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Analytic Regularity for the incompressible Navier-Stokes Equations in polygons

by C. Marcati and Ch. Schwab

(Report number 2019-12)

Abstract
In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed, corner-weighted spaces with homogeneous norms. Implications of this analytic regularity include exponential smallness of Kolmogorov $N$-widths of solutions, exponential convergence rates of mixed $hp$-discontinuous Galerkin finite element and spectral element discretizations and of model order reduction techniques.

Keywords: Navier-Stokes equations, analytic regularity, conical singularities, weighted Sobolev spaces.

@Techreport{MS19_816,
  author = {C. Marcati and Ch. Schwab},
  title = {Analytic Regularity for the incompressible Navier-Stokes Equations in polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-12.pdf },
  year = {2019}
}

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