$ L^1 $ local stability to a nonlinear shallow water wave model

Jun Meng; Shaoyong Lai; Jun Meng; Shaoyong Lai

doi:10.3934/era.2024251

Electronic Research Archive

2024, Volume 32, Issue 9: 5409-5423. doi: 10.3934/era.2024251

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$ L^1 $ local stability to a nonlinear shallow water wave model

Jun Meng ^{1
,
,},
Shaoyong Lai ²

1.
School of Mathematics and Statistics, Kashi University, Kashi 844006, China
2.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received: 03 June 2024 Revised: 31 August 2024 Accepted: 06 September 2024 Published: 26 September 2024

A nonlinear shallow water wave equation containing the famous Degasperis$ - $Procesi and Fornberg$ - $Whitham models is investigated. The novel derivation is that we establish the $ L^2 $ bounds of solutions from the equation if its initial value belongs to space $ L^2(\mathbb{R}) $. The $ L^{\infty} $ bound of the solution is derived. The techniques of doubling the space variable are employed to set up the $ L^1 $ local stability of short time solutions.
- local strong solutions,
- bounds of solution,
- shallow water wave model,
- $ L^1 $ local stability
Citation: Jun Meng, Shaoyong Lai. $ L^1 $ local stability to a nonlinear shallow water wave model[J]. Electronic Research Archive, 2024, 32(9): 5409-5423. doi: 10.3934/era.2024251

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Abstract

A nonlinear shallow water wave equation containing the famous Degasperis$ - $Procesi and Fornberg$ - $Whitham models is investigated. The novel derivation is that we establish the $ L^2 $ bounds of solutions from the equation if its initial value belongs to space $ L^2(\mathbb{R}) $. The $ L^{\infty} $ bound of the solution is derived. The techniques of doubling the space variable are employed to set up the $ L^1 $ local stability of short time solutions.

References

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