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Extended Divergence-Measure Fields, the Gauss-Green Formula and Cauchy Fluxes.

Gui-Qiang G Chen1, Christopher Irving2, Monica Torres3

  • 1Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.

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Summary

We developed the Gauss-Green formula for extended divergence-measure fields, proving the normal trace is supported on boundaries. This extends the balance law in continuum physics, generalizing Cauchy's Theorem for broader applications.

Keywords:
26B0526B1226B2026B3026B4028A0528A2535D3035L6535L67Primary: 28C05Secondary: 28A75

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Area of Science:

  • Mathematical Analysis
  • Continuum Physics
  • Vector Calculus

Background:

  • Classical Gauss-Green formula applies to smooth vector fields.
  • Extended divergence-measure fields require generalized formulations.
  • Continuum physics balance laws traditionally assume continuous fields.

Purpose of the Study:

  • Establish the Gauss-Green formula for extended divergence-measure fields.
  • Generalize the balance law in continuum physics.
  • Provide a rigorous mathematical foundation for physical balance laws.

Main Methods:

  • Utilizing the theory of vector-valued measures and distributional divergence.
  • Proving properties of normal traces for open sets and their boundaries.
  • Developing a framework for measure-valued normal traces and their limits.

Main Results:

  • The normal trace of extended divergence-measure fields is a measure supported on the boundary.
  • A representation for the normal trace is derived via limits of approximating sets.
  • The balance law is extended to a general framework using Radon measures for production and Cauchy flux.

Conclusions:

  • The study generalizes Cauchy's Theorem and extends previous Cauchy flux formulations.
  • Equivalence is established between entropy solutions of divergence form PDEs and physical balance laws.
  • This work provides a robust axiomatic foundation for Continuum Physics.