2017/01/30 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Computational Fluid Dynamics and Aerodynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #math.CA #math.DS
paper · pdf · doi:10.48550/arxiv.1701.08604
arxiv created 2017/01/30 · openalex publication_date 2017/01/30 · arxiv updated 2017/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a second order equation with a linear "elastic" part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero.We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fouriercomponents are asymptotic to solutions of the linearized equationwithout damping, and exhibit some sort of equipartition of theenergy among the components. Solutions with infinitely manyFourier components tend to zero weakly but not strongly. We showalso that the limit of the energy of solutions depends only on thenumber of their Fourier components.The proof of our results is inspired by the analysis of asimplified model which we devise through an averaging procedure,and whose solutions exhibit the same asymptotic properties as thesolutions to the original equation.