We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for \(3<p\le +\infty \) we prove the existence of weak \(L^p\) -solutions. Moreover, within the framework of a slightly modified system, we discuss the possibly non-existence of \(L^p-\) solutions for \(1\le p \le 3\) . Then, we use the more general setting of the \(L^{p,\infty }-\) spaces to show that weak solutions and their derivatives are Hölder continuous functions, where the maximum gain of regularity is determined by the initial regularity of the external forces and the gravitational acceleration. As a bi-product, we get a new regularity criterion for the steady-state Navier–Stokes equations. Furthermore, in the particular homogeneous case when the external forces are equal to zero; and for a range of values of the parameter p, we show that weak solutions are not only smooth enough, but also they are identical to the trivial (zero) solution. This result is of independent interest, and it is also known as the Liouville-type problem for the steady-state Boussinesq system.
