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Continuous dependence results for quasilinear evolution equations

Francesco Cellarosi, Anirban Dutta, Giusy Mazzone

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 $A(t,φ)$, with $t$ and $φ$ 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.

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 -regularity. For equations of the form we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators and . An important step for proving the main result is the fact that the maximal regularity constant of the operator , with and 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.
Paper Structure (5 sections, 6 theorems, 101 equations)

This paper contains 5 sections, 6 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Main abstract theorem on continuous dependence for strong solutions
  4. Viscosity limits of non-Newtonian fluid models
  5. Embedding-type theorem

Key Result

Lemma 3.1

Let $A_\infty: [t_0, \infty) \times X_p \rightarrow \mathcal{L}(X_1, X_0)$ be a continuous mapping that satisfies assumptions existence_reason1 and existence_reason4. Fix $R>0$. Let $t\in [t_0, t_0+R]$, and $u \in X_p$, and denote by $c_{\mathrm{M}}(t,u):=C(A_{\infty}(t,u);[t_0, t_0+R])$ the maximal is continuous.

Theorems & Definitions (16)

  • Definition 2.1: Maximal $L^p$-regularity
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 6 more