Glasnik Matematicki, Vol. 46, No.1 (2011), 215-231.

THE CAUCHY PROBLEM FOR ONE-DIMENSIONAL FLOW OF A COMPRESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION

Nermina Mujaković and Ivan Dražić

Faculty of Engineering, University of Rijeka, Vukovarska 58, 51 000 Rijeka, Croatia
e-mail: idrazic@riteh.hr

Abstract.   We analyze the Cauchy problem for non-stationary 1-D flow of a compressible viscous and heat-conducting fluid, assuming that it is in the thermodynamical sense perfect and polytropic. This problem has a unique generalized solution on R × ]0,T[ for each T>0. Supposing that the initial functions are small perturbations of the constants and using some a priori estimates for the solution independent of T, we prove a stabilization of the solution.

2000 Mathematics Subject Classification.   46E35, 35B40, 35B45, 76N10.

Key words and phrases.   Compressible viscous fluid, the Cauchy problem, stabilization.

DOI: 10.3336/gm.46.1.16

References:

1. S. N. Antontsev, A. V. Kazhykhov and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland, Amsterdam, 1990.
   MathSciNet    

2. R. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and techonology. Vol. 2, Springer-Verlag, Berlin, 1988.
   MathSciNet    

3. R. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and techonology. Vol. 5, Springer-Verlag, Berlin, 1992.
   MathSciNet    

4. Ya. I. Kanel', Cauchy problem for equations of gas dynamics with viscosity, Sibirsk. Mat. Zh. 20 (1979), 293-306.
   MathSciNet    

5. N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: The Cauchy problem, Math. Commun. 10 (2005), 1-14.
   MathSciNet    

6. N. Mujaković, Uniqueness of a solution of the Cauchy problem for one-dimensional compressible viscous micropolar fluid model, Appl. Math. E-Notes 6 (2006), 113-118.
   MathSciNet