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L2-convergence to nonlinear diffusion waves for Euler equations with time-dependent damping

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Abstract

In this paper, we are concerned with the asymptotic behavior of L∞ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping \( - \frac{m}{{{{(1 + t)}^\lambda }}}\). As \(\lambda \in (0,\tfrac{1}{7}]\), we prove that the L∞ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in \({L^2}(\mathbb{R})\). As \(\lambda \in (\tfrac{1}{7},1)\), we prove that the L∞ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in \({L^2}(\mathbb{R})\), which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.

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Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Shifeng Geng

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Feimin Huang

  3. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, China

    Xiaochun Wu

Authors
  1. Shifeng Geng
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  2. Feimin Huang
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  3. Xiaochun Wu
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Corresponding author

Correspondence to Xiaochun Wu.

Additional information

Dedicated to Professor Banghe LI on the Occasion of his 80th birthday

S. Geng’s research was supported in part by the National Natural Science Foundation of China (12071397) and Excellent Youth Project of Hunan Education Department (21B0165). F. Huang’s research was supported in part by the National Key R&D Program of China 2021YFA1000800 and the National Natural Science Foundation of China (12288201).

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Geng, S., Huang, F. & Wu, X. L2-convergence to nonlinear diffusion waves for Euler equations with time-dependent damping. Acta Math Sci 42, 2505–2522 (2022). https://doi.org/10.1007/s10473-022-0618-6

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  • DOI: https://doi.org/10.1007/s10473-022-0618-6

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