On the differentiability of weak solutions of a degenerate system of PDE's in fluid mechanics
Weak solutions, in problems of fluid mechanics, are generalized solutions usually constructed in order to describe the behaviour of phenomena such as discontinuities, for example. In the present paper, the author studies the differentiability of weak solutions for a nonnewtonian fluid (here, a pseudo-plastic fluid). Namely, the author derives a theorem setting that, under some restrictions about the deviatoric stress-tensor, then for any weak solution of the incompressible equations of motion, the velocity u possesses second-order derivatives which satisfy the properties: \[ u_{i,k\ell}\in L^{\sigma}_{loc}(\Omega),\quad (1+\| D(u)\|^ 2)^{(\sigma -2)/4} D_{ij}(u,k)\in L^ 2_{loc}(\Omega), \] \(\sigma\) being a real number related to constitutive law of the fluid. The paper deals only with mathematical results: an application to realistic materials and a physical interpretation of the results would be wellcome.
scientific article; zbMATH DE number 4086164