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The global Cauchy problem for the Euler-Riesz equations

Young-Pil Choi, Jinwook Jung, Yoonjung Lee

Abstract

We completely resolve the global Cauchy problem for the multi-dimensional Euler-Riesz equations, where the interaction forcing is given by $\nabla (-Δ)^{-σ/2}ρ$ for some $σ\in (0,2)$. We construct the global-in-time unique solution to the Euler-Riesz system in a $H^s$ Sobolev space under a smallness assumption on the initial density and a dispersive spectral condition on the initial velocity. Moreover, we investigate the algebraic time decay of convergences for the constructed solutions. Our results cover the both attractive and repulsive cases as well as the whole regime $σ\in (0,2)$.

The global Cauchy problem for the Euler-Riesz equations

Abstract

We completely resolve the global Cauchy problem for the multi-dimensional Euler-Riesz equations, where the interaction forcing is given by for some . We construct the global-in-time unique solution to the Euler-Riesz system in a Sobolev space under a smallness assumption on the initial density and a dispersive spectral condition on the initial velocity. Moreover, we investigate the algebraic time decay of convergences for the constructed solutions. Our results cover the both attractive and repulsive cases as well as the whole regime .
Paper Structure (19 sections, 22 theorems, 258 equations)

This paper contains 19 sections, 22 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Pressureless case
  4. The case with pressure
  5. Organization of the paper
  6. Preliminaries
  7. Pressureless Euler--Riesz system
  8. A priori estimates
  9. $H^s \times H^{s+\frac{\sigma}{2}}$ bound estimates
  10. Temporal decay estimates
  11. Proof of Theorem \ref{['main_thm']}
  12. Euler--Riesz system
  13. Repulsive interaction case
  14. Attractive interaction case
  15. Euler system with sub-Manev interactions: $1\leq\sigma<2$ case
  16. ...and 4 more sections

Key Result

Proposition 1.1

Suppose that $v_0 \in E^s$ satisfies where $\emph{Sp} A$ denotes the spectrum of a matrix $A$. Then the following holds:

Theorems & Definitions (40)

  • Proposition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3: repulsive case
  • Theorem 1.4: attractive case
  • Remark 1.3
  • Lemma 2.1
  • proof
  • ...and 30 more