Study of the stability of motions of incompressible fluids and the well-posedness of equations that govern their flow

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One of the investigator Tetsuro Miyakawa studied the asymptotic profiles of incompressible viscous flows in the whole space, in the half-space, or in exterior domains. As a result, he showed the following : The asymptotic behavior of flows depends on decaying properties of the initial velocities. However, under the boundary condition that the fluids rest at infinity, the first-order asymptotics of the velocities are given by the heat kernel or its first-order derivative. In particular, as for the flows that decay fast, the first-order derivative of the heat kernel appears in the first-order asymptotics. He also proved that upper bounds of decay rates for flows are actually the same as the ones for their first-order asymptotics. For instance, as for the flows in exterior domains, the net force exerted by the fluids must be zero, if the upper bounds of decay rates are attained by them. This fact corresponds to d'Alembert's paradox for ideal fluids. This study first showed that there exist corresponding solutions also for the equations of viscous fluids that were introduced in order to avoid the paradox. D'Alembert's paradox for ideal fluids arises under general boundary conditions. On the other hand, the paradox for viscous fluids seems to need special symmetry. This may imply the advantage of the theory for viscous fluids in comparison with the theory for ideal fluids.