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A current based approach for the uniqueness of the continuity equation

Tommaso Cortopassi

Abstract

We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L^1 (0,T; W^{1,p}(\mathbb{T}^d)) \cap L^\infty ((0,T) \times \mathbb{T}^d)]^d$ with $\operatorname{div}(u) ^- \in L^1 (0,T; L^\infty (\mathbb{T}^d))$ and an initial datum $ρ_0 \in L^q (\mathbb{T}^d)$, where $\mathbb{T}^d$ is the $d$-dimensional torus and $ 1 \leq p,q \leq +\infty$ such that $1/p + 1/q =1$ without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong $L^1$ estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.

A current based approach for the uniqueness of the continuity equation

Abstract

We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field with and an initial datum , where is the -dimensional torus and such that without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.
Paper Structure (4 sections, 6 theorems, 76 equations)

This paper contains 4 sections, 6 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Proof of uniqueness
  3. Appendix
  4. Acknowledgements

Key Result

Lemma 2.1

Let $T= f_t e_t + f_j e_j$ be a normal $1$-current such that $\partial T=0$ in $(- \infty , 1) \times \mathbb{T}^d$, supported in $[0,1) \times \mathbb{T}^d$ and such that $f_t , f_j \in L^\infty (0,1 ; L^1 (\mathbb{T}^d))$ for every $j=1, \dots,d$. Then $T$ is the boundary of a 2-current $S$ in $(- where As a consequence, it holds the flat norm estimate where $T^v = f_1 e_1 + \dots f_d e_d$ is

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2.1
  • proof
  • Remark 5
  • Lemma 2.2
  • ...and 18 more