Osaka Journal of Mathematics

Uniform boundedness of the radially symmetric solutions of the Navier-Stokes equations for isentropic compressible fluids

Abstract

We study the isentropic compressible Navier-Stokes equations with radially symmetric data and non-negative initial density in an annular domain. We prove the global existence of strong solutions for any $\gamma\geq 1$. Moreover, we obtain the uniform in time $L^{\infty}$-boundedness of the density and $H^{1}$-boundedness of the velocity, improving therefore the corresponding result in [2], where the condition $\gamma\geq 2$ is required to guarantee the existence.

Article information

Source
Osaka J. Math., Volume 46, Number 3 (2009), 863-876.

Dates
First available in Project Euclid: 26 October 2009

https://projecteuclid.org/euclid.ojm/1256564210

Mathematical Reviews number (MathSciNet)
MR2583333

Zentralblatt MATH identifier
05644241

Citation

Fan, Jishan; Jiang, Song; Ni, Guoxi. Uniform boundedness of the radially symmetric solutions of the Navier-Stokes equations for isentropic compressible fluids. Osaka J. Math. 46 (2009), no. 3, 863--876. https://projecteuclid.org/euclid.ojm/1256564210

References

• Y. Cho and H. Kim: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math. 120 (2006), 91--129.
• H.J. Choe and H. Kim: Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids, Math. Methods Appl. Sci. 28 (2005), 1--28.
• G.P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Linearized Steady Problem, I, Springer, New York, 1994.
• K. Higuchi: Global existence of the spherically symmetric solution and the stability of the stationary solution to compressible Navier-Stokes equation, Master thesis of Kanazawa Univ. (1992), in Japanese.
• D. Hoff: Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J. 41 (1992), 1225--1302.
• N. Itaya: On a certain temporally global solution, spherically symmetric, for the compressible NS equations, The Jinbun ronshu of Kobe Univ. Commun. 21 (1985), 1--10, in Japanese.
• S. Jiang and P. Zhang: On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys. 215 (2001), 559--581.
• S. Jiang and P. Zhang: Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data, Indiana Univ. Math. J. 51 (2002), 345--355.
• A. Matsumura: Large-time behavior of the spherically symmetric solutions of an isothermal model of compressible viscous gas, Transport Theory Statist. Phys. 21 (1992), 579--592.
• I. Straškraba and A. Zlotnik: Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys. 54 (2003), 593--607.
• R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York, 1988.
• V.A. Weigant: Example of non-existence in the large for the problem of the existence of solutions of Navier-Stokes equations for compressible viscous barotropic fluids, Dokl. Akad. Nauk. 339 (1994), 155--156, in Russian.