Well-posedness for mean-field evolutions arising in superconductivity

Duerinckx, Mitia; Fischer, Julian

doi:10.1016/j.anihpc.2017.11.004

Duerinckx, Mitia ; Fischer, Julian

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1267-1319.

Résumé

We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

MR Zbl

DOI : 10.1016/j.anihpc.2017.11.004

@article{AIHPC_2018__35_5_1267_0,
     author = {Duerinckx, Mitia and Fischer, Julian},
     title = {Well-posedness for mean-field evolutions arising in superconductivity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1267--1319},
     publisher = {Elsevier},
     volume = {35},
     number = {5},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.11.004},
     mrnumber = {3813965},
     zbl = {1393.35230},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.004/}
}

TY  - JOUR
AU  - Duerinckx, Mitia
AU  - Fischer, Julian
TI  - Well-posedness for mean-field evolutions arising in superconductivity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 1267
EP  - 1319
VL  - 35
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.004/
DO  - 10.1016/j.anihpc.2017.11.004
LA  - en
ID  - AIHPC_2018__35_5_1267_0
ER  -

%0 Journal Article
%A Duerinckx, Mitia
%A Fischer, Julian
%T Well-posedness for mean-field evolutions arising in superconductivity
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 1267-1319
%V 35
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.004/
%R 10.1016/j.anihpc.2017.11.004
%G en
%F AIHPC_2018__35_5_1267_0

Duerinckx, Mitia; Fischer, Julian. Well-posedness for mean-field evolutions arising in superconductivity. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1267-1319. doi : 10.1016/j.anihpc.2017.11.004. http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.004/

Bibliographie
Cité par

[1] Aftalion, A. Vortices in Bose–Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, vol. 67, Birkhäuser Boston, Inc., Boston, MA, 2006 | DOI | MR | Zbl

[2] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005 | MR | Zbl

[3] Ambrosio, L.; Mainini, E.; Serfaty, S. Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 2, pp. 217–246 | Numdam | MR | Zbl

[4] Ambrosio, L.; Serfaty, S. A gradient flow approach to an evolution problem arising in superconductivity, Commun. Pure Appl. Math., Volume 61 (2008) no. 11, pp. 1495–1539 | DOI | MR | Zbl

[5] Aranson, I.S.; Kramer, L. The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys., Volume 74 (2002), pp. 99–143 | DOI | MR | Zbl

[6] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[7] Bardos, K.; Titi, È.S. Euler equations for an ideal incompressible fluid, Usp. Mat. Nauk, Volume 62 (2007) no. 3(375), pp. 5–46 | MR | Zbl

[8] Bertozzi, A.L.; Laurent, Th.; Léger, F. Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions, Math. Models Methods Appl. Sci., Volume 22 (2012) no. suppl. 1 (39) | MR | Zbl

[9] Blatter, G.; Feigel'man, M.V.; Geshkenbein, V.B.; Larkin, A.I.; Vinokur, V.M. Vortices in high-temperature superconductors, Rev. Mod. Phys., Volume 66 (1994) no. 4 | DOI

[10] Bresch, D.; Métivier, G. Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for degenerate equations, Nonlinearity, Volume 19 (2006) no. 3, pp. 591–610 | DOI | MR | Zbl

[11] Brézis, H.; Gallouët, T. Nonlinear Schrödinger evolution equations, Nonlinear Anal., Volume 4 (1980) no. 4, pp. 677–681 | DOI | MR | Zbl

[12] Camassa, R.; Holm, D.D.; Levermore, C.D. Nonlinear Phenomena in Ocean Dynamics, Physica D, Volume 98 (1996) no. 2–4, pp. 258–286 (Los Alamos, NM, 1995) | MR | Zbl

[13] Camassa, R.; Holm, D.D.; Levermore, C.D. Long-time shallow-water equations with a varying bottom, J. Fluid Mech., Volume 349 (1997), pp. 173–189 | DOI | MR | Zbl

[14] Carrillo, J.A.; Choi, Y.-P.; Hauray, M. The derivation of swarming models: mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, Springer, 2014, pp. 1–46 | DOI | MR

[15] Chapman, S.J.; Richardson, G. Vortex pinning by inhomogeneities in type-II superconductors, Physica D, Volume 108 (1997), pp. 397–407 | DOI | MR | Zbl

[16] Chapman, S.J.; Rubinstein, J.; Schatzman, M. A mean-field model of superconducting vortices, Eur. J. Appl. Math., Volume 7 (1996) no. 2, pp. 97–111 | DOI | MR | Zbl

[17] Chemin, J.-Y. Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998 | MR | Zbl

[18] Coifman, R.R.; Meyer, Y. Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (With an English summary) | Numdam | MR | Zbl

[19] Coifman, R.R.; Meyer, Y. Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., Volume vol. 112, Princeton Univ. Press, Princeton, NJ (1986), pp. 3–45 (Beijing, 1984) | MR | Zbl

[20] Delort, J.-M. Existence de nappes de tourbillon en dimension deux, J. Am. Math. Soc., Volume 4 (1991) no. 3, pp. 553–586 | MR | Zbl

[21] Du, Q.; Zhang, P. Existence of weak solutions to some vortex density models, SIAM J. Math. Anal., Volume 34 (2003) no. 6, pp. 1279–1299 | MR | Zbl

[22] Duerinckx, M. Mean-field limits for some Riesz interaction gradient flows, SIAM J. Math. Anal., Volume 48 (2015) no. 3, pp. 2269–2300 | DOI | MR

[23] Duerinckx, M. Topics in the Mathematics of Disordered Media, Université Libre de Bruxelles and Université Pierre et Marie Curie, 2017 PhD thesis (supervised by A. Gloria and S. Serfaty)

