On the local existence for an active scalar equation in critical regularity setting

2016/06/14 by Walter Rusin, Fei Wang, Rusin, Walter +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1606.04525

openalex publication_date 2016/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we address the local well-posedness for the active scalar equation ∂_t θ+ u⋅ ∇ θ=0, where u = - ∇^⊥(-Δ)^-1+β/2θ. The local existence of solutions in the Sobolev class H^1+β+ε, where ε>0 and β∈ (1,2), has been recently addressed in \citeHKZ. The critical case ε=0 has remained open. Using a different technique, we prove the local well-posedness in the Besov space B^1+β_2,1, where β∈ (1,2). The proof is based on log-Lipschitz estimates for the transport equation.

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