2024/11/14 by Nikolay A. Gusev, Gusev, Nikolay A., Mikhail V. Korobkov +1
Computer Science · #35D30 #Advanced Algebra and Logic #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2411.09338
openalex publication_date 2024/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p≥ 1 and let \boldsymbolv \colon \mathbb Rd → \mathbb Rd be a compactly supported vector field with \boldsymbolv ∈ Lp(\mathbb Rd) and div \boldsymbolv = 0 (in the sense of distributions). It was conjectured by Nelson that it p=2 then the operator A(ρ) := \boldsymbolv ⋅ ∇ ρ with the domain D(\mathsf A)=C0^∞(\mathbb Rd) is essentially skew-adjoint on L2(\mathbb Rd). A counterexample to this conjecture for d≥ 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≥ 2 is necessary and sufficient for the following chain rule property of \boldsymbolv: for any ρ∈ L^∞(\mathbb R2) and any β∈ C1(\mathbb R) the equality div(ρ\boldsymbolv) = 0 implies that div(β(ρ) \boldsymbolv) = 0. Furthermore, for d=2 we prove that \boldsymbolv has the renormalization property if and only if the stream function (Hamiltonian) of \boldsymbolv 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=∞ by Alberti, Bianchini and Crippa.