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Well-posedness of non-autonomous transport equation on metric graphs

Christian Budde, Marjeta Kramar Fijavž

Abstract

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

Well-posedness of non-autonomous transport equation on metric graphs

Abstract

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.
Paper Structure (10 sections, 12 theorems, 46 equations)

This paper contains 10 sections, 12 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric graphs
  4. Non-autonomous abstract Cauchy problems
  5. The non-autonomous network transport problem
  6. The setting
  7. Operators with constant domains
  8. Well-posedness of the general non-autonomous problem
  9. The associated multiplication semigroup.
  10. The associated evolution semigroup and its generator

Key Result

Proposition 2.3

N1997 The Cauchy problem eqn:nACP is well-posed (on $Y_t$) if and only if there is an evolution family solving eqn:nACP (on $Y_t$).

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 12 more