Large-Eigenvalue Global Existence and Regularity Results for the Navier–Stokes Equation

Recently Raugel and Sell obtained global existence results for the Navier–Stokes equation requiring that certain products involving the size of the data and the thinness of the domain be small. Thus the initial and forcing data could actually be quite large if the domain was thin enough. These results were obtained for periodic, and a case of homogeneous mixed periodic-Dirichlet, boundary conditions. We develop integral-equation techniques that allow us to obtain similar results in the case of purely homogeneous-Dirichlet boundary conditions. Our results are fairly simple to state and hold in a general setting, whereby we replace the role of the thinness of the domain by the reciprocal of the first eigenvalue of the Laplacian. We show further utulity of the integral-equation techniques by bootstrapping globalH1-bounds, whenever available in 2-dor 3-d, into higher-order global bounds with slightly smoother forcing functions than those assumed by Guillope, but otherwise more general in thatLp-integrable singularities in time are allowed.