Ill-posedness of basic equations of fluid dynamics in Besov spaces

2009/04/14 by Alexey Cheskidov, Cheskidov, A., Roman Shvydkoy +1 · 2 citations
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.0904.2196

Abstract

We give a construction of a divergence-free vector field u₀ ∈ H^s ∩ B^-1_∞,∞, for all s<1/2, such that any Leray-Hopf solution to the Navier-Stokes equation starting from u₀ is discontinuous at t=0 in the metric of B^-1_∞,∞. For the Euler equation a similar result is proved in all Besov spaces B^s_r,∞ where s>0 if r>2, and s>n(2/r-1) if 1 ≤ r ≤ 2.

Cited by

• The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces
• Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

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