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February  2020, 13(1): 187-210. doi: 10.3934/krm.2020007

Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation

Marcel Braukhoff

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Received  March 2019 Revised  July 2019 Published  December 2019

Fund Project: The author was partially funded by the Austrian Science Fund (FWF) project F 65.

The semiconductor Boltzmann-Dirac-Benney equation
$ \partial_t f + \nabla\epsilon(p)\cdot\nabla_x f - \nabla \rho_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_\lambda(p)-f}\tau, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 $
is a model for ultracold atoms trapped in an optical lattice. The global existence of a solution is shown for small
$ \tau>0 $
 assuming that the initial data are analytic and sufficiently close to the Fermi-Dirac distribution
$ \mathcal F_\lambda $
. This system contains an interaction potential
$ \rho_f: = \int_B fdp $
 being significantly more singular than the Coulomb potential, which causes major structural difficulties in the analysis.
The key technique is based of the ideas of Mouhot and Villani by using Gevrey-type norms which vary over time. The global existence result for small initial data is also generalized to
$ \partial_t f + Lf = Q(f), $
where
$ L $
 is a generator of an
$ C^0 $
-group with
$ \|e^{tL}\|\leq Ce^{\omega t} $
 for all
$ t\in\mathbb R $
 and
$ \omega>0 $
 and, where further additional analytic properties of
$ L $
 and
$ Q $
 are assumed.
Keywords: Vlasov-Dirac-Benney equation, Vlasov equation, optical lattice, analytic norms.
Mathematics Subject Classification: Primary: 35F25, 35F20, 35Q20; Secondary: 35Q83.
Citation: Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic & Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007
References:
[1]

N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), P61.  doi: 10.1063/1.3037370.  Google Scholar

[2]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr and C. Lisdat, Noise and instability of an optical lattice clock, Phys. Rev. A, 92 (2015), 063814, 7 pages. doi: 10.1103/PhysRevA.92.063814.  Google Scholar

[3]

N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333.  doi: 10.1063/1.531567.  Google Scholar

[4]

E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30.  doi: 10.1038/nphys138.  Google Scholar

[5]

M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. Google Scholar

[6]

M. Braukhoff, Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness, Kinet. Relat. Models, 12 (2019), 445–482, arXiv 1711.06015 [math.AP]. doi: 10.3934/krm.2019019.  Google Scholar

[7]

M. Braukhoff and A. Jüngel, Energy-transport systems for optical lattices: Derivation, analysis, simulation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 579-614.  doi: 10.1142/S021820251850015X.  Google Scholar

[8]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits,, Kinet. Relat. Models, 6 (2013), 893-917.  doi: 10.3934/krm.2013.6.893.  Google Scholar

[9]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications, 1–30, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015. doi: 10.1007/978-1-4939-2950-4_1.  Google Scholar

[10]

C. Bardos and N. Besse, Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations,, Bull. Inst. Math., Acad. Sin. (N.S.), 11 (2016), 43-61.   Google Scholar

[11]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621, 16pp. doi: 10.1063/1.4765338.  Google Scholar

[12]

O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. Malomed, T. Sowinski and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys., 78 (2015), 066001, 47 pages. doi: 10.1088/0034-4885/78/6/066001.  Google Scholar

[13]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000.  Google Scholar

[14]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Rational Mech. Anal., 221 (2016), 1317-1344.  doi: 10.1007/s00205-016-0985-z.  Google Scholar

[15]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. cole Norm. Sup., 49 (2016), 1445-1495.  doi: 10.24033/asens.2313.  Google Scholar

[16]

P.-E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov equation,, C. R., Math., Acad. Sci. Paris, 349 (2011), 541-546.  doi: 10.1016/j.crma.2011.03.024.  Google Scholar

[17]

A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381.  doi: 10.1080/00107510410001705486.  Google Scholar

[18]

A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, 773. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[20]

S. Mandt, Transport and Non-Equilibrium Dynamics in Optical Lattices. From Expanding Atomic Clouds to Negative Absolute Temperatures, PhD thesis, University of Cologne, 2012. Google Scholar

