2023/02/02 by Fei Jiang, Fucai Li, Jiang, Fei +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2302.01013
openalex publication_date 2023/02/02 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28
Bresch-Desjardins-Gisclon-Sart have derived that the capillarity can slow down the growth rate of Rayleigh-Taylor (RT) instability in the capillary fluids based on the linearized two-dimensional (2D) Navier-Stokes-Korteweg equations in 2008. Motivated by their linear theory, we further investigate the nonlinear RT problem for the 2D incompressible case in a horizontally periodic slab domain with Navier boundary condition, and rigorously verify that the RT instability can be inhibited by capillarity under our 2D setting. More precisely, if the RT density profile ρ satisfies an additional stabilizing condition, then there is a threshold κC of capillarity coefficient, such that if the capillarity coefficient κ is bigger than κC, then the small perturbation solution around the RT equilibrium state is algebraically stable in time. In particular, if the RT density profile is linear, then the threshold κC can be given by the formula κC=g /(π2h-2+L-2)ρ', where 2πL denotes the length of a periodic cell of the slab domain in the horizontal direction, and h the height of the slab domain. In addition, we also provide a nonlinear instability result for κ∈[0,κC). The instability result presents that the capillarity can not inhibit the RT instability, if its strength is too small.