The Nelson conjecture and chain rule property
Nikolay A. Gusev, Mikhail V. Korobkov
TL;DR
This work resolves the chain-rule issue for divergence-free planar fields by showing CRP$_{\infty}$ holds in 2D precisely when $p\ge2$, while highlighting its separation from full renormalization RP$_{\infty}$. Central to the approach is a monotone decomposition of the stream function $f$ into components $f_i$, reducing complex 2D level-set questions to 1D analyses on each component. The authors establish a sharp equivalence between the weak Sard property of $f$ and RP$_{\infty}$, and between the weak Sard property and the uniqueness of bounded weak solutions to the continuity equation, thereby linking geometric level-set regularity to PDE well-posedness. They also develop refined fine properties of Sobolev level sets in the plane, including monotonicity-continuity equivalences and a detailed treatment of essential vs. standard level sets, extending ABC's results to the critical Sobolev range in 2D.
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 $\mathsf{A}(ρ) := \boldsymbol{v} \cdot \nabla ρ$ 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 $ρ\in L^\infty(\mathbb R^2)$ and any $β\in C^1(\mathbb R)$ the equality $\operatorname{div}(ρ\boldsymbol{v}) = 0$ implies that $\operatorname{div}(β(ρ) \boldsymbol{v}) = 0$. 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.