[21]

G. Metivier, Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric Analysis of PDE and Several Complex Variables, 337–356, Contemp. Math., 368, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/368/06790.  Google Scholar

[22]

C. Mouhot and C. Villani, On Landau damping,, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[23]

N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28.  doi: 10.1103/PhysRev.103.20.  Google Scholar

[24]

A. Rapp, S. Mandt and A. Rosch, Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices, Phys. Rev. Lett., 105 (2010), 220405, 4 pages. doi: 10.1103/PhysRevLett.105.220405.  Google Scholar

[25]

U. Schneider, L. Hackermüller, J. Ph. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nature Physics, 8 (2012), 213-218.  doi: 10.1038/nphys2205.  Google Scholar

show all references

References:
[1]

N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), P61.  doi: 10.1063/1.3037370.  Google Scholar

[2]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr and C. Lisdat, Noise and instability of an optical lattice clock, Phys. Rev. A, 92 (2015), 063814, 7 pages. doi: 10.1103/PhysRevA.92.063814.  Google Scholar

[3]

N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333.  doi: 10.1063/1.531567.  Google Scholar

[4]

E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30.  doi: 10.1038/nphys138.  Google Scholar

[5]

M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. Google Scholar

[6]

M. Braukhoff, Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness, Kinet. Relat. Models, 12 (2019), 445–482, arXiv 1711.06015 [math.AP]. doi: 10.3934/krm.2019019.  Google Scholar

[7]

M. Braukhoff and A. Jüngel, Energy-transport systems for optical lattices: Derivation, analysis, simulation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 579-614.  doi: 10.1142/S021820251850015X.  Google Scholar

[8]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits,, Kinet. Relat. Models, 6 (2013), 893-917.  doi: 10.3934/krm.2013.6.893.  Google Scholar

[9]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications, 1–30, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015. doi: 10.1007/978-1-4939-2950-4_1.  Google Scholar

[10]

C. Bardos and N. Besse, Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations,, Bull. Inst. Math., Acad. Sin. (N.S.), 11 (2016), 43-61.   Google Scholar

[11]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621, 16pp. doi: 10.1063/1.4765338.  Google Scholar

[12]

O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. Malomed, T. Sowinski and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys., 78 (2015), 066001, 47 pages. doi: 10.1088/0034-4885/78/6/066001.  Google Scholar

[13]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000.  Google Scholar

[14]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Rational Mech. Anal., 221 (2016), 1317-1344.  doi: 10.1007/s00205-016-0985-z.  Google Scholar

[15]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. cole Norm. Sup., 49 (2016), 1445-1495.  doi: 10.24033/asens.2313.  Google Scholar

[16]

P.-E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov equation,, C. R., Math., Acad. Sci. Paris, 349 (2011), 541-546.  doi: 10.1016/j.crma.2011.03.024.  Google Scholar

[17]

A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381.  doi: 10.1080/00107510410001705486.  Google Scholar

[18]

A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, 773. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[20]

S. Mandt, Transport and Non-Equilibrium Dynamics in Optical Lattices. From Expanding Atomic Clouds to Negative Absolute Temperatures, PhD thesis, University of Cologne, 2012. Google Scholar

[21]

G. Metivier, Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric Analysis of PDE and Several Complex Variables, 337–356, Contemp. Math., 368, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/368/06790.  Google Scholar

[22]

C. Mouhot and C. Villani, On Landau damping,, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[23]

N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28.  doi: 10.1103/PhysRev.103.20.  Google Scholar

[24]

A. Rapp, S. Mandt and A. Rosch, Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices, Phys. Rev. Lett., 105 (2010), 220405, 4 pages. doi: 10.1103/PhysRevLett.105.220405.  Google Scholar

[25]

U. Schneider, L. Hackermüller, J. Ph. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nature Physics, 8 (2012), 213-218.  doi: 10.1038/nphys2205.  Google Scholar

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