Critical functional framework and maximal regularity in action on systems of incompressible flows

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 L₁ 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 L₁ 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].