Existence of weak solutions for unsteady motions of generalized Newtonian fluids
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, p. 1-46
We prove the existence of weak solutions $𝐮:{Q}_{T}\to {ℝ}^{n}$ of the equations of unsteady motion of an incompressible fluid with shear-dependent viscosity in a cylinder ${Q}_{T}=\Omega ×\left(0,T\right)$, where $\Omega \subset {ℝ}^{n}\phantom{\rule{0.166667em}{0ex}}$ denotes a bounded domain. Under the assumption that the extra stress tensor $𝐒$ possesses a $q$-structure with $q>\frac{2n}{n+2}$, we are able to construct a weak solution $𝐮\in {L}^{q}\left(0,T;{W}_{0}^{1,q}\left(\Omega \right)\right)\cap {C}_{w}\left(\left[0,T\right];{L}^{2}\left(\Omega \right)\right)$ with $div𝐮=0$. Our approach is based on the Lipschitz truncation method, which is new in this context.
Classification:  76D03,  35D05,  35D46,  34A34
@article{ASNSP_2010_5_9_1_1_0,
author = {Diening, Lars and R\r u\v zi\v cka, Michael and Wolf, J\"org},
title = {Existence of weak solutions for unsteady motions of generalized Newtonian fluids},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {1},
year = {2010},
pages = {1-46},
zbl = {1253.76017},
mrnumber = {2668872},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_1_1_0}
}

Diening, Lars; Růžička, Michael; Wolf, Jörg. Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 1-46. http://www.numdam.org/item/ASNSP_2010_5_9_1_1_0/

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