2017/08/30 by Haibo Cui, Cui, Haibo, Haiyan Yin +5
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP
paper · pdf · doi:10.48550/arxiv.1708.09127
arxiv created 2017/08/30 · openalex publication_date 2017/08/30 · arxiv updated 2017/08/31 · openalex created_date 2017/09/15 · openalex updated_date 2026/07/28
This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant (x,t)∈ ℝ+×ℝ+, ∂t v - ∂x u=0, ∂t u + ∂x p(v) = -\fracα(1+t)λ u, with null-Dirichlet boundary condition or null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.