Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces

2024/10/24 by Marc Magaña, Magaña, Marc
Mathematics · #35B30 (Secondary) #35F20 #35Q35 (Primary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.19057

openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the solution map for a family of non-linear transport equations in ℝⁿ, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree -(n-1), is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.

Related

• C^γ well-posedness of some non-linear transport equations
• Transport equations in Hölder space by vanishing viscosity and applications
• Holder Continuous Solutions of Active Scalar Equations
• Smooth Solutions of the Surface Semi-Geostrophic Equations
• Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces