doi: 10.3934/krm.2021016

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

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

Received  October 2020 Revised  March 2021 Published  May 2021

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.

Citation: Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic & Related Models, doi: 10.3934/krm.2021016
References:
[1]

C. BardosF. GolseT. 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.  Google Scholar

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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.   Google Scholar

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G. BonhommeT. PierreG. 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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[6]

M. Griffin-Pickering and M. Iacobelli, Recent developments on the well-posedness theory for Vlasov-type equations, preprint, arXiv: 2004.01094. Google Scholar

[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.  Google Scholar

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L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, second edition, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

show all references

References:
[1]

C. BardosF. GolseT. 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.  Google Scholar

[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.   Google Scholar

[3]

G. BonhommeT. PierreG. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

M. Griffin-Pickering and M. Iacobelli, Recent developments on the well-posedness theory for Vlasov-type equations, preprint, arXiv: 2004.01094. Google Scholar

[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.  Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, second edition, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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