Propagation of regularity for the MHD system in optimal Sobolev space

2017/07/24 by Mimi Dai, Dai, Mimi
Engineering · Mathematics · #35D35 #76D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1707.07754

openalex publication_date 2017/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the problem of propagation of regularity of solutions to the incompressible viscous non-resistive magneto-hydrodynamics system. According to scaling, the Sobolev space H^{\frac n2-1}(\mathbb Rⁿ)× H^{\frac n2}(\mathbb Rⁿ) is critical for the system. We show that if a weak solution (u(t),b(t)) is in H^s(\mathbb Rⁿ)× H^s+1(\mathbb Rⁿ) with s>\frac n2-1 at a certain time t₀, then it will stay in the space for a short time, provided the initial velocity u(0)∈ H^s(\mathbb Rⁿ). In the case that the uniqueness of weak solution in H^s(\mathbb Rⁿ)× H^s+1(\mathbb Rⁿ) is known, the assumption of u(0)∈ H^s(\mathbb Rⁿ) is not necessary.

Citations

• Sur le mouvement d'un liquide visqueux emplissant l'espace

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