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Full regularity of weak solutions to a class of nonlinear fluids in two dimensions—stationary, periodic problem. (English) Zbl 0946.76006
The authors prove the existence of regular solution to a system of nonlinear equations describing steady motions of a class of incompressible non-Newtonian fluids in two dimensions. The non-Newtonian part of the equation of motion is given by \(\text{div}(T(D(v)))\), where \(T\) is nonlinear function with \(p\)-growth and with smooth enough potential. The right-hand side forces are denoted by \(f\). The equations are completed by the requirement that all considered functions are periodic. Starting from the assumption \(f\in L^{p'}\) if \(p\in (1,2)\) or \(f\in L^r\), \(r> 2\) if \(p\geq 2\), the authors show that there exists a weak solution to the problem, which is in fact as regular as data allow. This is a generalization of previously known results, since no restriction on the value of \(p\in (1,\infty)\) is imposed.

76A05 Non-Newtonian fluids
35Q35 PDEs in connection with fluid mechanics
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