The linearized non-stationary problem for the permeable boundary Navier–Stokes flows

摘要

We consider the following linearized model for the permeable boundary Navier–Stokes flows, representing the non-stationary state:ρ∂tv(x,t)=μΔv(x,t)−∇p(x,t)+f(x,t);v∈H2(Ω);(1a)x∈Ω⊂R3;f∈L2(Ω),t>0,subjectto(1b)∇·v(x,t)=0;γ0v(y,t)=−ηv(y,t)n(y);y∈Γ.σ∂tγ0v(y,t)+γ0p(y,t)+2μκηv(y,t)=p0(t).where, μ is the fluid viscosity; assumed constant, ρ, the fluid density; assumed constant, v(x,t), the fluid velocity field, p(x,t), the fluid pressure field, f(x,t), the external force; assumed to be of potential type.We study the existence and uniqueness of the solution; investigate the stability of the null solution using the Ljapunov's stability theory.This problem is analyzed by Ladyzhenskaya [O.L. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1963] in a different setting. In our setting of a permeable boundary, we introduce the second condition in (1b), which is the Sauer–Maritz boundary permeation model. This model was proposed by Sauer; used by Maritz and Sauer [R. Maritz, N. Sauer, in: H. Amman et al. (Eds.), Navier–Stokes Equations and Related Non-linear Problems, VSP/TEV, Utrecht/Vilnius, 1998, p. 153] in the study of incompressible second grade fluids. It was later used by Hlomuka and Sauer [J.V. Hlomuka, N. Sauer, in: R. Salvi (Ed.), Navier–Stokes Equations: Theory and Numerical Methods, Marcel Dekker, New York, 2001, p. 33] to confirm the stability of the permeable boundary Navier–Stokes flows for the non-linear problem. The third boundary condition is the dynamic boundary condition derived in [J.V. Hlomuka, N. Sauer, in: R. Salvi (Ed.), Navier–Stokes Equations: Theory and Numerical Methods, Marcel Dekker, New York, 2001, p. 33] by making use of the conservation laws.