A remark on the Stokes problem in Lorentz spaces
Paolo Maremonti
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.
keywords: Navier-Stokes equations Stokes problem well-posedeness.
A note on the Navier-Stokes IBVP with small data in $L^n$
Paolo Maremonti
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 $q$.
keywords: Navier-Stokes equations well posedeness asymptotic behavior.
On the Stokes problem in exterior domains: The maximum modulus theorem
Paolo Maremonti
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.
keywords: maximum modulus theorem stability. well-posedeness Stokes problem
A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem
Francesca Crispo Paolo Maremonti

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [1], as a corollary, under suitable assumptions of local character on the initial data, we investigate the behavior in time of the $L_{loc}^\infty$-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator.

keywords: Navier-Stokes equations suitable weak solutions partial regularity

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