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Maximal regularity and viscous incompressible flows with free interface

Senjo Shimizu — 2008

Banach Center Publications

We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time L p - L q maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.

On the Stokes equation with Neumann boundary condition

Yoshihiro Shibata;  Senjo Shimizu — 2005

Banach Center Publications

In this paper, we study the nonstationary Stokes equation with Neumann boundary condition in a bounded or an exterior domain in ℝⁿ, which is the linearized model problem of the free boundary value problem. Mainly, we prove L p - L q estimates for the semigroup of the Stokes operator. Comparing with the non-slip boundary condition case, we have the better decay estimate for the gradient of the semigroup in the exterior domain case because of the null force at the boundary.

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