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DCDS-S

We study the Stokes initial boundary
value problem with an initial data
in a Lorentz space. We develop a
suitable technique able to solve
the problem and to prove the
semigroup properties of the
resolving operator in the case of
the

*''limit exponents''*. The results are a completion of the ones related to the usual $L^p$-theory, of the ones already known and they are also useful tool to study some questions related to the Navier-Stokes equations.
DCDS-S

We study existence and uniqueness of regular solutions to the
Navier-Stokes initial boundary value problem in
bounded or exterior domains $\Omega$ ($\partial\Omega$ sufficiently smooth) under the assumption $v_\circ$ in $L^n(\Omega)$, sufficiently small, and we prove global in time existence. The results are known in literature (see Remark 3), however the proof proposed here seems shorter, and we give a result concerning the behavior in time of the $L^q$-norm ($q\in[n,\infty]$) of the solutions and of the $L^n$-norm of the time derivative, with a sort of continuous dependence on the data, which, as far as we know, are new, and are

*close*to the ones of the solution to the Stokes problem. Moreover, the constant for the $L^q$-estimate is independent of $q$.
DCDS

We study the Stokes initial boundary value problem, in
$(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exterior
domain, assuming that the initial data belongs to $L^\infty(Ω)$
and has null divergence in weak sense. We prove the maximum modulus
theorem for the corresponding solutions. Crucial for the proof of
this result is the analogous one proved by Abe-Giga for bounded
domains. Our proof is developed by duality arguments and employing
the semigroup properties of the resolving operator defined on
$L^1(Ω)$. Our results are similar to the ones proved by Solonnikov
by means of the potential theory.

DCDS

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [

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