On the linear problem arising in the study of a free boundary problem for the Navier-Stokes equations

Author:
V. A. Solonnikov

Original publication:
Algebra i Analiz, tom **22** (2010), nomer 6.

Journal:
St. Petersburg Math. J. **22** (2011), 1023-1049

MSC (2010):
Primary 35Q35, 76D27

Published electronically:
August 22, 2011

MathSciNet review:
2760093

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A problem under study arises as a result of linearization of a free boundary problem for Navier-Stokes equations governing the evolution of an isolated mass of a viscous incompressible capillary liquid.

**1.**V. A. Solonnikov,*Solvability of the problem of evolution of an isolated amount of a viscous incompressible capillary fluid*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**140**(1984), 179–186 (Russian, with English summary). Mathematical questions in the theory of wave propagation, 14. MR**765724****2.**V. A. Solonnikov,*An initial-boundary value problem for a Stokes system that arises in the study of a problem with a free boundary*, Trudy Mat. Inst. Steklov.**188**(1990), 150–188, 192 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 3, 191–239; Boundary value problems of mathematical physics, 14 (Russian). MR**1100542****3.**V. A. Solonnikov,*On the stability of uniformly rotating viscous incompressible self-gravitating liquid*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**348**(2007), no. Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 38, 165–208, 305 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.)**152**(2008), no. 5, 713–740. MR**2743018**, 10.1007/s10958-008-9090-7**4.**Vsevolod Solonnikov,*On problem of stability of equilibrium figures of uniformly rotating viscous incompressible liquid*, Instability in models connected with fluid flows. II, Int. Math. Ser. (N. Y.), vol. 7, Springer, New York, 2008, pp. 189–254. MR**2459267**, 10.1007/978-0-387-75219-8_5**5.**V. A. Solonnikov,*On the linear problem related to the stability of uniformly rotating self-gravitating liquid*, J. Math. Sci. (N. Y.)**144**(2007), no. 6, 4671–4695. Problems in mathematical analysis. No. 35. MR**2584390**, 10.1007/s10958-007-0303-2**6.**O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Lineinye i kvazilineinye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821**

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Linear and quasilinear equations of parabolic type*, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR**0241822****7.**M. S. Agranovič and M. I. Višik,*Elliptic problems with a parameter and parabolic problems of general type*, Uspehi Mat. Nauk**19**(1964), no. 3 (117), 53–161 (Russian). MR**0192188****8.**Gerd Grubb and Vsevolod A. Solonnikov,*Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods*, Math. Scand.**69**(1991), no. 2, 217–290 (1992). MR**1156428****9.**V. A. Solonnikov,*On the stability of uniformly rotating viscous incompressible self-gravitating liquid*, PDMI Preprint 03/2010, 1-101.**10.**J. Thomas Beale and Takaaki Nishida,*Large-time behavior of viscous surface waves*, Recent topics in nonlinear PDE, II (Sendai, 1984) North-Holland Math. Stud., vol. 128, North-Holland, Amsterdam, 1985, pp. 1–14. MR**882925**, 10.1016/S0304-0208(08)72355-7**11.**L. N. Slobodeckiĭ,*Generalized Sobolev spaces and their application to boundary problems for partial differential equations*, Leningrad. Gos. Ped. Inst. Učen. Zap.**197**(1958), 54–112. MR**0203222****12.**M. E. Bogovskiĭ,*Solutions of some problems of vector analysis, associated with the operators 𝑑𝑖𝑣 and 𝑔𝑟𝑎𝑑*, Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149 (Russian). MR**631691****13.**V. A. Solonnikov,*A priori estimates for solutions of second-order equations of parabolic type*, Trudy Mat. Inst. Steklov.**70**(1964), 133–212 (Russian). MR**0162065****14.**M. Padula and V. A. Solonnikov,*On local solvability of a problem with a free boundary for Navier-Stokes equations*, Probl. Mat. Anal., No. 50, Tamara Rozhkovskaya, Novosibirsk, 2010, pp. 87-112; English transl. in J. Math. Sci. (to appear).

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Additional Information

**V. A. Solonnikov**

Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Email:
solonnik@pdmi.ras.ru

DOI:
http://dx.doi.org/10.1090/S1061-0022-2011-01182-3

Keywords:
Navier–Stokes equations,
Laplace–Beltrami operator,
linearization,
free boundary problem

Received by editor(s):
July 8, 2010

Published electronically:
August 22, 2011

Dedicated:
Dedicated to Professor V.M.Babich on the occasion of his 80th birthday

Article copyright:
© Copyright 2011
American Mathematical Society