Applications of Fourier's transform to some resolvent estimates for the Stokes system
by
Senjo Shimizu
Shizuoka University, Hamamatsu, Japan
Coauthors: Yoshihiro Shibata (Waseda University, Tokyo, Japan)

When we analyze the solutions to some linear partial differential equations with constant coefficients, the Fourier transform plays an essential role in order to represent the solution. And then, applying the Fourier multiplier theorem, we obtain the optimal a priori estimates. This is one of the most standard and powerful methods in the theory of partial differential equations.

I will talk about the Lp estimate of solutions to the generalized Stokes resolvent problem in a bounded and exterior domain Rⁿ with Neumann boundary condition. The core of my approach is to estimate the solutions in the whole space and half-space case. Here, the Fourier transform and the Fourier multiplier theorem play a central role.

If we consider a properly elliptic problem, then we can get the required estimate by using the cut-off technique and compact perturbation argument from the estimates for the model problems in the whole space and half-space cases. But, the Stokes equation is a little bit different from the usual properly elliptic system. Namely, we have to handle with the divergence free condition under the cut-off process. This is a characteristic of the Stokes problem. In order to handle with this, we use W^-1p space. This idea is originally due to Farwig and Sohr [1].

The problem was already treated by Grubb and Solonnikov [2, 3]. Their approach heavily depends on the systematic use of theory of pseudo-differential operator to treat the divergence free condition, which does not seem to be simple. But, by introducing the W^-1p, we can reduce the problem to the half-space and the whole space model problem. Therefore the Fourier transform and the Fourier multiplier theorem enable us to solve the problem and to obtain the required estimates of solutions. Compared with Grubb and Solonnikov, our approach is simple and it is not necessary to use any technique of pseudo-differential operators.

References

[1] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643.

[2] G. Grubb and V. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential method, Math. Scand., 69 (1991), 217-290.

[3] G. Grubb, Parameter-elliptic and parabolic pseudodifferential boundary problems in global L_p Sobolev spaces, Math. Z., 218 (1995), 43-90.

Date received: May 23, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahk-91.