A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem
Francesca Crispo Paolo Maremonti
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1283-1294 doi: 10.3934/dcds.2017053

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
On the global regularity for nonlinear systems of the $p$-Laplacian type
Hugo Beirão da Veiga Francesca Crispo
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1173-1191 doi: 10.3934/dcdss.2013.6.1173
We are interested in regularity results, up to the boundary, for the second derivatives of the solutions of some nonlinear systems of partial differential equations with $p$-growth. We choose two representative cases: the ''full gradient case'', corresponding to a $p$-Laplacian, and the ''symmetric gradient case'', arising from mathematical physics. The domain is either the ''cubic domain'' or a bounded open subset of $\mathbb{R}^3$ with a smooth boundary. Depending on the model and on the range of $p$, $p<2$ or $p>2$, we prove different regularity results. It is worth noting that in the full gradient case with $p<2$ we cover the singular case and obtain $W^{2,q}$-global regularity results, for arbitrarily large values of $q$. In turn, the regularity achieved implies the Hölder continuity of the gradient of the solution.
keywords: full regularity. $p$-Laplacian systems regularity up to the boundary

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