Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918108851077120 |
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| author | Xuan, Pham Truong Ngoc, Tran Thi |
| author_facet | Xuan, Pham Truong Ngoc, Tran Thi |
| contents | We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with a generalized gravitational field on the framework of non-compact Riemannian manifolds. We work on some manifolds satisfying some bounded and negative conditions on curvature tensors. We consider a couple of Stokes and heat semigroups associated with the corresponding linear system which provides a vectorial matrix semigoup. By using dispersive and smoothing estimates of the vectorial matrix semigroup we establish the global-in-time existence and uniqueness of mild solutions for linear systems. Next, we can pass from the linear system to the semilinear systems to obtain the well-posedness by using fixed point arguments. Moreover, we will prove the exponential stability of such solutions by using Gronwall's inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07803 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds Xuan, Pham Truong Ngoc, Tran Thi Analysis of PDEs Mathematical Physics Differential Geometry Dynamical Systems Functional Analysis We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with a generalized gravitational field on the framework of non-compact Riemannian manifolds. We work on some manifolds satisfying some bounded and negative conditions on curvature tensors. We consider a couple of Stokes and heat semigroups associated with the corresponding linear system which provides a vectorial matrix semigoup. By using dispersive and smoothing estimates of the vectorial matrix semigroup we establish the global-in-time existence and uniqueness of mild solutions for linear systems. Next, we can pass from the linear system to the semilinear systems to obtain the well-posedness by using fixed point arguments. Moreover, we will prove the exponential stability of such solutions by using Gronwall's inequality. |
| title | Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry Dynamical Systems Functional Analysis |
| url | https://arxiv.org/abs/2209.07803 |