# American Institute of Mathematical Sciences

April 2019, 12(2): 445-482. doi: 10.3934/krm.2019019

## Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness

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

Received  June 2018 Published  November 2018

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

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice:
 $\partial _t f + \nabla _pε(p)·\nabla _x f - \nabla _x n_f·\nabla _p f = n_f(1- n_f)(\mathcal{F}_f-f),\;\;\;\; x∈\mathbb{R}^d, p∈\mathbb{T}^d, t>0.$
This system contains an interaction potential
 $n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$
being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore,
 $ε(p) = -\sum_{i = 1}^d$
 $\cos(2π p_i)$
is the dispersion relation and
 $\mathcal{F}_f$
denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on
 $f$
in this context.
In a dilute plasma—without collisions (r.h.s
 $. = 0$
)—this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.
Citation: Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
##### References:
 [1] N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3308-3333. doi: 10.1063/1.531567. [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. [3] N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), 61. doi: 10.1063/1.3037370. [4] 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. [5] C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications. Selected Papers Based on the Presentations at the Conference on Hamiltonian PDEs: Analysis, Computations and applications, Toronto, Canada, January 10–12, 2014, pages 1– 30. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer, 2015. doi: 10.1007/978-1-4939-2950-4. [6] 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. [7] 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. [8] E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30. [9] M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. [10] 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. [11] 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. [12] A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511575150. [13] 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. [14] 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. [15] 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. [16] A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381. [17] A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys., 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. [18] C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. [19] N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28. [20] 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. [21] 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.

show all references

##### References:
 [1] N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3308-3333. doi: 10.1063/1.531567. [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. [3] N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), 61. doi: 10.1063/1.3037370. [4] 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. [5] C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications. Selected Papers Based on the Presentations at the Conference on Hamiltonian PDEs: Analysis, Computations and applications, Toronto, Canada, January 10–12, 2014, pages 1– 30. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer, 2015. doi: 10.1007/978-1-4939-2950-4. [6] 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. [7] 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. [8] E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30. [9] M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. [10] 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. [11] 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. [12] A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511575150. [13] 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. [14] 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. [15] 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. [16] A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381. [17] A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys., 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. [18] C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. [19] N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28. [20] 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. [21] 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.
 [1] Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893 [2] Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002 [3] Adán J. Corcho. Ill-Posedness for the Benney system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965 [4] Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253 [5] Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565 [6] In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012 [7] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [8] Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727 [9] Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395 [10] Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009 [11] François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221 [12] Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic & Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309 [13] Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787 [14] Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13 [15] Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 [16] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [17] Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441 [18] Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863 [19] Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253 [20] Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341

2017 Impact Factor: 1.219