Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regular solutions to the dissipative Aw-Rascle system

Nilasis Chaudhuri, Tomasz Piasecki, Ewelina Zatorska

Abstract

In this paper we prove the local-in-time existence of regular solutions to dissipative Aw-Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier-Stokes equations to show the well-posedness of the system in the $L_2-L_2$ setting. We also discuss relevant existence results for offset involving singular or nonlocal functions of density.

Regular solutions to the dissipative Aw-Rascle system

Abstract

In this paper we prove the local-in-time existence of regular solutions to dissipative Aw-Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier-Stokes equations to show the well-posedness of the system in the setting. We also discuss relevant existence results for offset involving singular or nonlocal functions of density.

Paper Structure

This paper contains 11 sections, 12 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Linear theory
  4. Linear transport equation
  5. Linear continuity equation with dissipation
  6. Convergence of the iterative scheme
  7. Boundedness of the sequence of approximations
  8. Contraction argument. Proof of Theorem \ref{['thm:main']}.
  9. Existence theory for general velocity offsets
  10. Construction for the system with singular velocity offset
  11. Construction for the system with nonlocal velocity offset

Key Result

Theorem 1.1

Assume the initial data satisfies init1 or init2. Then there exists $T>0$ such that system sys3 admits a unique solution $(\varrho,{\bf w}) \in {\mathcal{V}}_3(T)\times {\mathcal{Y}}_3(T)$ with the estimate in case of init1 or in case of init2.

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • ...and 2 more