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Normal traces and applications to continuity equations on bounded domains

Gianluca Crippa, Luigi De Rosa, Marco Inversi, Matteo Nesi

Abstract

In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary to get uniqueness.

Normal traces and applications to continuity equations on bounded domains

Abstract

In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a assumption seems to be necessary to get uniqueness.
Paper Structure (20 sections, 29 theorems, 158 equations, 2 figures)

This paper contains 20 sections, 29 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The normal Lebesgue trace
  3. Applications to the continuity equation
  4. Plan of the paper
  5. Technical tools
  6. Notation
  7. Weak convergence of measures
  8. Functions of bounded variation
  9. Slicing and traces
  10. Measure-divergence vector fields
  11. Distance function and projection onto a closed set
  12. Rectifiable sets, Minkowski content, and Hausdorff measure
  13. Lipschitz sets and one-sided Minkowski contents
  14. Normal Lebesgue trace: Gauss-Green and further properties
  15. Gauss-Green identity
  16. ...and 5 more sections

Key Result

Theorem 1.4

Let $\Omega \subset \mathbb{R}^d$ be a bounded open set with Lipschitz boundary and let $u\in \mathcal{MD}^\infty(\Omega)$. Assume that $u$ has a normal Lebesgue trace on $\partial \Omega$ in the sense of D:Leb normal trace. Then, it holds $u_n^{\partial\Omega}\equiv \mathop{\rm Tr}\nolimits_n (u;\p

Figures (2)

  • Figure 1: Proof of \ref{['p: weak convergence to hausdorff']} in the case $d=2$ and $m=1$.
  • Figure 2: The "tiles" $Q_{i,j}$ become finer approaching the axis $\{y=0\}$.

Theorems & Definitions (63)

  • Definition 1.1: $L^p$ Measure-divergence vector fields
  • Definition 1.2: Distributional normal trace
  • Definition 1.3: Normal Lebesgue boundary trace
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7: Existence
  • Proposition 1.8
  • Definition 2.1
  • Proposition 2.2
  • ...and 53 more