Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

Abstract

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the L ∞ (H 1 ) norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.

Validity of boundary layer theory for the 3D plane‐parallel nonhomogeneous electrically conducting flows

“…Remark The convergence rate of magnetic field is in accord with velocity field and also the result for the Navier–Stokes system, 38 which implies the stabilizing effect of magnetic fields. In addition, it is very interesting to consider the other effect of the magnetic fields such as the Alfvén wave; see Alfvén, 39 for instance.…”

Stability of the boundary layer expansion for the 3D plane parallel MHD flow

“…Proof. The results of Corollary 4.1 is straightforward from the fact that (θ 0 , θ u,0 , h 0 , h u,0 , h 0 , h u,0 ) L ∞ (0,T ;L 2 (Ω∞)) ≈ ε 1 4 ,…”

“…It should be pointed out that the above bounds of order ε 1 4 can not be improved since we can not using integration by parts in the right-hand side involving second or mixed derivatives in z, as ∂ z u err 2 may not vanish on the boundaries. In addition, although one can apply the integration by parts in z for the magnetic fields, the convergence rates can not be improved due to the loss of √ ε resulted from the derivatives in z for the remainders, which will give the same convergence rates.…”

Boundary layer for 3D nonlinear parallel pipe flow of nonhomogeneous incompressible Navier‐Stokes equations

“…For example, the ( 0 , u 0 ) ∈ H m (Ω)×H m (Ω) and f ∈ L ∞ (0, T; H m (Ω)) for m > 5, then ( 0 , u 0 ) ∈ C(0, T; H m (Ω)) × C(0, T; H m (Ω)). The interested readers can see the literature 24,[28][29][30] for more details.…”

“…Han et al 23 verified the validity of the boundary layer theory for a class of nonlinear pipe flow. Recently, the first two authors and Niu 24 established the Prandtl theory for plane parallel channel flows in nonhomogeneous case.…”