# zbMATH — the first resource for mathematics

$$L^ 1$$ asymptotic behavior of compressible, isentropic, viscous 1-D flow. (English) Zbl 0807.35110
We study the large time behavior in $$L^ 1$$ of the compressible, isentropic, viscous 1-D flow. Under the assumption that the initial data are smooth and small, we show that the solutions are approximated by the solutions of a parabolic system, and in turn by diffusion waves, which are solutions of Burgers equations. Decay rates in $$L^ 1$$ are obtained. Our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35L65 Hyperbolic conservation laws
Full Text:
##### References:
 [1] Chern, Comm. Math. Phys. 110 pp 503– (1987) [2] Chern, Comm. Math. Phys. 120 pp 525– (1989) [3] Hopf, Comm. Pure Appl. Math. 3 pp 201– (1950) [4] Kawashima, Proc. Roy. Soc. Edinburgh Sect. A 106 pp 169– (1987) · Zbl 0653.35066 · doi:10.1017/S0308210500018308 [5] Nonlinear Stability of Shock Waves for viscous Conservation Laws, American Mathematical Society, Providence, 1985. · Zbl 0617.35058 [6] Interactions of nonlinear hyperbolic waves, pp. 171–184 in: Nonlinear Analysis, and , eds., World Scientific, Singapore, Teaneck, New Jersey, 1991. · doi:10.1142/1033 [7] Umeda, Japan J. Appl. Math. 1 pp 435– (1984) [8] Zheng, Scientia Sinica Ser. A 30 pp 1133– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.