2023/12/09 by Tien-Tai Nguyen, Nguyen, Tien-Tai
Earth and Planetary Sciences · Engineering · Mathematics · #35J40 #47A45 #47A55 #76D45 #76E09 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Oceanographic and Atmospheric Processes
paper · pdf · doi:10.48550/arxiv.2312.05536
openalex publication_date 2023/12/09 · openalex created_date 2023/12/13 · openalex updated_date 2026/07/28
Motivated by Bresch, Desjardins, Gisclon and Sart (2008), in this paper, we study the influence of capillary number on an instability result related to the Navier-Stokes-Korteweg equations. Precisely, we investigate the instability of a steady-state profile with a heavier fluid lying above a lighter fluid, i.e., to study the Rayleigh-Taylor instability problem if the capillary number is below the critical value. After writing the nonlinear equations in a perturbed form, the first part is to provide a spectral analysis showing that, there exist possibly multiple normal modes to the linearized equations by following the operator method of Lafitte-Nguyen (2022). Hence, we construct a wide class of initial data for which the nonlinear perturbation problem departs from the equilibrium, based on the finding of possibly multiple normal modes. Using a refined framework of Guo-Strauss (1995), we prove the nonlinear instability.