The Periodic Initial Value Problem and Initial Value Problem for the Modified Boussinesq Approximation

Abstract

" The Boussinesq approximation, where the viscosity depends polynomially on the shear rate,finds more and more frequent use in geological practice. In this paper, we consider the periodic initial value problem and initial value problem for this modified Boussinesq approximation with the viscous part of the stress tensor \u03c4^v = \u03c4(e)- 2\u03bc\u0394e, where the nonlinear function \u03c4(e) satisfies \u03c4\u001c_{ij}(e)e_{ij} \u2265 C|e|^p or \u001c\u03c4_{ij}(e)e_{ij} \u2265 C(|e|\u00b2+|e|^p). The existence, uniqueness and regularity of the weak solution is proved for p > \\frac{2n}{n + 2}."