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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 2

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

Pages: 383 – 400

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n2.a8

#### Author

#### Abstract

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