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

Without imposing the so-called compatibility condition on the initial data, we obtain an asymptotic expansion of the Stokes solutions at small viscosity $\epsilon$ 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 $\epsilon$ tends to zero, we obtain the uniform $L^2$ convergence of the Stokes solutions to the linearized Euler solution with rate of order $\epsilon^{1/4}$.

Keywords

boundary layers, singular perturbations, Stokes equations, non-compatible initial data, curvilinear coordinate system

2010 Mathematics Subject Classification

35B25, 35C20, 35K05, 76D07, 76D10

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Published 20 September 2013