Flow with A_∞(\mathbb R) density and transport equation in BMO(\mathbb R)

2018/05/04 by Renjin Jiang, K. Li, Jiang, Renjin +2
Mathematics · #Advanced Harmonic Analysis Research #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1805.01630

Abstract

We show that, if b∈ L¹(0,T;L¹_loc(ℝ)) has spatial derivative in the John-Nirenberg space BMO(ℝ), then it generalizes a unique flow ϕ(t,⋅) which has an A_∞(\mathbb R) density for each time t∈ [0,T]. Our condition on the map b is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in BMO(\mathbb R).

Citations

• Hp spaces of several variables
• Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

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