Abstract
In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling bypx-power law with Dirichlet boundary condition under the restriction3n/n+2n+2<px<2n+1/n−1. In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction2n+1/n+3<px<2n+1/n−1, which arises from the nonflatness of domain.
Highlights
We consider the steady flows of nonNewtonian fluids in Rn, n = 2, 3, which is modeled by the following system: 8 >>>
Global higher differentiability for weak solutions to the problem (1) with pðxÞ = const have been studied by several authors; for example, see [8,9,10,11,12,13,14,15,16,17,18,19] under the condition f ∈ Lp∧′ðΩÞ with p∧′ = p/ðp − 1Þ and p ≔ min fp, 2g
Global higher differentiability of weak solution to the problem (1) in 3D smooth domain is studied by us in [20] by using a global higher integrability condition, which holds under the condition f ∈ Lαp∧′ðxÞðΩÞ, where p∧′ ðxÞ = ðpðxÞÞ/ðpðxÞ − 1Þ, pðxÞ ≔ min fpðxÞ, 2g, and α > 1
Summary
Global higher differentiability for weak solutions to the problem (1) with pðxÞ = const have been studied by several authors; for example, see [8,9,10,11,12,13,14,15,16,17,18,19] under the condition f ∈ Lp∧′ðΩÞ with p∧′ = p/ðp − 1Þ and p ≔ min fp, 2g It was first established in [8] by Beirao da Veiga. Global higher differentiability of weak solution to the problem (1) in 3D smooth domain is studied by us in [20] by using a global higher integrability condition, which holds under the condition f ∈ Lαp∧′ðxÞðΩÞ, where p∧′ ðxÞ = ðpðxÞÞ/ðpðxÞ − 1Þ, pðxÞ ≔ min fpðxÞ, 2g, and α > 1 This is slightly stronger rather than the standard condition f ∈ Lp∧′ðΩÞ for the case p = const.
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