The vanishing viscosity limit for some symmetric flows
Gie, Gung-Min1 ; Kelliher, James P.2 ; Lopes Filho, Milton C.3 ; Mazzucato, Anna L.4 ; Nussenzveig Lopes, Helena J.3
1 Department of Mathematics, University of Louisville, 328 Natural Sciences Building, Louisville, KY 40292, USA
2 Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA
3 Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ, Brazil
4 Department of Mathematics, Penn State University, University Park, PA 16802, USA
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1237-1280.
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The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier–Stokes and the Euler solutions. Using properties of these correctors, we establish convergence of the Navier–Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of vanishing viscosity. Our work significantly extends prior work in the literature.

DOI : 10.1016/j.anihpc.2018.11.006
Classification : 35B25, 35C20, 76D05, 76D10
Mots-clés : Boundary layers, Singular perturbations, Navier–Stokes equations, Euler equations, Inviscid limit
Gie, Gung-Min 1 ; Kelliher, James P. 2 ; Lopes Filho, Milton C. 3 ; Mazzucato, Anna L. 4 ; Nussenzveig Lopes, Helena J. 3

1 Department of Mathematics, University of Louisville, 328 Natural Sciences Building, Louisville, KY 40292, USA
2 Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA
3 Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ, Brazil
4 Department of Mathematics, Penn State University, University Park, PA 16802, USA 
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     title = {The vanishing viscosity limit for some symmetric flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Gie, Gung-Min; Kelliher, James P.; Lopes Filho, Milton C.; Mazzucato, Anna L.; Nussenzveig Lopes, Helena J. The vanishing viscosity limit for some symmetric flows. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1237-1280. doi : 10.1016/j.anihpc.2018.11.006. http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.006/

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