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Anisotropic parabolic problems with slowly or rapidly growing terms

100%
Świerczewska-Gwiazda A.
Colloquium Mathematicum
|
2014
|
tom 134
|
nr 1
113-130
EN
We consider an abstract parabolic problem in a framework of maximal monotone graphs, possibly multi-valued, with growth conditions formulated with the help of an x-dependent N-function. The main novelty of the paper consists in the lack of any growth restrictions on the N-function combined with its anisotropic character, namely we allow the dependence on all the directions of the gradient, not only on its absolute value. This leads to using the notion of modular convergence and studying in detail the question of density of compactly supported smooth functions with respect to modular convergence.
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Well-posedness for a class of non-Newtonian fluids with general growth conditions

51%
Gwiazda P., Świerczewska-Gwiazda A., Wróblewska A., Warzyński A.
Banach Center Publications
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2009
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tom 86
|
nr 1
115-128
EN
The paper concerns uniqueness of weak solutions to non-Newtonian fluids with nonstandard growth conditions for the Cauchy stress tensor. We recall the results on existence of weak solutions and additionally provide the proof of existence of measure-valued solutions. Motivated by the fluids of strongly inhomogeneous behaviour and having the property of rapid shear thickening we observe that the described situation cannot be captured by power-law-type rheology. We describe the growth conditions with the help of general x-dependent convex functions. This formulation yields the existence of solutions in generalized Orlicz spaces. These considerations are motivated by e.g. electrorheological fluids, magnetorheological fluids, and shear thickening fluids.
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