American Institute of Mathematical Sciences

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September  2014, 7(3): 591-604. doi: 10.3934/krm.2014.7.591

On a regularized system of self-gravitating particles

René Pinnau 1, and  Oliver Tse 1,

1. 

Department of Technomathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany

Received  April 2014 Revised  May 2014 Published  July 2014

We consider a regularized macroscopic model describing a system of self-gravitating particles. We study the existence and uniqueness of nonnegative stationary solutions and allude the differences to results obtained from classical gravitational models. The system is analyzed on a convex, bounded domain up to three spatial dimensions, subject to Neumann boundary conditions for the particle density, and Dirichlet boundary condition for the self-interacting potential. Finally, we show numerical simulations underlining our analytical results.
Keywords: macroscopic quantum models., self-gravitating particles, Bohm potential, Second order elliptic systems.
Mathematics Subject Classification: Primary: 35D30, 35J57; Secondary: 76Y0.
Citation: René Pinnau, Oliver Tse. On a regularized system of self-gravitating particles. Kinetic and Related Models, 2014, 7 (3) : 591-604. doi: 10.3934/krm.2014.7.591
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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