This memoir is devoted to endpoint maximal regularity in Besov spaces

for the evolutionary Stokes system in bounded or exterior domains of Rⁿ.

We strive for time independent a priori estimates with L ₁ time integrability.

In the whole space case, endpoint maximal regularity estimates are

well known and have proved to be spectacularly powerful to investigate

the well-posedness issue of PDEs related to fluid mechanics. They have

been extended recently by the authors to the half-space setting [ 15 ].

The present work deals with the bounded and exterior domain cases.

Although in both situations the Stokes system may be localized and

reduced up to low order terms to the half-space and whole space cases,

the exterior domain case is more involved owing to a bad control on

the low frequencies of the solution (no Poincaré inequality is available

whatsoever). In order to glean some global-in-time integrability, we adapt

to the Besov space setting the approach introduced by P. Maremonti

and V.A. Solonnikov in [ 39 ]. The price to pay is that we end up with

estimates in intersections of Besov spaces, rather than in a single Besov

space.

As a first application of our work, we solve locally for large data or globally

for small data, the (slightly) inhomogeneous incompressible Navier-Stokes

equations in critical Besov spaces, in an exterior domain. After

observing that the L ₁ time integrability allows to determine globally the

streamlines of the flow, the whole system is recast in the Lagrangian coordinates

setting. This, in particular, enables us to consider discontinuous

densities, as in [ 17 ], [ 19 ].

The second application concerns a low Mach number system that has

been studied recently in the whole space setting by the first author and

X. Liao [ 14 ].