A Lagrangian approach for inhomogeneous Boussinesq equations

This paper investigates the Cauchy problem for inhomogeneous Boussinesq equations with the data in the scaling invariant Besov spaces: (θ0,u0)∈M(Ḃp,1n/p−1)×Ḃp,1n/p−1(Rn). We shall prove under some smallness assumption on the data: ‖ν(θ0)−ν(1)‖M(Ḃp,1n/p)+ν_−1‖u0‖Ḃp,1n/p−1 +ν_−1‖θ0‖M(Ḃp,1n/p−1)‖g...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2020-04, Vol.43 (6), p.3556-3568
Main Authors: Huang, Jingchi, Wei, Zhengzhen, Yao, Zheng‐an
Format: Article
Language:English
Subjects:
Boussinesq equations
Cauchy problems
Function space
inhomogeneous boussinesq
lagrangian coordinates
Mapping
Mathematical analysis
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Summary:This paper investigates the Cauchy problem for inhomogeneous Boussinesq equations with the data in the scaling invariant Besov spaces: (θ0,u0)∈M(Ḃp,1n/p−1)×Ḃp,1n/p−1(Rn). We shall prove under some smallness assumption on the data: ‖ν(θ0)−ν(1)‖M(Ḃp,1n/p)+ν_−1‖u0‖Ḃp,1n/p−1 +ν_−1‖θ0‖M(Ḃp,1n/p−1)‖g‖L1(Ḃp,1n/p−1)⩽c, Boussinesq equations with partial viscosity exists a unique global solution, where we extend recent results as regards the conditions for uniqueness. Using Lagrangian coordinates enables us to solve the nonhomogeneous Boussinesq equations by means of the contraction mapping theorem.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6137