This paper studies the vanishing vertical viscosity limit for the two-dimensional Navier–Stokes equations with anisotropic viscosity. The fluid domain is bounded above and below by impenetrable walls, with a no-slip boundary condition imposed on the bottom wall and a slip boundary condition imposed on the top wall. We derive global L^p energy estimates with explicit dependence on the horizontal and vertical viscosities. By introducing a decomposition of the pressure into a flow component and a boundary-layer component, we isolate and quantify boundary effects associated with the vertical viscosity. Using these estimates, we prove that solutions converge strongly in L^p(Ω ) , for any 2 ≤ p < ∞ , to solutions of the horizontally viscous limit system as the vertical viscosity tends to zero, provided that the initial data are sufficiently regular. The main difficulty arises from the mismatch in boundary conditions, namely, the no-slip condition in the original problem versus the slip condition in the limiting system. The purpose of this paper is to understand the associated boundary layer phenomena under stronger assumptions. Our model captures the essential features of the boundary layer behavior.
The geometric property of σ -superstarlikeness, along with its related notions– σ -superstarlikeness of order α and σ -superstarlikeness of complex order–are introduced and studied in this paper. These concepts are shown not only to cover but also to generalize a wide range of known time-invariant geometric properties of the moving boundary in the inner Hele-Shaw flow problem for bounded domains, i.e., two-dimensional bounded viscous fluid flows under injection in the absence of surface tension. We prove that, under suitable parameter constraints, these properties are preserved in time. Consequently, this work extends, unifies, and completes recent research on the Hele-Shaw flow problem in the absence of surface tension.
This work analyzes the finite element approximation to a viscoelastic flow model, which generalizes the Navier–Stokes equation and Oldroyd’s model by introducing the tempered power-law memory kernel. We prove regularity and long-time exponential decay of the solutions, as well as a long-time convolution-type Grönwall inequality to support numerical analysis. A Volterra–Stokes projection is developed and analyzed to facilitate the parabolic-type duality argument, leading to the long-time error estimates and exponential decay of velocity and pressure. A benchmark problem of planar four-to-one contraction flow is simulated to substantiate the generality of the proposed model in comparison with the Navier–Stokes equation and Oldroyd’s model.
This paper is concerned with the time-decay rates of the viscous contact wave in 1D heat-conductive ideal gas without viscosity. For the case without the zero mass condition, although the heat-conductive ideal gas without viscosity studied in this paper lacks dissipation in terms of velocity, we can still prove the same optimal decay rate as the compressible Navier–Stokes equation. Precisely, we can obtain the optimal decay rate (1+t)^-1/2 for the perturbation in L^∞ -norm. The crucial point of the analysis is as follows. We refine the estimates for both anti-derivatives and the original perturbations in the first, and then, we use an innovative transformation to ensure the structural conditions continue to hold for the system of derivatives. With this approach, we obtain the optimal decay rate.
