### Abstract

We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit, ε → 0, with ε being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution μ of Vlasov-Poisson systems in arbitrarily high Sobolev norms, but become of order one away from μ in arbitrary negative Sobolev norms within time of order |log ε|. Second, we deduce the invalidity of the quasineutral limit in L^{2} in arbitrarily short time.

Original language | English (US) |
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Pages (from-to) | 3444-3466 |

Number of pages | 23 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 48 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

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*SIAM Journal on Mathematical Analysis*, vol. 48, no. 5, pp. 3444-3466. https://doi.org/10.1137/15M1028765

**Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits.** / Han-Kwan, Daniel; Nguyen, Toan T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

AU - Han-Kwan, Daniel

AU - Nguyen, Toan T.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit, ε → 0, with ε being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution μ of Vlasov-Poisson systems in arbitrarily high Sobolev norms, but become of order one away from μ in arbitrary negative Sobolev norms within time of order |log ε|. Second, we deduce the invalidity of the quasineutral limit in L2 in arbitrarily short time.

AB - We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit, ε → 0, with ε being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution μ of Vlasov-Poisson systems in arbitrarily high Sobolev norms, but become of order one away from μ in arbitrary negative Sobolev norms within time of order |log ε|. Second, we deduce the invalidity of the quasineutral limit in L2 in arbitrarily short time.

UR - http://www.scopus.com/inward/record.url?scp=84994140783&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994140783&partnerID=8YFLogxK

U2 - 10.1137/15M1028765

DO - 10.1137/15M1028765

M3 - Article

AN - SCOPUS:84994140783

VL - 48

SP - 3444

EP - 3466

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -