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On integrability up to the boundary of the weak solutions to a non-Newtonian fluid. (English) Zbl 1499.35499

Summary: This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equations as \[ \mathbb{S} = -P(\varrho) + 2\mu_0(1 + |\mathbb{D}^d(\mathbf{u})|^2)^{(p - 2)/2} \mathbb{D}^d(\mathbf{u}) + \frac{c\operatorname{div}\mathbf{u}}{(1 - c^a|\operatorname{div}\mathbf{u}|^a)^{1/a}}\mathbb{I}. \] The existence result of weak solutions can be get based on Galerkin approximation. With the linear operator \(\mathcal{B}\) constructed by M. E. Bogovskij [Tr. Semin. S. L. Soboleva 1, 5–40 (1980; Zbl 0479.58018)], we show that the density \(\varrho\) is square integrable up to the boundary.

MSC:

35Q35 PDEs in connection with fluid mechanics
35D30 Weak solutions to PDEs
76A05 Non-Newtonian fluids
76N99 Compressible fluids and gas dynamics

Citations:

Zbl 0479.58018
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References:

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