In these notes we will present (a part of) the parabolic tent spaces theory and then apply it in solving some PDEs originated from the fluid mechanics. In more details, to our most interest are the incompressible homogeneous Navier-Stokes equations.
These equations have been investigated mathematically for almost one century. Yet, the question of proving well-posedness (i.e. existence, uniqueness and regularity of solutions) lacks satisfactory answer. A large part of the known positive results in connection with Navier-Stokes equations are those in which the initial data $u_0$ is supposed to have a small norm in some critical or scaling invariant functional space. All those spaces are embedded in the homogeneous Besov space $dot B^{-1}_{infty,infty}$. A breakthrough was made in the paper [16] by Koch and Tataru, where the authors showed the existence and the uniqueness of solutions to the Navier-Stokes system in case when the norm $|u_0|_{mathrm{BMO}^{-1}}$ is small enough. The principal goal of these notes is to present a new proof of the theorem by Koch and Tataru on the Navier-Stokes system, namely the one using the tent spaces theory. We also hope that after having read these notes, the reader will be convinced that the theory of tent spaces is highly likely to be useful in the study of other equations in fluid mechanics. These notes are mainly based on the content of the article [1] by P. Auscher and D. Frey. However, in [1] the authors deal with a slightly more general system of parabolic equations of Navier-Stokes type. Here we have chosen to write down a self-contained text treating only the relatively easier case of the classical incompressible homogeneous Navier-Stokes equations.
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with
area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.
We prove a number of textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Diri
chlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability a la {v{S}}ne{ui}berg. Another one is stability of solvability in perturbing the coefficients in $L^infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.
We give a new proof of a well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in $R^n$ with small initial data in $BMO^{-1}(R^n)$. The proof is formulated operator theoretically and does not make use of self-adjointness of the Laplacian.
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in approp
riate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.
We show that, under general conditions, the operator $bigl (- abla cdot mu abla +1bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Omega)$ and $L^p(Omega)$, for $p in {]1,2[}$ if one presupposes that
this isomorphism holds true for $p=2$. The domain $Omega$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $overline {partial Omega setminus D}$ the existence of bi-Lipschitzian boundary charts is required.
The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $RR^n$. The analysis interestingly combines use of Poincar
e inequalities and of some Hardy type inequalities.
