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On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. (English) Zbl 1050.35080
Let \(d\in\mathbb{N}\), \(d\geq 2\), \(S\) be the space of the symmetric real \(d\times d\) matrices (in \(S\) we denote with \(\bullet\) the inner product of \(\mathbb{R}^{d^2})\), \(C^\infty_{0,\sigma}(\Omega,\mathbb{R}^d)\) the space of \(\Phi\in C_0^\infty (\Omega, \mathbb{R}^d)\) such that \(\text{div}\,\Phi =0\), \(p>1\), \(V_p\) the closure in \(W^{1,p} (\Omega,\mathbb{R}^d)\) of \(C_{0,\sigma}^\infty (\Omega,\mathbb{R}^d)\). Let \(\Omega\) be an open bounded subset of \(\mathbb{R}^d\) with \(\partial\Omega\) of the class \(C^{1, 1}\), \(T:\Omega\times S\to S\) a Caratheodory function such that there are \(c_1>0\) and \(\varphi_1>0\) and \(\varphi_1\in L^1(\Omega)\) such that \(T(x,\eta)\bullet \eta\geq c_1|\eta|^p -\varphi_1(x)\) for almost all \(x\in\Omega\) and for all \(\eta \in S\), there are \(c_2>0\) and \(\varphi_2\in L^{\frac{p}{p-1}} (\Omega)\) such that \(| T(x,\eta) |\leq c_1| \eta|^{p-1}+ \varphi_2(x)\) for almost all \(x\in \Omega\) and for all \(\eta\in S\), \((T(x,\eta)- T(x,\xi))\bullet (\eta-\xi) >0\) for almost all \(x\in\Omega\) and for all \(\eta\), \(\xi\in S\) with \(\eta \neq \xi\); \(f\in W^{-1,p'}(\Omega,\mathbb{R}^d)\).
Theorem. Let \(p>2d/(d+2)\), then there exists \(v\in V_p\) such that \[ \begin{split} \int_\Omega T\biggl(x,\bigl( \nabla v(x)+\nabla v(x)^T \bigr)/2\biggr)\bullet \bigl(\nabla\Phi(x)+ \nabla\Phi(x)^T\bigr)\,dx =\\ 2\langle f,\Phi \rangle_{1,p}+ \int_\Omega(v \otimes v)\bullet\bigl(\nabla \Phi(x)+\nabla \Phi(x)^T \bigr)\,dx \end{split} \] for all \(\Phi\in C^\infty_{0,\sigma} (\Omega,\mathbb{R}^d)\), where \((v\otimes v)_{i,j}=v_iv_j\).

MSC:
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35J70 Degenerate elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
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