2020/05/31 by Shanming Ji, Ji, Shanming, Ming Mei +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.00403
openalex publication_date 2020/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent over-damping of the form -\fracμ(1+t)λρ\boldsymbol u in \mathbb Rn, where n≥2, μ>0, and λ∈[-1,0). This continues our previous work dealing with the under-damping case for λ∈[0,1). We show the optimal decay estimates of the solutions such that for λ∈(-1,0) and n≥2, ‖ρ-1‖L2(\mathbb Rn)≈(1+t)-(1+λ)/(4)n and ‖\boldsymbol u‖L2(\mathbb Rn)≈ (1+t)-(1+λ)/(4)n-(1-λ)/(2), which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of λ=-1, we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that ‖ρ-1‖L2(\mathbb Rn)≈ |ln(e+t)|-(n)/(4) and ‖\boldsymbol u‖L2(\mathbb Rn)≈ (1+t)-1⋅|ln(e+t)|-(n)/(4)-(1)/(2) for n≥7. The over-damping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.