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Inverse problems for the Boussinesq system

We obtain two results on inverse problems for a 2D Boussinesq system. One is that we prove the Lipschitz stability for the inverse source problem of identifying a time-independent external force in the system with observation data in an arbitrary sub-domain over a time interval of the velocity and the data of velocity and temperature at a fixed positive time t0 > 0 over the whole spatial domain. The other one is that we prove a conditional stability estimate for an inverse problem of identifying the two initial conditions with a single observation on a sub-domain.