Continuous dependence results for quasilinear evolution equations

Abstract

We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), \] we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators $A$ and $f$. An important step for proving the main result is the fact that the maximal regularity constant of the operator $A(t,\ddot{I}•)$, with $t$ and $\ddot{I}•$ fixed, admits a uniform bound over compact subsets of the relevant Banach spaces. As an application, we consider a class of non-Newtonian fluid models with a Carreau-type viscosity and mixed boundary conditions. We show that, as the nonlinear contribution in the viscosity vanishes and the initial data converge, solutions of the non-Newtonian fluid model converge to those of the classical Navier--Stokes equations.