Abstract
We study a nonstationary initial–boundary value problem on the motion of a viscous incompressible fluid in the case of small viscosity. We prove the convergence of solutions to the corresponding limit relations as the viscosity tends to zero.
Similar content being viewed by others
References
Kochin, N.E., Kibel’, I.A., and Roze, N.V., Teoreticheskaya gidromekhanika (Theoretical Hydromechanics), Moscow: Fizmatgiz,1963, part 2.
Golovkin, K.K, Vanishing viscosity in the Cauchy problem for equations of hydrodynamics, Tr. Mat. Inst. Steklova, 1966, vol. 92, pp. 31–49.
Ladyzhenskaya, O.A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti (Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid), Moscow: Nauka, 1970.
Lions, J.-L., Quelques methodes de résolution des problémes aux limites nonlinéaires, Paris: Dunod, 1969.
Maslov, V.P., Asimptoticheskie metody i teoriya vozmushchenii (Asymptotic Methods and Perturbation Theory), Moscow: Nauka, 1988.
Kato, T, Remarks on zero-viscosity limit for nonstationary Navier–Stokes plows with boundary, in Seminar on Nonlinear Partial Differential Equations, Math. Sciences Research Institute Publications, 2. New York, 1983, pp. 85–98.
Alekseenko, S.N, On vanishing viscosity in three-dimensional boundary value problems of dynamics of incompressible fluid, Doctoral (Phys.–Math.) Dissertation, Bishkek, 1994.
Temam, R. and Wang, X, On the behavior of the solutions of the Navier–Stokes equations at vanishing viscosity, Ann. Sc. Norm. Super. Pisa Cl. Sci. bis e mat. 4, 1997, vol. 25, no. 3, pp. 807–828.
Sandrakov, G.V, Asymptotic behavior with respect to small velocity of solutions of a system of Stokes equations, Dokl. Math., 2007, vol. 75, no. 3, pp. 377–380.
Chernous’ko, F.L, Motion of a rigid body with cavities filled with viscous fluid at large Reynolds numbers, Prikl. Mat. Mekh., 1966, vol. 30, pp. 476–494.
Temam, R., Navier–Stokes Equations. Theory and Numerical Analysis, Amsterdam: North-Holland, 1977.
Yudovich, V.I., A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region, Mat. Sb., 1964, vol. 64, no. 4, pp. 562–588.
Antontsev, S.N., Kazhikhov, A.V., and Monakhov, V.I., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei (Boundary Value Problems of the Mechanics of Inhomogeneous Fluids), Novosibirsk: Nauka, 1983.
Gyunter, N.M, On Main Problem of Hydrodynamics, Izv. Fiz. Mat. Inst. Steklova Akad. Nauk SSSR, 1927, vol. 2, pp. 1–168.
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 2004.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.L. Khatskevich, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 6, pp. 830–840.
Rights and permissions
About this article
Cite this article
Khatskevich, V.L. On the asymptotics of motion of a viscous incompressible fluid for small viscosity. Diff Equat 53, 825–835 (2017). https://doi.org/10.1134/S0012266117060131
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266117060131