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arXiv · 2209.07803

Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds

Pham Truong Xuan·Tran Thi Ngoc
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Abstract

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.

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Pham Truong Xuan, Tran Thi Ngoc. 2022-09-16. Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds. https://doi.org/10.1063/5.0238189

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