Global strong solutions in $  {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems

Megan Griffin-Pickering; Mikaela Iacobelli

doi:10.3934/krm.2021016

Article Contents

2021, Volume 14, Issue 4: 571-597. Doi: 10.3934/krm.2021016

This issue Previous Article Next Article Incompressible Navier-Stokes-Fourier limit from the Landau equation

Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems

Megan Griffin-Pickering^1, and
Mikaela Iacobelli^2, ,

1.
Durham University, Department of Mathematical Sciences, Lower Mountjoy, Stockton Road, Durham DH1 3LE, UK
2.
ETH Zurich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

^* Corresponding author: Mikaela Iacobelli
^* Corresponding author: Mikaela Iacobelli

Received: September 30, 2020

Revised: February 28, 2021

Early access: May 2021

Published: August 2021

Abstract Full Text(HTML) Related Papers Cited by

Abstract

Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space $ \mathbb{R}^3 $, under minimal assumptions on the initial data and the confining potential.

Keywords:

Vlasov-Poisson systems,
well-posedness theory.

Mathematics Subject Classification: Primary: 35Q83, 35A01, 35Q82; Secondary: 82C70, 82C40, 82D10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. Bardos, F. Golse, T. T. Nguyen and R. Sentis, The Maxwell-Boltzmann approximation for ion kinetic modeling, Phys. D, 376/377 (2018), 94-107. doi: 10.1016/j.physd.2017.10.014.
[2]	J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411-416.
[3]	G. Bonhomme, T. Pierre, G. Leclert and J. Trulsen, Ion phase space vortices in ion beam-plasma systems and their relation with the ion acoustic instability: Numerical and experimental results, Plasma Physics and Controlled Fusion, 33 (1991), 507-520. doi: 10.1088/0741-3335/33/5/009.
[4]	F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm. Partial Differential Equations, 16 (1991), 1337-1365. doi: 10.1080/03605309108820802.
[5]	M. Griffin-Pickering and M. Iacobelli, Global well-posedness for the Vlasov-Poisson system with massless electrons in the 3-dimensional torus, preprint, to appear on Comm. Partial Differential Equations, arXiv: 1810.06928.
[6]	M. Griffin-Pickering and M. Iacobelli, Recent developments on the well-posedness theory for Vlasov-type equations, preprint, arXiv: 2004.01094.
[7]	D. Han-Kwan and M. Iacobelli, The quasineutral limit of the Vlasov-Poisson equation in Wasserstein metric, Commun. Math. Sci., 15 (2017), 481-509. doi: 10.4310/CMS.2017.v15.n2.a8.
[8]	L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, second edition, 1990. doi: 10.1007/978-3-642-61497-2.
[9]	E. H. Lieb and M. Loss, Analysis: Second Edition, volume 14 of Graduate Studies in Mathematics, American Mathematical Society, 2001. doi: 10.1090/gsm/014.
[10]	P. L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.
[11]	G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9), 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.
[12]	R. J. Mason, Computer simulation of ion-acoustic shocks. The diaphragm problem, The Physics of Fluids, 14 (1971), 1943-1958. doi: 10.1063/1.1693704.
[13]	Y. V. Medvedev, Ion front in an expanding collisionless plasma, Plasma Physics and Controlled Fusion, 53 (2011), 125007. doi: 10.1088/0741-3335/53/12/125007.
[14]	R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.
[15]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.
[16]	P. H. Sakanaka, C. K. Chu and T. C. Marshall, Formation of ion-acoustic collisionless shocks, The Physics of Fluids, 14 (1971), 611. doi: 10.1063/1.1693480.
[17]	L. Tartar., An Introduction to Sobolev Spaces and Interpolation Spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana., Springer-Verlag Berlin Heidelberg, 2007.
[18]	S. Ukai and T. Okabe., On classical solutions in the large in time of two-dimensional Vlasov's equation., Osaka Math. J., 15 (1978), 245-261.