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

Xu Zhao; Wenshu Zhou; Xu Zhao; Wenshu Zhou

doi:10.3934/era.2023329

Electronic Research Archive

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

Xu Zhao ¹,
Wenshu Zhou ^{2
,
,}

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 $ L^2 $ convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer.
- Hyperbolic system,
- global large solution,
- vanishing diffusion limit,
- boundary layer,
- BL-thickness
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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Abstract

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 $ L^2 $ convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer.

References

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