Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

2025/07/03 by Jinlu Li, Xing Wu, Li, Jinlu +3
Mathematics · Earth and Planetary Sciences · #Navier-Stokes equation solutions #Aquatic and Environmental Studies #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2507.02247

Abstract

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of B^s_p,∞ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \citeCS and Misiołek-Yoneda \citeMY. Secondly, we prove the failure of the B^s_p,∞-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.

Citations

• Non-existence and strong ill-posedness in C^k,β for the generalized Surface Quasi-geostrophic equation
• Non existence and strong ill-posedness in C^k and Sobolev spaces for\n SQG
• Inviscid limit of vorticity distributions in Yudovich class
• Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces
• Remarks on the inviscid limit for the Navier-Stokes equations for\n uniformly bounded velocity fields
• On the inviscid limit of the Navier-Stokes equations
• Ill-posedness of basic equations of fluid dynamics in Besov spaces
• On the Well-posedness of the Ideal MHD Equations in the Triebel–Lizorkin Spaces
• Nonstationary flows of viscous and ideal fluids in R3

Related

• Non-convergence of the Navier-Stokes equations toward the Euler equations in the endpoint Besov spaces
• Ill-posedness of basic equations of fluid dynamics in Besov spaces
• Local well-posedness of the incompressible Euler equations in B¹_∞,1 and the inviscid limit of the Navier-Stokes equations
• Ill-posedness for the Euler equations in Besov spaces
• Ill-posedness in B^s_p,∞ of the Euler equations: Non-continuous dependence