On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces

2024/01/09 by Yanghai Yu, Jinlu Li, Yu, Yanghai +1
Mathematics · Physics and Astronomy · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Cosmology and Gravitation Theories

paper · pdf · doi:10.48550/arxiv.2401.04387

Abstract

It is proved in \citeIO21 that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in B_{p, q}^{2 / p}(ℝ²) × B_{p, q}^{2 / p-1}(ℝ²) × B_{p, q}^{2 / p-2}(ℝ²) with 1≤ p≤ ∞ and 1≤ q<∞. In this paper, we aim to solve the end-point case left in \citeIO21 and prove that the Cauchy problem is ill-posed in B_{p, ∞}^{d / p}(ℝ^d) × B_{p, ∞}^{d / p-1}(ℝ^d) × B_{p, ∞}^{d / p-2}(ℝ^d) with 1≤ p≤∞ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in B_p,∞^d/p-1(ℝ^d), which is an interesting open problem in itself.

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