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

Arch. Rational Mech. Anal. (2025) 249:79 Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-025-02135-7 Extended Divergence-Measure Fields, the Gauss-Green Formula and Cauchy Fluxes Gui-Qiang G. Chen, Christopher Irving & Monica Torres Communicated by T.-P. Liu Abstract We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the boundary of the open set as the limit of measure-valued normal traces over the boundaries of approximating sets. Furthermore, using this theory, we extend the balance law from classical continuum physics to a general framework in which the production on any open set is measured with a Radon measure and the associated Cauchy flux is bounded by a Radon measure concentrated on the boundary of the set. We prove that there exists an extended divergence-measure field such that the Cauchy flux can be recovered through the field, locally on almost every open set and globally on every open set. Our results generalize the classical Cauchy’s Theorem (that is only valid for continuous vector fields) and extend the previous formulations of the Cauchy flux (that generate vector fields within L p ). Thereby, we establish the equivalence between entropy solutions of the multidimensional nonlinear partial differential equations of divergence form and of the mathematical formulation of physical balance laws via the Cauchy flux through the constitutive relations in the axiomatic foundation of Continuum Physics. Mathematics Subject Classification: Primary: 28C05, 26B20, 28A05, 26B12, 35L65, 35L67 Secondary: 28A75, 28A25, 26B05, 26B30, 26B40, 35D30 Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Extended Divergence-Measure Fields and Distributional Normal Traces . . . . 3. Representation and Limit Formula for the Normal Trace via Disintegration . . . 2 6 16 79 Page 2 of 82 Arch. Rational Mech. Anal. (2025) 249:79 4. 5. 6. 7. 8. Properties of the Disintegration . . . . . . . . . . . . . . . . . . . . . . . . . . Localization of the Normal Trace . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Flux I: Main Results and Connections . . . . . . . . . . . . . . . . . . Cauchy Flux II: Properties of the Cauchy Flux . . . . . . . . . . . . . . . . . . Cauchy Flux III: Construction and Uniqueness of the Representing DivergenceMeasure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Cauchy Flux IV: Local Recovery and Applications . . . . . . . . . . . . . . . 10. Extension of the Normal Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Remarks on the Existence of Divergence-Measure Fields . . . . . . . . . . . . 12. Equivalence between Entropy Solutions of Nonlinear PDEs of Divergence Form and the Mathematical Formulation of Physical Balance Laws . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 26 30 37 53 57 67 71 74 79 1. Introduction Divergence-measure fields are defined as vector-valued fields F = (F1 , F2 , · · · , Fn ) whose distributional divergences are represented by (signed) Radon measures. An underlying connection between divergence-measure fields and hyperbolic conservation laws was first observed in [10], and such vector fields over domains with Lipschitz boundary were analyzed in [10,11]. Since then, the analysis of divergencemeasure fields has depended essentially on the regularity of F. For example, the divergence-measure fields were extensively analyzed first in L ∞ in [19] and then in L p in [9]. See also [1,4,14,16,20–25,58,59,62–64] and the references therein for further developments for the theory of divergence-measure fields. In this paper, we focus on the case when F is only a vector-valued Radon measure. More precisely, we analyze extended divergence-measure fields which are defined as vector-valued Radon measures whose distributional divergences are Radon measures. Our approach in this paper is motivated by the previous results in the L p setting in [9,19]. However, the case of extended divergence-measure fields is more delicate, since F may concentrate on lower dimensional sets (for instance, rectifiable curves). We prove that, for almost every open set, the normal trace of an extended divergencemeasure field is a Radon measure supported on the boundary of the set. Moreover, for every open set, the normal trace distribution can be computed as the limit of measure-valued normal traces over the boundaries of approximating sets. Equipped with these results, we further develop a theory of Cauchy fluxes, starting from the balance law and establishing a one-to-one correspondence {Cauchy fluxes F in } ←→ {Extended divergence-measure fields F in } via the normal trace. The precise statement is given in Sect. 6.2. In the development of a theory of divergence-measure fields, one of the fundamental issues is whether a Gauss-Green formula involving these weakly differentiable vector fields can still be provided. We refer the reader to [18] for a detailed exposition on the development of this fundamental formula, starting from Lagrange (1762) and culminating with the classical formula    φ div F dx + ∇φ · F dx = − φ F · ν dHn−1 , (1.1) U U ∂U Arch. Rational Mech. Anal. (2025) 249:79 Page 3 of 82 79 valid for any smooth vector field F, smooth test function φ, and open set U with smooth boundary and interior unit normal ν. A first extension of this formula was achieved by Federer and De-Giorgi [28,29,36,37] for the cases of Lipschitz vector fields and sets with irregular boundaries (sets of finite perimeter) by using tools of geometric-measure theory. Further extensions of (1.1) to divergence-measure fields require a notion of normal trace on the boundary ∂U of any open U . For the case of bounded vector fields and sets of finite perimeter, the approach in [19] consisted in constructing essentially interior and exterior approximations of the sets of finite perimeter with smooth sets and then obtaining the normal traces as the limits of classical normal traces on the smooth approximating sets, which leads to the existence of interior and exterior traces for every set of finite perimeter (see also [23]). This approach is consistent with applications to hyperbolic conservation laws since solutions to these equations have jumps across the shock waves (cf. [8,10,27,45]). For a divergencemeasure field whose underlying field is bounded, given any set of finite perimeter, it was shown in [19] that the normal trace is a bounded function supported on the reduced boundary of the set. For the case of an unbounded vector field, the normal trace is classical for al (...truncated)