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Minimizers of the $ p $-oscillation functional
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 |
$ \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 $ |
$ \Omega $ |
$ {\mathbb R}^d $ |
$ d\geq 2 $ |
$ \epsilon>0 $ |
$ u = \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}} $ |
$ \rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $ |
$ \Phi^\delta (x) $ |
$ \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 $ |
$ |\delta | <\delta_{*} $ |
$ d = 2 $ |
$ |\nabla \Phi^\delta (x)|^2 $ |
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. 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. |
[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. 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. |
[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. Google Scholar |
[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. |
show all references
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. 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. |
[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. 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. |
[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. Google Scholar |
[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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