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

We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with generalized gravitational field on the non-compact Riemannian manifold (M,g) with dimension d ⩾ 2. The curvature tensors of manifold (M,g) satisfy some bounded and negative conditions. We consider a couple of Stokes (or vectorial heat) semigroup and scalar heat semigroup 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 well-posedness of mild solutions for linear systems in framework of Lp-spaces for p > d. Then, we use results of linear systems and fixed point arguments to establish the well-posedness of solutions for the semilinear systems. Moreover, we establish an exponential stability of such solutions by using Gronwall’s inequality. Besides, in the case of a manifold with dimension d ⩾ 3 and small Ricci curvature, we also establish the well-posedness and stability of mild solutions to the Boussinesq system in the framework of weak-Ld space, i.e., Ld,∞-space.