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Critical thresholds in pressureless Euler--Poisson equations with background states

Young-Pil Choi, Dong-ha Kim, Dowan Koo, Eitan Tadmor

Abstract

We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces $H^s({\mathbb R})$, $s \geq 2$. Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.

Critical thresholds in pressureless Euler--Poisson equations with background states

Abstract

We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces , . Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.
Paper Structure (19 sections, 15 theorems, 155 equations, 4 figures)

This paper contains 19 sections, 15 theorems, 155 equations, 4 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Systems and notions
  3. History and review
  4. Main results
  5. Local well-posedness and neutrality condition
  6. The case of attractive forcing
  7. The case of repulsive forcing
  8. Application to the EP system for cold plasma ion dynamics
  9. Outline of the paper
  10. Local well-posedness and the neutrality condition
  11. Local well-posedness and Non-existence
  12. Neutrality condition and Anomalous solution
  13. Attractive forcing and the restricted borderline threshold
  14. Constant background case
  15. Variable background case
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose that the initial data $(\rho_0, u_0)$ satisfy Then there exists a positive constant $T>0$ such that the system main_sys--ini admits a unique solution $(\rho, u)$ satisfying Moreover, if the solutions $\rho$ and $\partial_x u$ are bounded over the time interval $[0,T]$, i.e. then there is propagation of higher order regularity of $(\rho, u)$ expressed in terms of $|\|\{\rho(t,\cdot),u(t,

Figures (4)

  • Figure 1: Ilustration of the super-critical regions
  • Figure 2: The evolution of $s$ (left) and the phase plane plot (right) of the solution trajectory of \ref{['test_sys']}.
  • Figure 3: Illustration of the sub/super-critical regions for the repulsive EP system
  • Figure 4: Illustration of the sub-critical$\{\mathcal{L}_{s}^-(w,s) \geqslant 0 \}$ and super-critical$\{\mathcal{L}_{s}^+(w,s) < 0 \}$ criteria for the attractive case

Theorems & Definitions (35)

  • Theorem 1.1: Local well-posedness
  • Theorem 1.2: Non-existence
  • Theorem 1.3: Well-posedness with neutrality
  • Theorem 1.4: Non-propagation of neutrality: Anomalous solution
  • Theorem 1.5: Reduction to the borderline
  • Remark 1.6
  • Theorem 1.7: Finite-time breakdown
  • Remark 1.8
  • Theorem 1.9: Global-in-time regularity
  • Remark 1.10
  • ...and 25 more