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Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory

Kohei Soga

Abstract

Co-moving volumes and Reynolds transport theorem along a fluid flow are fundamental tools to derive balance laws in fluid mechanics, where the classical theory on flow maps of ODEs associated to smooth vector fields plays a central role. Related to weak solutions of Navier-Stokes equations in Sobolev classes, DiPerna-Lions (Invent. Math. 1989) generalized the classical notion of ODEs and flow maps in the case of vector fields belonging to Sobolev classes. DiPerna-Lions theory also clarifies evolution of measure of the inverse image of each Borel measurable set under generalized flow maps in terms of the divergence of vector fields. On the other hand, the image of each measurable set under generalized flow maps, which corresponds to co-moving volumes in the classical theory, is not necessarily measurable. Hence, formulation of Reynolds transport theorem would not make sense. In this paper, we show that the image of each Borel measurable set trimmed with a suitable null set is measurable possessing measure consistent with the classical case without trimming. Then, defining co-moving volumes with such trimming, we prove Reynolds transport theorem for generalized flow maps. We also formulate Reynolds transport theorem in terms of the inverse image.

Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory

Abstract

Co-moving volumes and Reynolds transport theorem along a fluid flow are fundamental tools to derive balance laws in fluid mechanics, where the classical theory on flow maps of ODEs associated to smooth vector fields plays a central role. Related to weak solutions of Navier-Stokes equations in Sobolev classes, DiPerna-Lions (Invent. Math. 1989) generalized the classical notion of ODEs and flow maps in the case of vector fields belonging to Sobolev classes. DiPerna-Lions theory also clarifies evolution of measure of the inverse image of each Borel measurable set under generalized flow maps in terms of the divergence of vector fields. On the other hand, the image of each measurable set under generalized flow maps, which corresponds to co-moving volumes in the classical theory, is not necessarily measurable. Hence, formulation of Reynolds transport theorem would not make sense. In this paper, we show that the image of each Borel measurable set trimmed with a suitable null set is measurable possessing measure consistent with the classical case without trimming. Then, defining co-moving volumes with such trimming, we prove Reynolds transport theorem for generalized flow maps. We also formulate Reynolds transport theorem in terms of the inverse image.

Paper Structure

This paper contains 5 sections, 17 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Classical theory on ODEs and linear transport equation
  3. DiPerna-Lions theory and Reynolds transport theorem
  4. Weak solution of linear transport equation
  5. Construction of flow map and its properties

Key Result

Theorem 1.1

Suppose v. Then, there exists a flow map $X:{\mathbb R}\times{\mathbb R}\times\Omega\to\overline{\Omega}$ of the ODE $\gamma'(s)=v(s,\gamma(s))$ in the following sense:

Theorems & Definitions (34)

  • Theorem 1.1: DiPerna-Lions theorem on a bounded domain
  • Theorem 1.2: Reynolds transport theorem in DiPerna-Lions theory
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3: Classical Reynolds transport theorem
  • proof
  • Proposition 3.1
  • proof
  • ...and 24 more