2021/08/16 by Yin Li, Huaqiao Wang, Li, Yin +5
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2108.06974
openalex publication_date 2021/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate global existence and optimal decay rates of a generic non-conservative compressible two--fluid model with general constant viscosities and capillary coefficients.The main novelty of this work is three--fold: First, for any integer ℓ≥3, we show that the densities and velocities converge to their corresponding equilibrium states at the L2 rate (1+t)-(3)/(4), and the k(∈ [1, ℓ])--order spatial derivatives of them converge to zero at the L2 rate (1+t)-(3)/(4)-(k)/(2), which are the same as ones of the compressible Navier--Stokes system, Navier--Stokes--Korteweg system and heat equation. Second, the linear combination of the fraction densities (β+α+ρ++β-α-ρ-) converges to its corresponding equilibrium state at the L2 rate (1+t)-(3)/(4), and its k(∈ [1, ℓ])--order spatial derivative converges to zero at the L2 rate (1+t)-(3)/(4)-(k)/(2), but the fraction densities (α^±ρ^±) themselves converge to their corresponding equilibrium states at the L2 rate (1+t)-(1)/(4), and the k(∈ [1, ℓ])--order spatial derivatives of them converge to zero at the L2 rate (1+t)-(1)/(4)-(k)/(2), which are slower than ones of their linear combination (β+α+ρ++β-α-ρ-) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well--chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two--fluid model.