TY - CHAP

T1 - Partial Invariance and Problems with Free Boundaries

AU - Pukhnachev, V. V.

N1 - Publisher Copyright: © 2021, Higher Education Press.

PY - 2021

Y1 - 2021

N2 - The foundations of group analysis of differential equations were laid by S. Lie. This theory was essentially developed in works of L. V. Ovsiannikov, N. H. Ibragimov, their students, and followers. Notion of the partially invariant solution to the system of differential equations (Ovsiannikov 1964) substantially extended possibilities of exact solutions construction for multidimensional systems of differential equations admitting the Lie group. It is important to note that fundamental equations of continuum mechanics and physics fall in this class a priori as invariance principle of space, time, and moving medium there with respect to some group (Galilei, Lorenz, and others) are situated in the base of their derivation. It should be noticed that classical group analysis of differential equations has a local character. To apply this approach to initial boundary value problems, one need to provide the invariance properties of initial and boundary conditions. Author (1973) studied these properties for free boundary problems to the Navier–Stokes equations. Present chapter contains an example of partially invariant solution of these equations describing the motion of a rotating layer bounded by free surfaces.

AB - The foundations of group analysis of differential equations were laid by S. Lie. This theory was essentially developed in works of L. V. Ovsiannikov, N. H. Ibragimov, their students, and followers. Notion of the partially invariant solution to the system of differential equations (Ovsiannikov 1964) substantially extended possibilities of exact solutions construction for multidimensional systems of differential equations admitting the Lie group. It is important to note that fundamental equations of continuum mechanics and physics fall in this class a priori as invariance principle of space, time, and moving medium there with respect to some group (Galilei, Lorenz, and others) are situated in the base of their derivation. It should be noticed that classical group analysis of differential equations has a local character. To apply this approach to initial boundary value problems, one need to provide the invariance properties of initial and boundary conditions. Author (1973) studied these properties for free boundary problems to the Navier–Stokes equations. Present chapter contains an example of partially invariant solution of these equations describing the motion of a rotating layer bounded by free surfaces.

UR - http://www.scopus.com/inward/record.url?scp=85121729746&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/13b489a0-aa02-379d-a8a8-7a101654f41f/

U2 - 10.1007/978-981-16-4683-6_9

DO - 10.1007/978-981-16-4683-6_9

M3 - Chapter

AN - SCOPUS:85121729746

SN - 978-981-16-4682-9

SN - 978-981-16-4685-0

T3 - Nonlinear Physical Science

SP - 251

EP - 267

BT - Nonlinear Physical Science

A2 - Luo, Albert C. J.

A2 - Gazizov, Rafail K.

PB - Springer Science and Business Media Deutschland GmbH

ER -