Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 09 Dec 2019 10:40:07 GMT2019-12-09T10:40:07Z5021Local Hölder continuity for doubly nonlinear parabolic equationshttp://hdl.handle.net/10316/13697Title: Local Hölder continuity for doubly nonlinear parabolic equations
Authors: Kuusi, Tuomo; Siljander, Juhana; Urbano, José Miguel
Abstract: We give a proof of the Hölder continuity of weak solutions of certain degenerate
doubly nonlinear parabolic equations in measure spaces. We only assume
the measure to be a doubling non-trivial Borel measure which supports a Poincaré
inequality. The proof discriminates between large scales, for which a Harnack inequality
is used, and small scales, that require intrinsic scaling methods.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/136972010-01-01T00:00:00ZOn the interior regularity of weak solutions to the 2-D incompressible Euler equationshttp://hdl.handle.net/10316/44405Title: On the interior regularity of weak solutions to the 2-D incompressible Euler equations
Authors: Siljander, Juhana; Urbano, José Miguel
Abstract: We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result \begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned} for weak solutions in the energy space L_t^\infty L_x^2, satisfying appropriate vorticity estimates. We also obtain improved integrability for the vorticity—which is to be compared with the DiPerna–Lions assumptions. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier–Stokes equations in any dimension.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/444052017-01-01T00:00:00Z