# The Cauchy problem for the Stokes equations

## Authors

• G. Dairbayeva al-Farabi Kazakh National University, Almaty, Republic of Kazakhstan

49 45

## Keywords:

Cauchy Problem, Stokes equations, inverse problem, FEM, optimization method

## Abstract

In this paper we consider the Cauchy problem for the Stokes equations in domain with curvilinear
boundary, the solution is not known on a part of the boundary. This problem is ill-posed. Direct
and conjugate problems are constructed for the initial equations, the definition of generalized
solutions are introduced for these problems in the Sobolev space. It is shown that the solution for
the initial problem is reduced to the solution of the inverse problem for the direct problem. The
inverse problem is represented in operator form, objective functional is constructed, its gradient is
calculated. Computational algorithm is developed for solving the inverse problem for the Stokes
equations on the basis of the combination of optimization method and the finite element method
(FEM).

## References

[1] Bastay G., Johansson T., Lesnik D., Kozlov K., "An Alternating Method for the Stationary Stokes System" , ZAMM (Z.
Angew. Math. Mech) 86 (2006.): 268-280.
[2] Kabanikhim S.I., Dairbaeva G., "The Cauchy problem for Laplace equation on plane" , Notes on Numerical Fluid Mechanics
and Multidisciplinary Design (Springer-Verlag Berlin Heidelberg. Germany) 93 (2006): 89-102.
[3] Larry J. Segerlind. Applied finite elements analysis ( New York: United States Copyright, 1984), 411 р.
[4] Kabanikhin S.I. Inverse and ill-posed problems (Novosibirsk: Siberian scientific publishing, 2009), 457 p.
[5] Ladyzhenskaya O.A. Matematicheskie voprosy dinamiki vyazkoj neszhimaemoj zhidkosti [Mathematical problems of dynamics
of viscous incompressible fluid]. (M.: Gosudarstvennoe izdatel’stvo fizik-matematicheskoj literatury, 1961), 310 s.

2018-08-24

## How to Cite

Dairbayeva, G. (2018). The Cauchy problem for the Stokes equations. Journal of Mathematics, Mechanics and Computer Science, 95(3), 78–89. https://doi.org/10.26577/jmmcs-2017-3-477

## Section

Applied Mathematics