This paper aims to study the so-called Boussinesq system in critical function spaces over bounded Lipschitz and C1,α domains of IR³. We prove existence for small times and arbitrary data, or globally-in-time for small initial data. This model describes a heated fluid flow, whose velocity transport the temperature, and the temperature acts then as a force on the system. Two different frameworks are considered.

The first framework concerns an Lp-theory with velocity in L2(W1,3) and temperature in L2(L3/2), taking advantage of L2-maximal regularity, and where one has to assume C1,α regularity for the boundary. In this case, one obtains uniqueness even in the class of weak solutions.

The second framework has to do with existence and uniqueness in Besov spaces Bsp,1 by L1-maximal regularity. One obtains existence and uniqueness in the class of mild solutions for arbitrary bounded Lipschitz domains. Again, when the domain has C1,α regularity for the boundary, one obtains uniqueness in the class of weak solutions.

In both cases, we show that for large times, the fluid velocity and the temperature stabilize exponentially to an equilibrium. The limit temperature is then the averaged initial temperature of the fluid, which is physically very relevant.

In order to provide such results -which conceptually combine rough bounded domains and critical function spaces- one has to provide a deep preparation from the side of the linear theory. Both frameworks rely then on operator theoretic arguments and product rules in Besov spaces, even in the case of the Lp-theory! In this case new additional knowledge for the regularity of the Stokes operator and Neumann Laplacian on Lipschitz domains was needed. Fortunately, once well prepared, this linear theory and such appropriate product rules simplify notably previous similar approaches for non-linear estimates of the system.

This memoir focuses on the elliptic and parabolic regularity of the Stokes problem in low-regularity domains, providing an essentially optimal description of gains of regularity. It also includes a treatment of pathological spaces such as the space of bounded functions. All the classical properties are studied (such as the bounded holomorphic functional calculus of the Stokes operator and the description of the domain of its fractional powers), relying primarily on an analysis of the resolvent problem through the method of Sobolev multipliers introduced by Maz’ya and Shaposhnikova. This work is intended to be essentially exhaustive and treats, as a particular case, domains with Hölder regularity arbitrarily close to C¹, and even certain domains whose boundary regularity lies between C¹ and Lipschitz in a certain sense.

Surprisingly, we obtain new results even in the case of the flat half-space. For example, we provide an explicit characterization of the generator of the Stokes semigroup on L∞ which had been an open problem for more than 20 years. In the case of domains with Hölder-continuous boundary, we also establish L¹-decay results for the semigroup.

Moreover, the method deployed here is completely general and can be applied as is to a large class of elliptic operators with low-regularity coefficients and subject to various boundary conditions. Compared to most comparable results in the literature, our method essentially saves one derivative in terms of the boundary regularity required for the domain.

This document also provides a largely comprehensive toolkit for tackling the main problems of incompressible viscous fluid mechanics in low-regularity domains, possibly unbounded.

This article focuses on a construction of homogeneous Sobolev and Besov spaces on rough bent half spaces for which an optimal trace theorem is proved. The particular cases p = 1 and p = ∞ are also treated. Results on real interpolation, density, and even, to some extent, duality results are no longer contingent on the completeness of the spaces involved. The construction and analysis aim to be almost exhaustive (with a few technical remnants), and consequently, and unfortunately, the lack of completeness makes certain proofs quite technical and not so much accessible.

Let us note that, to my knowledge, and up to this day, even the case of smooth half spaces appeared to be untreated in the literature.

Soon.

This article focuses on the global-in-time regularity of abstract linear parabolic problems in Banach spaces, where the operator playing the role of the Laplacian in the heat equation is not necessarily invertible. Here, the Lebesgue space is replaced by a Sobolev space. In order to preserve global-in-time control, the Sobolev space must be homogeneous, and the abstract spaces corresponding to the spatial variable must also have a “homogeneous version.” This implies that, in concrete applications, the normed spaces involved may no longer be complete nor admit a meaningful completion, which the abstract theory must take into account. This is reflected even in the description of the set of admissible initial data. The main standard result for the general theory, especially concerning the choice of initial data, even in the case of Lq spaces for non-invertible operators, did not appear to be well understood until now. 

This paper provides an elementary construction of homogeneous Sobolev and Besov spaces on the flat half space, removing ambiguities in the definition of certain quantities or notions. For example, one can define the trace on the boundary and use well-defined product laws, but this comes at the cost of losing completeness for spaces of high regularity. The framework is mostly restricted to essentially reflexive spaces, but some results can dispense with the notion of completeness under certain conditions. This construction extends the one initiated by H. Bahouri, J.-Y. Chemin, and R. Danchin.