On weak solutions of nonstationary Boussinesq equations (Q1261637)

Weak solutions of the initial-boundary value problem for the nonstationary Boussinesq equations for a viscous incompressible fluid for a bounded or unbounded exterior domain are studied. The results are (i) an existence and regularity result for a new class of weak solutions with initial velocity in \(L^ 2\) and initial temperature in \(L^ 1\); a posteriori the temperature function is Hölder continuous of order 1/2; (ii) an analogon of uniqueness results for weak solutions of the Navier-Stokes system where one assumes that the solutions are in certain \(L^ p\)-spaces. For the proof of the existence results the author uses a nonstandard approximation scheme introduced by Miyakawa and Sohr; for the deduction of a priori estimates a Nash type procedure is applied to the Green functions of a linearization of the system where discontinuous coefficient functions are allowed; The proof of regularity uses a recent result by Dore and Venni, whereas uniqueness is shown by using \(L^ p- L^ q\) estimates.