Abstract
We prove that any weak space-time \(L^2\) vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of \({\mathbb R}^2\) satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that \(t-a.e.\) weak \(L^2\) inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.
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Bardos, C., Székelyhidi, L., Wiedemann, E.: Non-uniqueness for the Euler equations: the effect of boundaries. Russ. Math. Surv. 69, 3–22 (2014)
Bardos, C., Titi, E.S.: Euler equations for an ideal incompressible fluid. Uspekhi Mat. Nauk 62(3):5–46 (2007). (Russian) Translation. Russ. Math. Surv. 62(3), 409–451 (2007)
Bardos, C., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)
Bardos, C., Titi, E.S., Wiedemann, E.: The vanishing viscosity as a selection principle for the Euler equations: the case of 3D shear flow. Comptes Rend. Math. Acad. Sci. Paris 350, 757–760 (2012)
Constantin, P., W, E., Titi, E.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165, 207–209 (1994)
Constantin, P., Elgindi, T., Ignatova, M., Vicol, V.: Remarks on the inviscid limit for the Navier–Stokes equations for uniformly bounded velocity fields. SIAM J. Math. Anal. (2017, to appear)
Constantin, P., Kukavica, I., Vicol, V.: On the inviscid limit of the Navier–Stokes equations. Proc. Am. Math. Soc. 143(7), 3075–3090 (2015)
Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995)
Gie, G.-M., Kelliher, J.P., Lopes Filho, M.C., Nussenzveig Lopes, H.J., Mazzucato, A.L.: The vanishing viscosity limit for some symmetric flows. (2015, Preprint)
Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations (Berkeley, California, 1983), vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ. Springer, New York (1984)
Kelliher, J.P.: On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J. 56(4), 1711–1721 (2007)
Kelliher, J.P.: Vanishing viscosity and the accumulation of vorticity on the boundary. Commun. Math. Sci. 6(4), 869–880 (2008)
Kelliher, J.P.: On the vanishing viscosity limit in a disk. Math. Ann. 343(3), 701–726 (2009)
Lions, J.L.: Quelque Methodes de Résolution des Problemes aux Limites Non linéaires. Dunod-Gauth, Paris (1969)
Lopes Filho, M.C., Mazzucato, A.L., Nussenzveig Lopes, H.J.: Vanishing viscosity limit for incompressible flow inside a rotating circle. Phys. D 237(10–12), 1324–1333 (2008)
Lopes Filho, M.C., Mazzucato, A.L., Nussenzveig Lopes, H.J., Taylor, M.: Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. (N.S.) 39(4), 471–513 (2008)
Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Comm. Pure Appl. Math. 67(7), 1045–1128 (2014)
Mazzucato, A., Taylor, M.: Vanishing viscosity plane parallel channel flow and related singular perturbation problems. Anal. PDE 1(1), 35–93 (2008)
Prandtl, L.: Uber Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des dritten internationalen Mathematiker-Kongresses (Heidelberg 1904):484–491, (1905)
Sammartino, M., Caflisch, R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys. 192(2), 463–491 (1998)
Temam, R., Wang, X.: On the behavior of the solutions of the Navier–Stokes equations at vanishing viscosity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(3–4):807–828 (1998), 1997. Dedicated to Ennio De Giorgi
Wang, X.: A Kato type theorem on zero viscosity limit of Navier–Stokes flows. Indiana Univ. Math. J., 50(Special Issue):223–241. Dedicated to Professors Ciprian Foias and Roger Temam(2001)
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The research of PC is partially funded by NSF Grant DMS-1209394, and the research if VV is partially funded by NSF Grant DMS-1652134.
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Communicated by Edriss S. Titi.
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Constantin, P., Vicol, V. Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit. J Nonlinear Sci 28, 711–724 (2018). https://doi.org/10.1007/s00332-017-9424-z
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DOI: https://doi.org/10.1007/s00332-017-9424-z