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2023, Volume 31, Issue 10: 6505-6524. doi: 10.3934/era.2023329
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Research article

Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion

  • 1.
    School of Mathematics, Jilin University, Changchun 130012, China
  • 2.
    Department of Mathematics, Dalian Minzu University, Dalian 116600, China

  • Academic Editor: Ronghua Pan
  • Received: 12 July 2023 Revised: 27 August 2023 Accepted: 19 September 2023 Published: 09 October 2023

  • We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and diffusion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some diffusion coefficient tending to zero is justified. In addition, the L2 convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer.

    Citation: Xu Zhao, Wenshu Zhou. Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion[J]. Electronic Research Archive, 2023, 31(10): 6505-6524. doi: 10.3934/era.2023329

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  • We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and diffusion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some diffusion coefficient tending to zero is justified. In addition, the L2 convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer.




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