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One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space
On a regularized system of self-gravitating particles
1. | Department of Technomathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany |
References:
[1] |
N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift diffusion model, Z. angew. Math. Phys., 49 (1998), 251-275.
doi: 10.1007/s000330050218. |
[2] |
C. Bennett and R. C. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, Boston, 1988. |
[3] |
P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, Colloq. Math., 66 (1994), 319-334. |
[4] |
E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525.
doi: 10.1007/BF02099262. |
[5] |
E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, II, Comm. Math. Phys., 174 (1995), 229-260.
doi: 10.1007/BF02099602. |
[6] |
D. Cassani, B. Ruf and C. Tarsi, Best constants in a borderline case of second-order Moser type inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 73-93.
doi: 10.1016/j.anihpc.2009.07.006. |
[7] |
A. Dall'Aglio, D. Giachetti and J. -P. Puel, Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., 181 (2002), 407-426.
doi: 10.1007/s102310100046. |
[8] |
V. Ferone, M. R. Posteraro and J. M. Rakotoson, $L^\infty$-estimates for nonlinear elliptic problems with $p$-growth in the gradient, J. Inequal. Appl., 3 (1999), 109-125.
doi: 10.1155/S1025583499000077. |
[9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[10] |
A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM J. Math. Anal, 32 (2000), 760-777.
doi: 10.1137/S0036141099360269. |
[11] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, Vol. 88, Academic Press, New York, 1980. |
[12] |
J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, (French) [Some results of Višik on nonlinear elliptic problems by the methods of Minty-Browder], Bull. Soc. Math. France, 93 (1965), 97-107. |
[13] |
M. Montenegro and M. Montenegro, Existence and nonexistence of solutions for quasilinear elliptic equations, J. Math. Anal. Appl., 245 (2000), 303-316.
doi: 10.1006/jmaa.1999.6697. |
[14] |
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana. Univ. Math. J., 20 (1971), 1077-1092. |
[15] |
R. Pinnau and A. Unterreiter, The stationary current-voltage characteristics of the quantum drift-diffusion model, SIAM J. Numer. Anal., 37 (1999), 211-245.
doi: 10.1137/S0036142998341039. |
[16] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, Vol. 62, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[17] |
O. Tse, On the effects of the Bohm potential on a macroscopic system of self-interacting particles, J. Math. Anal. Appl., 418 (2014), 796-811.
doi: 10.1016/j.jmaa.2014.04.021. |
[18] |
J. Winter, Wigner transformation in curved space-time and the curvature correction of the Vlasov equation for semiclassical gravitating systems, Phys. Rev. D, 32 (1985), 1871-1888.
doi: 10.1103/PhysRevD.32.1871. |
show all references
References:
[1] |
N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift diffusion model, Z. angew. Math. Phys., 49 (1998), 251-275.
doi: 10.1007/s000330050218. |
[2] |
C. Bennett and R. C. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, Boston, 1988. |
[3] |
P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, Colloq. Math., 66 (1994), 319-334. |
[4] |
E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525.
doi: 10.1007/BF02099262. |
[5] |
E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, II, Comm. Math. Phys., 174 (1995), 229-260.
doi: 10.1007/BF02099602. |
[6] |
D. Cassani, B. Ruf and C. Tarsi, Best constants in a borderline case of second-order Moser type inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 73-93.
doi: 10.1016/j.anihpc.2009.07.006. |
[7] |
A. Dall'Aglio, D. Giachetti and J. -P. Puel, Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., 181 (2002), 407-426.
doi: 10.1007/s102310100046. |
[8] |
V. Ferone, M. R. Posteraro and J. M. Rakotoson, $L^\infty$-estimates for nonlinear elliptic problems with $p$-growth in the gradient, J. Inequal. Appl., 3 (1999), 109-125.
doi: 10.1155/S1025583499000077. |
[9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[10] |
A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM J. Math. Anal, 32 (2000), 760-777.
doi: 10.1137/S0036141099360269. |
[11] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, Vol. 88, Academic Press, New York, 1980. |
[12] |
J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, (French) [Some results of Višik on nonlinear elliptic problems by the methods of Minty-Browder], Bull. Soc. Math. France, 93 (1965), 97-107. |
[13] |
M. Montenegro and M. Montenegro, Existence and nonexistence of solutions for quasilinear elliptic equations, J. Math. Anal. Appl., 245 (2000), 303-316.
doi: 10.1006/jmaa.1999.6697. |
[14] |
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana. Univ. Math. J., 20 (1971), 1077-1092. |
[15] |
R. Pinnau and A. Unterreiter, The stationary current-voltage characteristics of the quantum drift-diffusion model, SIAM J. Numer. Anal., 37 (1999), 211-245.
doi: 10.1137/S0036142998341039. |
[16] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, Vol. 62, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[17] |
O. Tse, On the effects of the Bohm potential on a macroscopic system of self-interacting particles, J. Math. Anal. Appl., 418 (2014), 796-811.
doi: 10.1016/j.jmaa.2014.04.021. |
[18] |
J. Winter, Wigner transformation in curved space-time and the curvature correction of the Vlasov equation for semiclassical gravitating systems, Phys. Rev. D, 32 (1985), 1871-1888.
doi: 10.1103/PhysRevD.32.1871. |
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