By Michael Ruzicka

This can be the 1st ebook to provide a version, in keeping with rational mechanics of electrorheological fluids, that takes under consideration the complicated interactions among the electromagnetic fields and the relocating liquid. numerous constitutive relatives for the Cauchy tension tensor are mentioned. the most a part of the ebook is dedicated to a mathematical research of a version owning shear-dependent viscosities, proving the lifestyles and area of expertise of vulnerable and powerful options for the regular and the unsteady case. The PDS structures investigated own so-called non-standard development stipulations. life effects for elliptic structures with non-standard development stipulations and with a nontrivial nonlinear r.h.s. and the 1st ever effects for parabolic platforms with a non-standard progress stipulations are given for the 1st time. Written for complicated graduate scholars, in addition to for researchers within the box, the dialogue of either the modeling and the math is self-contained.

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For the remaining terms we proceed as follows. 52) are bounded from below by (cf. 56) - (I~I[(P - 1) ÷ 21/~- I ~ ( p - 2)1)IBEI IBIIEI. 57) where p = p(lEI2). 58) / ( z , y) > c 2 z ~ /'Z for all x E [0, cx~) and y E [0, ~/~x]. 40l51) if ~3 ~ 3 3 ~ < 7(P), -- must be satisfied for all p E ~ooo,p0]. The above condition looks very compact, but both sides of these inequalities and the conditions depend on p E ~o~, P0] and it must be ensured that t h e right-hand side is larger than the left-hand side.

In recent years the interest in the study of elliptic equations and systems with nonstandard growth conditions has increased. Most investigations are concerned with the 2Here we restrict ourselves to this case in order to make the computations not more complicated, but it is clear that the methods can also be applied in the case of two-dimensional domains with appropriate changes. 3Of course in the case Pc¢ = Po we have standard growth conditions. 1. SETTING OF THE PROBLEM AND INTRODUCTION 41 regularity of minimizers of variational integrals and solutions of elliptic and parabolic equations and systems.

55) > o~3iT(pc~) (1 + JD[2)a-~IBI 2 -= Cl(1 + JDJ2) 2e-~JBI 2 is valid, where C1 is independent of B, D and E and 7(:o) = P - 1, i f p E (1, 2), whereas 7(P) = 1 if p >_ 2. For the remaining terms we proceed as follows. 52) are bounded from below by (cf. 56) - (I~I[(P - 1) ÷ 21/~- I ~ ( p - 2)1)IBEI IBIIEI. 57) where p = p(lEI2). 58) / ( z , y) > c 2 z ~ /'Z for all x E [0, cx~) and y E [0, ~/~x]. 40l51) if ~3 ~ 3 3 ~ < 7(P), -- must be satisfied for all p E ~ooo,p0]. The above condition looks very compact, but both sides of these inequalities and the conditions depend on p E ~o~, P0] and it must be ensured that t h e right-hand side is larger than the left-hand side.