Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

Necessary conditions for blow-up solutions to the restricted Euler–Poisson equations

Pages: 327 – 357

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n2.a2

Authors

Hailiang Liu (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Jaemin Shin (Department of Mathematical Sciences & Institute for Applied Mathematics and Optics, Hanbat National University, Daejeon, South Korea)

Abstract

In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler–Poisson equations which are the localized version of the full Euler–Poisson system. We provide necessary conditions for the existence of finite-time blow-up solutions in terms of the initial data, and describe the asymptotic behavior of the solutions near blow-up times. We also identify a rich set of the initial data which yields global bounded solutions.

Keywords

restricted Euler–Poisson dynamics, blow-up solutions, asymptotic behaviors

2010 Mathematics Subject Classification

34C11, 35Q35

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Liu was partially supported by the National Science Foundation under Grant DMS1812666.

Shin was supported by newly appointed professor research fund of Hanbat National University and the National Research Foundation under Grant NRF-2017R1E1A1A03070498.

Received 29 October 2019

Accepted 30 June 2021

Published 28 January 2022