Asymptotic expansion of the stokes solutions at small viscosity: The case of non-compatible initial data

Abstract

Without imposing the so-called compatibility condition on the initial data, we obtain an asymptotic expansion of the Stokes solutions at small viscosity ε as the sum of the linearized Euler solution and a corrector function, which balances the discrepancy on the boundary of the Stokes and the linearized Euler solutions. Using such an expansion and smallness of the corrector, as the viscosity ε tends to zero, we obtain the uniform L 2 convergence of the Stokes solutions to the linearized Euler solution with rate of order ε 1/4 .

Remarks on the Inviscid Limit for the Navier--Stokes Equations for Uniformly Bounded Velocity Fields

“…The main difference between the corrector u K we consider, and the one considered in [Kat84], is its characteristic length scale: we let u K obey a Prandtl √ νt scaling (see also [TW95,Gie14]). Roughly speaking, u K 1 is a lift of the Euler boundary condition which obeys the heat equation (∂ t − ν∂ x 2 x 2 )u K 1 = 0 to leading order in ν.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…Therefore, one expects that in the context of the Navier-Stokes equations linearized around a non-trivial profile, i.e., Oseen-type equations, it should be possible to establish the zero-viscosity limit. This is indeed the case at least if the Oseen profile is regular enough and under some compatibility conditions between the initial and boundary data [2,3,135,136,96] (see also [52] for incompatible data for the Stokes equation).…”

Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

“…Equation 13with initial and boundary conditions 14is a parabolic problem (heat equation), in which − is the (positive) time like variable so that the boundary conditions at = 1 are the analogue of the initial conditions for an evolution problem, and the possible inconsistencies in (15) correspond to the (analogue of ) the compatibility conditions between the initial and boundary values for a parabolic equation; see [7][8][9] and the references therein. Due to the lack of consistency at the corners (1, 0) and (1, 1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at = 1 and = 0, 1, similar to what is done in [10,11] or [12] for the compatibility issue in parabolic problems; see [13] as well.…”

“…Due to the lack of consistency at the corners (1, 0) and (1, 1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at = 1 and = 0, 1, similar to what is done in [10,11] or [12] for the compatibility issue in parabolic problems; see [13] as well.…”