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January  2022, 42(1): 211-237. doi: 10.3934/dcds.2021113

The relaxation limit of bipolar fluid models

Nuno J. Alves , and  Athanasios E. Tzavaras

CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

* Corresponding author: Nuno J. Alves

Received  January 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid models, and it is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.

Keywords: Bipolar Euler-Poisson, bipolar drift-diffusion, relaxation limit, high-friction limit, relative energy.
Mathematics Subject Classification: Primary: 35Q35, 35Q81; Secondary: 35L65, 35K55, 76W05.
Citation: Nuno J. Alves, Athanasios E. Tzavaras. The relaxation limit of bipolar fluid models. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 211-237. doi: 10.3934/dcds.2021113
References:
[1]

M. Bessemoulin-Chatard, C. Chainais-Hillairet and A. Jüngel, Uniform $ L^{\infty} $ estimates for approximate solutions of the bipolar drift-diffusion system, in International Conference on Finite Volumes for Complex Applications, Springer Proc. Math. Stat., Springer, Cham, 199 (2017), 381–389.

[2]

R. Bianchini, Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method, J. Math. Pures Appl., 132 (2019), 280-307.  doi: 10.1016/j.matpur.2019.04.004.

[3]

J. A. Carrillo, A. Wróblewska-Kamińksa and Y. Peng, Relative entropy method for the relaxation limit of hydrodynamic models, Netw. Heterog. Media, 15 (2020), 369-387.  doi: 10.3934/nhm.2020023.

[4]

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, New York: Plenum press, 1 (1984).

[5]

C. M. Dafermos, Stability of motions of thermoelastic fluids, Journal of Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.

[6]

M. Di Francesco and M. Wunsch, Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models, Monatsh. Math., 154 (2008), 39-50.  doi: 10.1007/s00605-008-0532-6.

[7]

H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl., 113 (1986), 12-35.  doi: 10.1016/0022-247X(86)90330-6.

[8]

J. Giesselmann, C. Lattanzio and A. E. Tzavaras, Relative energy for the Korteweg-theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., 223 (2017), 1427-1484.  doi: 10.1007/s00205-016-1063-2.

[9]

X. Huo, A. Jüngel and A. E. Tzavaras, High-friction limits of Euler flows for multicomponent systems, Nonlinearity, 32 (2019), 2875-2913.  doi: 10.1088/1361-6544/ab12a6.

[10]

A. Jüngel, Transport Equations for Semiconductors, Vol. 773, Springer, 2009. doi: 10.1007/978-3-540-89526-8.

[11]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703.  doi: 10.1142/S0218202594000388.

[12]

C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Vlue Problems, Vol. 83, American Mathematical Society, 1994. doi: 10.1090/cbms/083.

[13]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM Control Optim. Calc. Var., 23 (2017), 137-164.  doi: 10.1051/cocv/2015043.

[14]

C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dynam. Systems, 5 (1999), 449-455.  doi: 10.3934/dcds.1999.5.449.

[15]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261-290.  doi: 10.1080/03605302.2016.1269808.

[16]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.

[17]

A. Lenard and I. B. Bernstein, Plasma oscillations with diffusion in velocity space, Phys. Rev., 112 (1958), 1456-1459.  doi: 10.1103/PhysRev.112.1456.

[18]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145.  doi: 10.1007/BF00379918.

[19]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[20]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281.  doi: 10.1006/jmaa.1996.0081.

[21]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[22] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Vol. 2, Princeton University Press, 1970. 

show all references

References:
[1]

M. Bessemoulin-Chatard, C. Chainais-Hillairet and A. Jüngel, Uniform $ L^{\infty} $ estimates for approximate solutions of the bipolar drift-diffusion system, in International Conference on Finite Volumes for Complex Applications, Springer Proc. Math. Stat., Springer, Cham, 199 (2017), 381–389.

[2]

R. Bianchini, Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method, J. Math. Pures Appl., 132 (2019), 280-307.  doi: 10.1016/j.matpur.2019.04.004.

[3]

J. A. Carrillo, A. Wróblewska-Kamińksa and Y. Peng, Relative entropy method for the relaxation limit of hydrodynamic models, Netw. Heterog. Media, 15 (2020), 369-387.  doi: 10.3934/nhm.2020023.

[4]

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, New York: Plenum press, 1 (1984).

[5]

C. M. Dafermos, Stability of motions of thermoelastic fluids, Journal of Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.

[6]

M. Di Francesco and M. Wunsch, Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models, Monatsh. Math., 154 (2008), 39-50.  doi: 10.1007/s00605-008-0532-6.

[7]

H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl., 113 (1986), 12-35.  doi: 10.1016/0022-247X(86)90330-6.

[8]

J. Giesselmann, C. Lattanzio and A. E. Tzavaras, Relative energy for the Korteweg-theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., 223 (2017), 1427-1484.  doi: 10.1007/s00205-016-1063-2.

[9]

X. Huo, A. Jüngel and A. E. Tzavaras, High-friction limits of Euler flows for multicomponent systems, Nonlinearity, 32 (2019), 2875-2913.  doi: 10.1088/1361-6544/ab12a6.

[10]

A. Jüngel, Transport Equations for Semiconductors, Vol. 773, Springer, 2009. doi: 10.1007/978-3-540-89526-8.

[11]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci., 4 (1994), 677-703.  doi: 10.1142/S0218202594000388.

[12]

C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Vlue Problems, Vol. 83, American Mathematical Society, 1994. doi: 10.1090/cbms/083.

[13]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM Control Optim. Calc. Var., 23 (2017), 137-164.  doi: 10.1051/cocv/2015043.

[14]

C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dynam. Systems, 5 (1999), 449-455.  doi: 10.3934/dcds.1999.5.449.

[15]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261-290.  doi: 10.1080/03605302.2016.1269808.

[16]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.

[17]

A. Lenard and I. B. Bernstein, Plasma oscillations with diffusion in velocity space, Phys. Rev., 112 (1958), 1456-1459.  doi: 10.1103/PhysRev.112.1456.

[18]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145.  doi: 10.1007/BF00379918.

[19]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[20]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281.  doi: 10.1006/jmaa.1996.0081.

[21]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[22] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Vol. 2, Princeton University Press, 1970. 
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