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Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions

D. Amadori, F. A. Chiarello, C. Christoforou

Abstract

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in a periodic domain in one-space dimension with linear pressure term. The main result is the global existence of periodic entropy weak solutions, for periodic initial data having finite total variation and initial density bounded away from zero.

Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions

Abstract

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in a periodic domain in one-space dimension with linear pressure term. The main result is the global existence of periodic entropy weak solutions, for periodic initial data having finite total variation and initial density bounded away from zero.

Paper Structure

This paper contains 14 sections, 12 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set up in Lagrangian coordinates
  3. Characteristic curves
  4. Global existence of weak solutions
  5. Front-tracking approximate solutions
  6. Step 1.
  7. Step 2.
  8. Step 3.
  9. The linear functional
  10. Uniform bounds and global existence
  11. A weighted functional
  12. Estimates on vertical traces
  13. Proof of Theorem 2
  14. Proof of Theorem \ref{['Th-1']}

Key Result

Theorem 1

Assume (P), $\bf{(\Psi)}$ and that the initial data satisfy hyp-init_data. Then the Cauchy problem eq:system-eq:initial_datum admits an entropy weak solution $(\rho,\mathtt v)$ in the sense of Definition entropy-sol, with $(\rho,\mathtt v)(\cdot, t)\in \hbox{BV}\,(\mathbb{T}_\ell)$$\forall\, t,$ an where $C=C(q)$ and $q\doteq\frac{1}{2}\mathop{\mathrm{TV}}\nolimits{\{\ln(\rho_0)\}}+\frac{1}{2\alp

Figures (2)

  • Figure 1: The interaction of waves in case c within the triangle $D$ defined in \ref{['triangleD']}. The difference in colours indicates approaching (blue) and not approaching (red) wave fronts.
  • Figure 2: The interaction of waves in case d within the triangle $D$ defined in \ref{['triangleD']}. The difference in colours indicates approaching (blue) and not approaching (red) wave fronts.

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 11 more