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$$L_p$$-theory for a class of non-Newtonian fluids. (English) Zbl 1172.35052
In this original long research article, the authors make an attempt to develop $$L_p$$-theory for a class of non-Newtonian fluid flows, and prove many abstract theorems. Based on the maximal regularity results for the generalized Stokes problem, they prove local well-posedness for the nonlinear boundary value problems for the Navier-Stokes equations of the non-Newtonian fluid flows with the help of contraction argument. With homogeneous slip and non-slip boundary conditions, the boundary problems are discussed in great detail. They also prove that the problem is well-posed locally in time, for any space dimension without any restrictions on the size of the initial velocity. The authors also discuss the generalized Stokes problem on $$\mathbb{R}^{n}$$:
$\partial_t \mathbf{u} + A (D) \mathbf{u} + \nabla \pi=0,$ where $$J$$ is a compact interval, and $$A (D) = \sum_{k,l=1}^{n} A^{kl} D_{k} D_{l}$$ is the differential operator with constant coefficient matrices $$A^{kl}$$ acting on $$C^{n}$$-valued functions. They prove an interesting theorem concerning well posedness and then extends this theorem. They also generalize this Stokes problem on $$\mathbb{R}_{+}^{n} = \mathbb{R}^{n-1} \times \mathbb{R}_{+}$$ with different boundary conditions. Included is a detailed mathematical analysis of this problem. This is followed by the nonlinear problem with homogeneous slip and nonslip conditions, and then the same problem with general boundary conditions. The authors also add an appendix with normally elliptic boundary value problems and then summarize some auxiliary results on maximal regularity of completely inhomogeneous vector-valued parabolic initial value problems. Although the paper deals with a rigorous mathematical $$L_p$$-theory, it is too technical for a general audiance.

MSC:
 35Q30 Navier-Stokes equations 76A05 Non-Newtonian fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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