doi: 10.1007/s100510050178.  Google Scholar

[14]

M. del PinoM. 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.  Google Scholar

[15]

M. del PinoM. 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.  Google Scholar

[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.  Google Scholar

[17]

M. del PinoP. 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.  Google Scholar

[18]

Q. DuJ. 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.  Google Scholar

[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.  Google Scholar

[20]

T. FrischY. 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.  Google Scholar

[21]

P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal., 45 (2005), 227-299.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. AbidC. HuepeS. MetensC. NoreC. T. PhamL. 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.  Google Scholar

[28]

C. A. JonesS. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. JosserandY. 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.  Google Scholar

[32]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj, 8 (1935), 153.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. Liu and J. Wei, Adler-Moser polynomials and traveling waves solutions of Gross-Pitaevskii, preprint. Google Scholar

[36]

P. I. Lizorkin, Multipliers of Fourier integrals, Proc. Steklov Inst. Math., 89 (1967), 269-290.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

C.-T. PhamC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Royal Soc. Edin. A, 131 (2001), 1457-1480.  doi: 10.1017/S0308210500001487.  Google Scholar

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