Abstract
In the following we consider a class of non-linear systems that covers some well known generalized Navier-Stokes systems with shear dependent viscosity of power law type, p < 2 . We show that weak solutions to our class of systems have integrable gradient up to the boundary, with any finite exponent. This result extends, up the the boundary, some of the interior regularity results known in the literature for systems of the above power law type. Boundedness of the gradient, up to the boundary, remains an open problem.
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References
Apushkinskaya D., Bildhauer, M., Fuchs, M.: Steady states of anisotropic generalized Newtonian fluids. J. Math. Fluid Mech. 7, 261–297 (2005)
Bildhauer, M.: Convex Variational Problems. Lecture Notes in Mathematics, 1818. Springer-Verlag, Berlin (2003)
Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)
Crispo, F.: A note on the global regularity of steady flows of generalized Newtonian fluids. Portugaliae Math. 66, 211–223 (2009)
Fuchs, M.: Regularity for a class of variational integrals motivated by nonlinear elasticity. Asymp. Analysis 9, 23–38 (1994)
Fuchs, M.: Seregin G.: Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids. Lecture Notes in Mathematics, 1749. Springer-Verlag, Berlin (2000)
Kaplický, P., Máalek, J., Stará, J.: C 1 α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem. POMI, 259 , 89–121 (1999)
Ružičza M.: Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. of Math. 49, 565–609 (2004)
Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Soviet Math. Dokl. 2 , 746–749 (1961)
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da Veiga, H.B. (2010). On the Global Integrability for Any Finite Power of the Full Gradient for a Class of Generalized Power Law Models p < 2. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_3
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DOI: https://doi.org/10.1007/978-3-642-04068-9_3
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