Hiroshima Mathematical Journal

The initial value problem for motion of incompressible viscous and heat-conductive fluids in Banach spaces

Ryôhei Kakizawa

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Abstract

We consider the abstract initial value problem for the system of evolution equations which describe motion of incompressible viscous and heat-conductive fluids in a bounded domain. It is difficulty of our problem that we do not neglect the viscous dissipation function in contrast to the Boussinesq approximation. This problem has uniquely a mild solution locally in time for general initial data, and globally in time for small initial data. Moreover, a mild solution of this problem can be a strong or classical solution under appropriate assumptions for initial data. We prove the above properties by the theory of analytic semigroups on Banach spaces.

Article information

Source
Hiroshima Math. J., Volume 40, Number 3 (2010), 371-402.

Dates
First available in Project Euclid: 8 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1291818851

Digital Object Identifier
doi:10.32917/hmj/1291818851

Mathematical Reviews number (MathSciNet)
MR2766668

Zentralblatt MATH identifier
1223.35258

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35K90: Abstract parabolic equations 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

Keywords
Incompressible viscous and heat-conductive fluids abstract initial value problem analytic semigroups on Banach spaces

Citation

Kakizawa, Ryôhei. The initial value problem for motion of incompressible viscous and heat-conductive fluids in Banach spaces. Hiroshima Math. J. 40 (2010), no. 3, 371--402. doi:10.32917/hmj/1291818851. https://projecteuclid.org/euclid.hmj/1291818851


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References

  • R. A. Adams and J. J. F. Fournier, Sobolev Spaces (Second Edition), Academic Press, 2003.
  • J. Boussinesq, Théorie Analytique de la Chaleur. II, Gauthier-Villars, 1903.
  • H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Ration. Mech. Anal. 16 (1964), 269--315.
  • D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 24 (1977), 685--700.
  • Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ spaces, Math. Z. 178 (1981), 297--329.
  • Y. Giga, Domains of fractional powers of the Stokes operator in $L_r$ spaces, Arch. Ration. Mech. Anal. 89 (1985), 251--265.
  • Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal. 89 (1985), 267--281.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics 840), Springer-Verlag, 1981.
  • T. Hishida, Existence and regularizing properties of solutions for the nonstationary convection problem, Funkcial. Ekvac. 34 (1991), 449--474.
  • Y. Kagei and M. Skowron, Nonstationary flows of nonsymmetric fluids with thermal convection, Hiroshima Math. J. 23 (1993), 343--363.
  • Y. Kagei, Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function, Hiroshima Math. J. 25 (1995), 251--311.
  • H. Lamb, Hydrodynamics (Sixth Edition), Cambridge University Press, 1932.
  • G. Łukaszewicz and P. Krzyżanowski, On the heat convection equations with dissipation term in regions with moving boundaries, Math. Methods Appl. Sci. 20 (1997), 347--368.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Applied Mathematical Sciences 44), Springer-Verlag, 1983.
  • J. Serrin, Mathematical Principles of Classical Fluid Mechanics (Fluid Dynamics I, Encyclopedia of Physics VIII/1), Springer-Verlag, 1959.
  • Y. Shibata and R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition, J. Math. Soc. Japan 59 (2007), 469--519.