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Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

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Abstract

This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u t + f(u) x = u x,x on [0,1], with the boundary condition u(0,t)=u −,u(1t)=u + and the initial data u(x,0)= u 0(x), where u − ≠ u + and f is a given function satisfying f"u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When u − < u +, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u − > u +, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |u − − u +| is small. Moreover, exponential decay rates are both given.

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, 100037, China

    Quansen Jiu*

  2. Department of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Tao Pan**

  3. Faculty of Bioresources, Mie University, Tsu, Mie 514-8507, Japan

    Tao Pan**

Authors
  1. Quansen Jiu*
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  2. Tao Pan**
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Correspondence to Quansen Jiu*.

Additional information

*Partially supported by the National Natural Sciences Foundation of China (No. 10101014), the Key Project of Natural Sciences Foundation of Beijing and Beijing Education Committee Foundation.

**Supported by the National Natural Science Foundation of China (No. 10061001) and Guangxi Natural Science Foundation (No. 9912020).

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Jiu*, Q., Pan**, T. Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval. Acta Mathematicae Applicatae Sinica, English Series 19, 297–306 (2003). https://doi.org/10.1007/s10255-003-0105-3

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  • DOI: https://doi.org/10.1007/s10255-003-0105-3

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