On the stability of a uniformly rotating viscous incompressible self-gravitating liquid

Solonnikov, V. A.

doi:10.1007/s10958-008-9090-7

Cited by 14 publications

(12 citation statements)

References 3 publications

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“…A similar assertion is proved in [3] for the problem governing the evolution of an isolated mass of a viscous incompressible self-gravitating liquid bounded only by a free surface Γ t . This problem is more complicated technically, in particular, because of the presence of nonlocal terms in the form of the Newtonian and single layer potentials.…”

Section: The Main Resultssupporting

confidence: 70%

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On the nonstationary motion of a viscous incompressible liquid over a rotating ball

Solonnikov

2009

J Math Sci

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We consider the free boundary problem for the Navier-Stokes equations governing a nonstationary motion of a layer of a viscous incompressible liquid that covers the surface of a rigid ball rotating around a fixed axis with constant angular velocity ω. The liquid is subject to the gravitation force generated by the mass of the ball. The self-gravitation forces between the liquid particles and capillary forces on the free surface are not taken into account. We consider the problem of stability of the regime of the rigid rotation of the liquid with the same angular velocity and prove that it is stable if |ω| is less than a certain constant. Bibliography: 10 titles.

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Section: The Main Resultssupporting

confidence: 70%

“…When the point y moves along the curve (2.1) from the point y (1) = (0, k/a 2 ) to the point y (2) = (t 1 , 0), the function k/|y| 3 − ω 2 decreases from a 6 /k 2 − ω 2 > 0 to k/t 3 1 − ω 2 0. It can be shown that k/t 3 1 − ω 2 > 0. This follows from the inequality…”

Section: On the Equilibrium Figurementioning

confidence: 99%

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

Solonnikov

2009

J Math Sci

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“…Finally let us consider the following free boundary problem modeling the motion of liquid drops (cf. [11,14,15]):…”

Section: Standard Local Chart Of D M+µ (R N )mentioning

confidence: 99%

“…These results have already been proved in the literature by using classical method for dealing with free boundary problems, cf. [11,14,15] for instance. However, the proofs are hard.…”

Section: Standard Local Chart Of D M+µ (R N )mentioning

confidence: 99%

On the Banach Manifold of Simple Domains in the Euclidean Space and Applications to Free Boundary Problems

Cui

2019

Acta Appl Math

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In this paper we study the Banach manifold made up of simple C m+µ -domains in the Euclidean space R. This manifold is merely a topological or a C 0 Banach manifold. It does not possess a differentiable structure. We introduce the concept of differentiable point in this manifold and prove that it is still possible to introduce the concept of tangent vector and tangent space at a differentiable point. Consequent, it is possible to consider differential equations in this Banach space. We show how to reduce some important free boundary problems into differential equations in such a manifold and then use the abstract result that we established earlier to study these free boundary problems. AMS 2000 Classification: 35R35, 47J35, 58B99. Key words and phrases: Banach manifold; manifold of simple domains; free boundary problem.

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“…A typical example in this line is the following free boundary problem modeling the motion of liquid drops (cf. [34,35]):…”

Section: Introductionmentioning

confidence: 99%

Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems

Cui

2016

Preprint

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If a differential equation in a Banach manifold is invariant or quasi-invariant under the action of one or more Lie groups, then its stationary points cannot be isolated, so that classical linearized stability theorem does not apply to it. The first main purpose of this paper is to establish a linearized stability theorem for parabolic differential equations in Banach manifolds which are either invariant or quasi-invariant under actions of a number of Lie groups. The second purpose of this paper is to apply this theorem to analyze stability of stationary solutions of some free boundary problems. In order to apply the abstract result to concrete free boundary problems, Banach manifold made up of certain kind of domains such as simple domains in R n is a fundamental tool which seems to have not been well-studied in the literature yet. Hence in this paper we also make some basic investigation to a such manifold. In Section 5 we use Nash-Moser implicit function theorem to prove an interesting result for an obstacle problem which says that if the domain Ω of this obstacle problem is a small perturbation of a sphere then its interface Γ is smooth and depends on Ω smoothly. By using these results, in the last section we prove asymptotic stability of radial stationary solution of a free boundary problem modeling the growth of necrotic tumors, which has been kept open for over ten years.

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On the stability of a uniformly rotating viscous incompressible self-gravitating liquid

Cited by 14 publications

References 3 publications

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

On the Banach Manifold of Simple Domains in the Euclidean Space and Applications to Free Boundary Problems

Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems

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