Abstract
The global existence of solutions to the equations of one-dimensional compressible flow with density-dependent viscosity is proved. Specifically, the assumptions on initial data are that the modulo constant is stated at x=+∞ and x=-∞ which may be different, the density and velocity are in L 2, and the density is bounded above and below away from zero. The results also show that even under these conditions, neither vacuum states nor concentration states can be formed in finite time.
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References
Amosov A A, Zlotnik A A. Solvability “in the large” of a class of quasilinear systems of equations of composite type with nonsmooth data, Differential Equations, 1994, 30:545–558.
Amosov A A, Zlotnik A A. On quasi-averaged equations of the one-dimensional motion of a viscous barotropic medium with rapidly oscillating data, Compute Math Math Phys, 1996, 36:203–220.
Amosov A A, Zlotnik A A. On properties of generalized solutions of one-dimensional linear parabolic problems with nonsmooth coefficients, Differential Equations, 1997, 33:83–96.
Becker E. Gasdynamik, Stuttgart: Teubner Verlag, 1966.
Hoff D. Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc Roy Soc Edinburgh Sect A, 1986, 103:301–315.
Hoff D. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans AMS, 1987, 303(1):169–181.
Hoff D. Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states, Z Angew Math Phys, 1998, 49:774–785.
Simon J. Compact sets in the space L p(0,T,B), Ann Math Pura Appl, 1987, 146:65–96.
Smoller J: Shock Waves and Reaction-diffusion Equations, New York: Springer-Verlag, 1994.
Strakraba I. Zlotnik A. Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity, Z Angew Math Phys, 2003, 54(4):593–697.
Straskraba I, Zlotnik A. Global behavior of 1D-viscous compressible barotropic fluid with a free boundary and large data, J Math Fluid Mech, 2003, 5(2):119–143.
Zel'dovich Y B, Raizer Y P. Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Vol II, New York: Academic Press, 1967.
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Supported by the National Natural Science Foundation of China (10271108).
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Ting, Z. Compressible navier-stokes equations with density-dependent viscosity. Appl. Math.- J. Chin. Univ. 21, 165–178 (2006). https://doi.org/10.1007/BF02791354
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DOI: https://doi.org/10.1007/BF02791354