Abstract
The paper contains a justification of the principle of minimum of potential energy in the problem of stability of a rotating viscous incompressible self-gravitating liquid bounded only by a free surface. It is assumed that the domain occupied by a rotating liquid that is referred to as an equilibrium figure is not symmetric with respect to the axis of rotation. The surface tension is not taken into account. The proof of stability is based on analysis of the evolution free boundary problem for perturbations of the velocity and pressure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.A. Solonnikov, On the stability of uniformly rotating viscous incompressible self-gravitating liquid, J.Math. Sci. 152, No. 5 (2008), 4343–4370.
A.M. Lyapunov, On the stability of ellipsoidal equilibrium forms of rotating fluid, Collected works, vol. 3, Moscow (1959).
A.M. Lyapunov, On the equilibrium figures of rotating homogeneous liquid mass slightly different from ellipsoids, Collected works, vol. 4, Moscow (1959), 5–645.
P. Appell, Figures d’équilibre d’une masse liquide homogène en rotation, Paris, 1932.
L. Lichtenstein, Equilibrium figures of rotating liquid, M., “Nauka”, 1965.
Y. Hataya, Decaying solutions of the Navier-Stokes flow without surface tension, submitted to J. Math. Kyoto Univ.
V.A. Solonnikov, On the stability of nonsymmetric equilibrium figures of rotating viscous incompressible liquid, Interfaces and free boundaries 6 (2004), 461–492.
V.A. Solonnikov, On the problem of evolution of self-gravitating isolated liquid mass not subjected to capillary forces, J. Math. Sci. 122, No. 3 (2004), 3310–3330.
V.A. Solonnikov, On the nonstationary motion of a viscous incompressible liquid over a rotating ball, J. Math. Sci, 157 (2009), 885–947.
V.A. Solonnikov, On the linear problem related to the stability of uniformly rotating self-gravitating liquid, J. Math. Sci. 144, No 6 (2007), p. 4671.
V.A. Solonnikov, On the estimates of potentials related to the problem of stability of rotating self-gravitating liquid, J. Math. Sci. 154 No. 1 (2008), 90–124.
V.A. Solonnikov, On estimates for volume and surface potentials in domains with boundary of class W 1 2, J. Math. Sci. 150, No. 1 (2008), 1890–1916.
V.A. Solonnikov, On the stability of axially symmetric equilibrium figures of a rotating viscous incompressible fluid, St. Petersburg Math. J. 16 (2005) No. 2, 377–400.
V.A. Solonnikov, Estimate of generalized energy in the free boundary problem for viscous incompressible liquid, J. Math. Sci. 120 No. 5 (2004), 1766–1783.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
To the memory of Professor A.V. Kazhikhov
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Solonnikov, V.A. (2009). On the Stability of Non-symmetric Equilibrium Figures of Rotating Self-gravitating Liquid not Subjected to Capillary Forces. In: Fursikov, A.V., Galdi, G.P., Pukhnachev, V.V. (eds) New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0152-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0152-8_19
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0151-1
Online ISBN: 978-3-0346-0152-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)