Abstract
The paper presents some coercive a priori estimates of the solution of the Dirichlet problem for the linear Stokes equation relating vorticity and the stream function of an axially symmetric flow of an incompressible fluid. This equation degenerates on the axis of symmetry. The method used to obtain the estimates is based on a differential substitution transforming the Stokes equation into the Laplace equation and on the subsequent transition from cylindrical to Cartesian coordinates in three-dimensional space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Leray, “Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique,” J. Math. Pures Appl. 12, 1–82 (1933).
M. V. Korobkov, K. Pileckas, V. V. Pukhnachev, and R. Russo, “The flux problem for the Navier-Stokes equations,” Uspekhi Mat. Nauk 69 (6 (420)), 115–176 (2014) [Russian Math. Surveys 69 (6), 1065–1122 (2014)].
M. V. Korobkov, K. Pileckas, and R. Russo, “Solution of Leray problem for stationary Navier–Stokes equations in plane and axially symmetric spatial domains,” Ann. of Math. (2) 181 (2), 769–807 (2015).
H. Fujita, “On the existence and regularity of the steady-state solutions of the Navier–Stokes equations,” J. Fac. Sci. Univ. Tokyo Sec. I 9, 59–102 (1961).
R. Finn, “On steady-state solutions of the Navier–Stokes equations. III,” ActaMath. 105, 197–244 (1961).
H. Fujita and H. Morimoto, “A remark on the existence of the Navier–Stokes flowwith non-vanishing outflow conditions,” in Nonlinear Waves, GAKUTO Int. Ser. Math. Sci. Appl. (Gakkotosho, Tokyo, 1997), Vol. 10, pp. 53–61.
V. V. Pukhnachev, “Three-dimensional symmetric flow-through problem for the Navier–Stokes equations,” Vestnik YuUrGU, Ser. Matem. Modelir. i Programmir. 8 (2), 95–104 (2015).
V. I. Polezhaev, A. V. Buné, N. A. Verezub, et al., Mathematical Modeling of Convective Heat Exchange Based on the Navier–Stokes Equations (Nauka, Moscow, 1987) [in Russian].
G. V. Alekseev and V. V. Pukhnachev, “The axially symmetric flow-through problem for the Navier-Stokes equations in ‘Vorticity–Stream Function’ Variables,” Dokl. Ross. Akad. Nauk 445 (4), 402–406 (2012) [Dokl. Math. 57 (8), 301–306 (2012)].
O. A. Ladyzhenskaya, Mathematical Questions of Dynamics of a Viscous Incompressible Fluid (Nauka, Moscow, 1970) [in Russian].
A. A. Kovalevskii, I. I. Skrypnik, and A. E. Shishkov, Singular Solutions of Nonlinear Elliptic and Parabolic Equations (Naukova Dumka, Kiev, 2010) [in Russian].
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the Boundaries for Solutions of Elliptic Partial Differential Equations Satisfying General Boundaries (New Yourk, 1959; Inostr. Lit., Moscow, 1962).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. V. Pukhnachev, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 1, pp. 110–115.
Rights and permissions
About this article
Cite this article
Pukhnachev, V.V. Dirichlet problem for the Stokes equation. Math Notes 101, 132–136 (2017). https://doi.org/10.1134/S0001434617010138
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617010138