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The relaxation limit of bipolar fluid models
CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia |
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.
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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