The Nelson conjecture and chain rule property

TL;DR

This paper investigates the properties of divergence-free vector fields in two dimensions, establishing conditions under which chain rule and renormalization properties hold, and relates these to the uniqueness of solutions to the continuity equation.

Contribution

It proves that for two-dimensional divergence-free vector fields, the chain rule property holds if and only if the vector field's integrability exponent is at least 2, and links renormalization to the weak Sard property of the stream function.

Findings

Chain rule property holds for p≥2 in 2D.

Renormalization property is equivalent to the weak Sard property.

Uniqueness of solutions is characterized by these properties.

Abstract

Let $p\ge 1$ and let $\boldsymbol{v} \colon \mathbb R^d \to \mathbb R^d$ be a compactly supported vector field with $\boldsymbol{v} \in L^p(\mathbb R^d)$ and $\operatorname{div} \boldsymbol{v} = 0$ (in the sense of distributions). It was conjectured by Nelson that it $p=2$ then the operator $\mathsf{A}(\rho) := \boldsymbol{v} \cdot \nabla \rho$ with the domain $D(\mathsf A)=C_0^\infty(\mathbb R^d)$ is essentially skew-adjoint on $L^2(\mathbb R^d)$. A counterexample to this conjecture for $d\ge 3$ was constructed by Aizenmann. From recent results of Alberti, Bianchini, Crippa and Panov it follows that this conjecture is false even for $d=2$. Nevertheless, we prove that for $d=2$ the condition $p\ge 2$ is necessary and sufficient for the following chain rule property of $\boldsymbol{v}$: for any $\rho \in L^\infty(\mathbb R^2)$ and any $\beta\in C^1(\mathbb R)$ the equality $\operatorname{div}(\rho \boldsymbol{v}) = 0$ implies that $\operatorname{div}(\beta(\rho) \boldsymbol{v}) = 0$. Furthermore, for $d=2$ we prove that $\boldsymbol{v}$ has the renormalization property if and only if the stream function (Hamiltonian) of $\boldsymbol{v}$ has the weak Sard property, and that both of the properties are equivalent to uniqueness of bounded weak solutions to the Cauchy problem for the corresponding continuity equation. These results generalize the criteria established for $d=2$ and $p=\infty$ by Alberti, Bianchini and Crippa.