Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces

2022/03/01 by Qionglei Chen, Yao Nie, Chen, Qionglei +1
Mathematics · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.2203.00309

arxiv created 2022/03/01 · arxiv updated 2022/03/02

Abstract

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B^(2)/(p)_p,1(ℝ²)× B^(2)/(p)-1_p,q(ℝ²). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B^(2)/(p)_p,1(ℝ²)× B^(2)/(p)-1_p,1(ℝ²) for p<4 and ill-posed in B^(2)/(p)_p,1(ℝ²)× B^(2)/(p)-1_p,1(ℝ²) for p>4. In this paper, we prove that this system is ill-posed for the critical case p=4 in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in B^(1)/(2)_4,1(ℝ²)× B^-(1)/(2)_4,q(ℝ²) for any q≠ 2

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