Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 


On the solvability of initial-boundary value problems for a viscous compressible fluid in an infinite time interval

Author: V. A. Solonnikov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 523-546
MSC (2010): Primary 35Q30
Published electronically: March 30, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The solution of the first boundary-value problem for the Navier-Stokes equations is estimated in the case of a compressible fluid in an infinite time interval; the solvability of the problem is proved, together with the exponential decay of the solution as $ t\to \infty $. The proof is based on the ``free work'' method due to Prof. M. Padula. It is shown that the method is applicable to the analysis of free boundary problems.

References [Enhancements On Off] (What's this?)

  • 1. Mariarosaria Padula, On the exponential stability of the rest state of a viscous compressible fluid, J. Math. Fluid Mech. 1 (1999), no. 1, 62–77. MR 1699019, 10.1007/s000210050004
  • 2. Mariarosaria Padula, Asymptotic stability of steady compressible fluids, Lecture Notes in Mathematics, vol. 2024, Springer, Heidelberg, 2011. MR 2848254
  • 3. Irina Vlad. Denisova, On energy inequality for the problem on the evolution of two fluids of different types without surface tension, J. Math. Fluid Mech. 17 (2015), no. 1, 183–198. MR 3313115, 10.1007/s00021-014-0197-y
  • 4. O. A. Ladyženskaja and V. A. Solonnikov, Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 81–116, 256 (Russian, with English summary). Boundary value problems of mathematical physics and related questions in the theory of functions, 9. MR 0467031
  • 5. 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
  • 6. Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
  • 7. K. K. Golovkin, On equivalent normalizations of fractional spaces, Trudy Mat. Inst. Steklov. 66 (1962), 364–383 (Russian). MR 0154028
  • 8. V. I. Smirnov, A course of higher mathematics. Vol. IV [Integral equations and partial differential equations], Translated by D. E. Brown; translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. MR 0177069
  • 9. 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
  • 10. M. Padula and V. A. Solonnikov, On the free boundary problem of magnetohydrodynamics, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), no. Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 41, 135–186, 236 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 178 (2011), no. 3, 313–344. MR 2749373, 10.1007/s10958-011-0550-0
  • 11. Irina V. Denisova, Global 𝐿₂-solvability of a problem governing two-phase fluid motion without surface tension, Port. Math. 71 (2014), no. 1, 1–24. MR 3194642, 10.4171/PM/1938
  • 12. V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with a surface tension, Constantin Carathéodory: an international tribute, Vol. I, II, World Sci. Publ., Teaneck, NJ, 1991, pp. 1270–1303. MR 1130887
  • 13. Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 337–342. MR 555060
  • 14. Akitaka Matsumura and Takaaki Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. MR 713680
  • 15. Gerhard Ströhmer, About the resolvent of an operator from fluid dynamics, Math. Z. 194 (1987), no. 2, 183–191. MR 876229, 10.1007/BF01161967
  • 16. Gerhard Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. Math. 143 (1990), no. 2, 359–375. MR 1051082
  • 17. W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. (Rozprawy Mat.) 324 (1993), 101. MR 1218047
  • 18. W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface, SIAM J. Math. Anal. 25 (1994), no. 1, 1–84. MR 1257142, 10.1137/S0036141089173207
  • 19. Yuko Enomoto and Yoshihiro Shibata, On the ℛ-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcial. Ekvac. 56 (2013), no. 3, 441–505. MR 3157151, 10.1619/fesi.56.441
  • 20. 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

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35Q30

Retrieve articles in all journals with MSC (2010): 35Q30

Additional Information

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

Keywords: Navier--Stokes equations, viscosity, anisotropic Sobolev--Slobodetski spaces
Received by editor(s): December 2, 2014
Published electronically: March 30, 2016
Additional Notes: The author is thankful to E. V. Frolova, W. M. Zajaczkowski, and V. Kalantarov for useful suggestions and discussions. The work was partially supported by the EU project FLUX 319012
Dedicated: Dedicated to Nina Nicolaevna Ural’tseva with great admiration
Article copyright: © Copyright 2016 American Mathematical Society