Sharp well-posedness and ill-posedness in Fourier-Besov spaces for the viscous primitive equations of geophysics

2015/10/24 by Jinyi Sun, Sun, Jinyi, Shangbin Cui +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1510.07134

arxiv created 2015/10/24 · openalex publication_date 2015/10/24 · arxiv updated 2015/10/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner's localization method, we prove that the Cauchy problem with Prandtl number P=1 is locally well-posed in the Fourier-Besov spaces [FB^2-(3)/(p)_p,r(ℝ³)]⁴ for 1<p≤∞,1≤ r<∞ and [FB^-1_1,r(ℝ³)]⁴ for 1≤ r≤ 2, and globally well-posed in these spaces when the initial data (u₀,θ₀) are small. We also prove that such problem is ill-posed in [FB^-1_1,r(ℝ³)]⁴ for 2<r≤∞, showing that the results stated above are sharp.

Citations

• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions

Related

• Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier-Besov spaces
• Global Well--posedness of the Three-dimensional Viscous Primitive Equations of Large Scale Ocean and Atmosphere Dynamics
• Ill-posedness for the 2D viscous shallow water equations in the critical Besov spaces
• Sharp well-posedness and ill-posedness of the Navier-Stokes initial value problem in Besov-type spaces
• From anisotropic Navier-Stokes equations to primitive equations for the ocean and atmosphere