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

其中一位研究者Tetsuro Miyakawa研究了不可压缩粘性流在整个空间、半空间和外部域的渐近分布。结果，他证明了以下几点：流动的渐近行为取决于初始速度的衰减性质。然而，在流体停留在无穷远处的边界条件下，速度的一阶渐近性由热核或其一阶导数给出。特别地，对于衰减快的流，热核的一阶导数出现在一阶渐近中。他还证明了流的衰减率的上界实际上与一阶渐近的上界相同。例如，对于外部域的流动，如果流体达到衰减率的上界，则流体施加的净力必须为零。这一事实符合理想流体的达朗贝尔悖论。本研究首先证明了为避免悖论而引入的粘性流体方程也存在相应的解。理想流体的达朗贝尔悖论是在一般边界条件下产生的。另一方面，粘性流体的悖论似乎需要特殊的对称性。这可能意味着与理想流体理论相比，粘性流体理论具有优势。

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.