On the Well-posedness of the Magnetic, Semi-relativistic Schrödinger-Poisson System
DOI:
https://doi.org/10.5890/DNC.2018.09.002Abstract
We prove global existence and uniqueness of strong solutions for the Schrödinger-Poisson system in the repulsive Coulomb case with relativistic, magnetic kinetic energy.References
[1] Barbaroux, J.M. and Vougalter, V. (2016), Existence and nonlinear stability of stationary states for the magnetic Schrödinger-Poisson system, J.Math. Sci. (N.Y.), 219(6), Problems in mathematical analysis, 87, (Russian), 874-898.
[2] Anapolitanos, I. (2011), Rate of convergence towards the Hartree-von Neumann limit in the mean-field regime, Lett. Math. Phys., 98(1), 1-31.
[3] Anapolitanos, I. and Sigal, I.M., The Hartree-von Neumann limit of many body dynamics, Preprint[http://arxiv.org/abs/0904.4514].
[4] Brezzi, F. and Markowich, P.A. (1991), The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation,Math. Methods Appl. Sci., 14(1), 35-61.
[5] Markowich, P.A., Rein, G., and Wolansky, G. (2002), Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system, J. Statist. Phys., 106(5-6), 1221-1239.
[6] Abou Salem, W., Chen, T., and Vougalter, V. (2012), On the well-posedness of the semi-relativistic Schrödinger- Poisson system, Dyn. Partial Differ. Equ., 9(2), 121-132.
[7] Abou Salem, W., Chen, T., and Vougalter, V. (2014), Existence and nonlinear stability of stationary states for the semi-relativistic Schrödinger-Poisson system, Ann. Henri Poincare, 15(6), 1171-1196.
[8] Barbaroux, J.M. and Vougalter, V. (2017), On the well-posedness of the magnetic Schrödinger-Poisson system in R3, Math. Model. Nat. Phenom., 12(1), 15-22.
[9] Steinrück, H. (1991), The one-dimensionalWigner-Poisson problem and its relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal., 22(4), 957C972.
[10] Aki, G.L.,Markowich, P.A., and Sparber, C. (2008), Classical limit for semirelativistic Hartree systems, J. Math. Phys., 49(10), 102110, 10pp.
[11] Lenzmann, E. (2007),Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10(1), 43-64.
[12] Lieb, E.H., Loss, M., and Siedentop, H. (1996), Stability of relativistic matter via Thomas-Fermi theory, Helv. Phys. Acta, 69(5-6), 974-984.
[13] Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, Springer: Berlin.
[14] Reed, M. and Simon, B. (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 361pp.
[15] Lieb, E.H. and Loss, M. (1997), Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 278pp.
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