Continuous dependence results for quasilinear evolution equations

TL;DR

This paper proves that solutions to certain quasilinear parabolic equations depend continuously on initial data and parameters, with applications to non-Newtonian fluids converging to Navier-Stokes solutions.

Contribution

It establishes uniform bounds for maximal regularity constants and demonstrates continuous dependence for a class of nonlinear evolution equations.

Findings

Solutions depend continuously on initial data and nonlinear operators.

Uniform bounds for maximal regularity constants are established.

Non-Newtonian fluid solutions converge to Navier-Stokes solutions under certain limits.

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 \[ \frac{d\phi}{dt} + A(t,\phi)\phi = f(t,\phi), \] 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 $A(t,\phi)$, with $t$ and $\phi$ 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.