[24] Duerinckx, M.; Serfaty, S. Mean-field dynamics for Ginzburg–Landau vortices with pinning and applied force, 2017 (Preprint) | arXiv | MR

[25] E, W. Dynamics of vortex liquids in Ginzburg–Landau theories with applications to superconductivity, Phys. Rev. B, Volume 50 (1994), pp. 1126–1135 | MR | Zbl

[26] Greenspan, H.P. The Theory of Rotating Fluids, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, New York, 1980 (Reprint of the 1968 original) | MR

[27] Gulisashvili, A.; Kon, M.A. Exact smoothing properties of Schrödinger semigroups, Am. J. Math., Volume 118 (1996) no. 6, pp. 1215–1248 | MR | Zbl

[28] Han, Q.; Lin, F. Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 1997 | MR | Zbl

[29] Jerrard, R.L.; Smets, D. Vortex dynamics for the two dimensional non homogeneous Gross–Pitaevskii equation, Ann. Sc. Norm. Super. Pisa, Volume 14 (2015) no. 3, pp. 729–766 | MR | Zbl

[30] Jerrard, R.L.; Spirn, D. Hydrodynamic limit of the Gross–Pitaevskii equation, Commun. Partial Differ. Equ., Volume 40 (2015) no. 2, pp. 135–190 | DOI | MR | Zbl

[31] Jian, H.-Y.; Song, B.-H. Vortex dynamics of Ginzburg–Landau equations in inhomogeneous superconductors, J. Differ. Equ., Volume 170 (2001) no. 1, pp. 123–141 | MR | Zbl

[32] Jordan, R.; Kinderlehrer, D.; Otto, F. The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., Volume 29 (1998) no. 1, pp. 1–17 | DOI | MR | Zbl

[33] Kato, T.; Ponce, G. Commutator estimates and the Euler and Navier–Stokes equations, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 891–907 | DOI | MR | Zbl

[34] Kurzke, M.; Marzuola, J.L.; Spirn, D. Gross–Pitaevskii vortex motion with critically-scaled inhomogeneities, SIAM J. Math. Anal., Volume 49 (2017) no. 1, pp. 471–500 | DOI | MR | Zbl

[35] Kurzke, M.; Spirn, D. Vortex liquids and the Ginzburg–Landau equation, Forum Math. Sigma, Volume 2 (2014) (63) | DOI | MR | Zbl

[36] Levermore, C.D.; Oliver, M.; Titi, E.S. Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J., Volume 45 (1996) no. 2, pp. 479–510 | DOI | MR | Zbl

[37] Levermore, C.D.; Oliver, M.; Titi, E.S. Global well-posedness for the lake equations, Physica D, Volume 98 (1996), pp. 492–509 | DOI | Zbl

[38] Li, D. On Kato–Ponce and fractional Leibniz, 2016 (Preprint) | arXiv | Zbl

[39] Lin, F.; Zhang, P. On the hydrodynamic limit of Ginzburg–Landau vortices, Discrete Contin. Dyn. Syst., Volume 6 (2000) no. 1, pp. 121–142 | MR | Zbl

[40] Lions, P.-L. Mathematical Topics in Fluid Mechanics: Vol. 2: Compressible Models, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, Oxford Science Publications, New York, 1998 | MR | Zbl

[41] Loeper, G. Uniqueness of the solution to the Vlasov–Poisson system with bounded density, J. Math. Pures Appl. (9), Volume 86 (2006) no. 1, pp. 68–79 | DOI | MR | Zbl

[42] Masmoudi, N.; Zhang, P. Global solutions to vortex density equations arising from sup-conductivity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 4, pp. 441–458 | DOI | Numdam | MR | Zbl

[43] Meyers, N.G. An $L^{p}$ -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa, Volume 17 (1963) no. 3, pp. 189–206 | Numdam | MR | Zbl

[44] Oliver, M. Justification of the shallow water limit for a rigid lid flow with bottom topography, Theor. Comput. Fluid Dyn., Volume 9 (1997), pp. 311–324 | DOI | Zbl

[45] Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Commun. Partial Differ. Equ., Volume 26 (2001) no. 1–2, pp. 101–174 | MR | Zbl

[46] Rougerie, N. La théorie de Gross–Pitaevskii pour un condensat de Bose–Einstein en rotation: vortex et transitions de phase, Université Pierre et Marie Curie, 2010 (PhD thesis)

[47] Sandier, E.; Serfaty, S. Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and their Applications, vol. 70, Birkhäuser Boston Inc., Boston, MA, 2007 | DOI | MR | Zbl

[48] Serfaty, S. Mean-field limits of the Gross–Pitaevskii and parabolic Ginzburg–Landau equations, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 713–768 | DOI | MR | Zbl

[49] Serfaty, S.; Tice, I. Ginzburg–Landau vortex dynamics with pinning and strong applied currents, Arch. Ration. Mech. Anal., Volume 201 (2011) no. 2, pp. 413–464 | DOI | MR | Zbl

[50] Serfaty, S.; Vázquez, J.L. A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differ. Equ., Volume 49 (2014) no. 3–4, pp. 1091–1120 | MR | Zbl

[51] Tice, I. Ginzburg–Landau vortex dynamics driven by an applied boundary current, Commun. Pure Appl. Math., Volume 63 (2010) no. 12, pp. 1622–1676 | DOI | MR | Zbl

[52] Tilley, D.; Tilley, J. Superfluidity and Superconductivity, Adam Hilger, 1986

[53] Tinkham, M. Introduction to Superconductivity, McGraw-Hill Inc., 1996

[54] Yudovich, V.I. Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. Mat. Fiz., Volume 3 (1963), pp. 1032–1066 | MR | Zbl

Cité par Sources :