Sobolev Stability of Prandtl Expansions for the Steady Navier–Stokes Equations

  • David Gerard-VaretEmail author
  • Yasunori Maekawa


We show the H1 stability of shear flows of Prandtl type: \({U^\nu = \left(U_s(y/\sqrt{\nu}), 0\right)}\), in the steady two-dimensional Navier–Stokes equations, under the natural assumptions that \({U_s(Y) > 0}\) for \({Y > 0}\), \({U_s(0) = 0}\), and \({U'_s(0) > 0}\). Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. This provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier–Stokes flows with no-slip boundary condition.


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David Gerard-Varet is supported by the Sing Flows project, Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). He also acknowledges the support of the Institut Universitaire de France.Yasunori Maekawa is partially supported by JSPS Grant 17K05320, 16H06339, 16H03947 and 17H02853.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586Université Paris Diderot and IUFParisFrance
  2. 2.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan

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