By Frédéric Alliot, Cherif Amrouche (auth.), Adélia Sequeira, Hugo Beirão da Veiga, Juha Hans Videman (eds.)
This e-book is intended as a gift to honor Professor at the th party of his 70 birthday. It collects refereed contributions from sixty-one mathematicians from 11 nations. They disguise many alternative parts of analysis relating to the paintings of Professor together with Navier-Stokes equations, nonlinear elasticity, non-Newtonian fluids, regularity of strategies of parabolic and elliptic difficulties, operator idea and numerical tools. the belief of this e-book couldn't were made attainable with out the beneficiant aid of Centro de Matemática Aplicada (CMA/IST) and Fundação Calouste Gulbenkian. distinctive thank you are because of Dr. Ulrych for the cautious practise of the ultimate model of this ebook. final yet now not least, we want to convey our gratitude to Dr. for her priceless the help of the very starting. This venture couldn't were effectively concluded with out her enthusiasm and loving deal with her father. On behalf of the editors ADÉLIA SEQUEIRA v venerated by way of the Order of advantage of the Czech Republic by means of Václav Havel, President of the Czech Republic, at the October 28, 1998, Professor Emeritus of arithmetic on the Charles college in Prague, Presidential learn Professor on the Northern Illinois collage and health practitioner Honoris Causa on the Technical collage of Dresden, has been enriching the Czech and international arithmetic together with his new rules within the components of partial differential equations, nonlinear useful research and purposes of the either disciplines in continuum mechanics and hydrodynamics for greater than 40 years.
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One way to achieve this is through adopting a gradient structure for which is motivated by the diffusive terms of molecular viscosity in the equations; an artificial viscosity is introduced both to simulate the intensified momentum transfer in turbulence and to dampen the fluctuation as intended by the averaging. This scheme has been widely used in turbulence modeling, such as the Smagorinsky eddy viscosity model in LES (Smagorinsky ), the Boussinesq assumption concerning the Reynolds stress in zero-equation models and two-equation models and the flux form in Reynolds stress equation models in engineering turbulence modeling (Rodi ).
Definition 2. A function if is said to belong to - regularity for nonlinear elliptic systems of second order and by we denote subspace of all 35 such that as where Some basic properties of above mentioned spaces are formulated in the following proposition (for the proofs see , ,  and ). Proposition 1. For a domain following (i) of the class we have the is isomorphic to the (ii) and (iii) then is a Banach space with norm (iv) and are not empty. (v) For and set Then we have for each 3.
Amrouche C. 6) yield that is bounded in which is reflexive. Therefore, extracting subsequences if necessary, we have: Let us now check that u is a weak solution. 7), we can pass to limit in the first integral. Moreover, extracting a subsequence if necessary, we know that converges strongly to u in since the imbedding is compact. 1). 1) and from a well-known consequence of a very general theorem of G. de Rham. Moreover, is unique up to a constant because is connected. Besides, the local regularity of can be deduced from standard local properties of the distribution and from a result of L.