[Paper Review] A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators

Shunsuke Kurima, Tomomi Yokota|arXiv (Cornell University)|May 2, 2017

Advanced Mathematical Modeling in Engineering8 citations

TL;DR

This paper establishes existence and uniqueness of solutions to quasilinear parabolic equations—specifically porous media and Cahn–Hilliard type systems—on unbounded domains using Brézis's theory for subdifferential operators. By applying the theory directly without compactness methods, it extends prior results to unbounded domains and proves an error estimate of order $\varepsilon^{1/2}$ between the Cahn–Hilliard approximation and the original equation.

ABSTRACT

This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion type equations. The other is the Cahn--Hilliard type system. The present paper applies Brézis theory directly to both equations and gives existence results for these two equations even if the domain is unbounded. Moreover, an error estimate is also proved via apriori estimates obtained directly.

Motivation & Objective

• To extend Brézis's theory for subdifferential operators to quasilinear parabolic equations on unbounded domains, where prior compactness-based methods fail.
• To establish existence and uniqueness of solutions for the porous media-type equation $\frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g$ on unbounded $\Omega$.
• To apply the same theory directly to the Cahn–Hilliard-type system $\frac{\partial u_\varepsilon}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_\varepsilon + \beta(u_\varepsilon) + \pi_\varepsilon(u_\varepsilon)) = g$ on unbounded domains.
• To derive an error estimate between the approximate system $ (P)_\varepsilon $ and the original problem $ (P) $, showing convergence at rate $ \varepsilon^{1/2} $.

Proposed method

• Direct application of Brézis's abstract theory for subdifferential operators to evolution equations of the form $ u'(t) + \partial\psi(u(t)) \ni \tilde{f}(t) $ in Hilbert spaces.
• Use of the operator $ -\Delta + 1 $ on unbounded domains $ \Omega \subset \mathbb{R}^N $ with Neumann boundary conditions.
• Construction of initial data $ u_{0\varepsilon} $ via the resolvent $ (I + \varepsilon(-\Delta+1))^{-1}u_0 $ to satisfy compatibility conditions.
• Verification of structural conditions (C1), (C2), (C4), and (C5) on $ \beta $ and $ \pi_\varepsilon $, including convexity, Lipschitz continuity, and uniform boundedness.
• Use of apriori estimates derived from the subdifferential framework to prove convergence and error control without compactness arguments.
• Application of the theory to both degenerate and non-degenerate diffusion cases via $ \beta(u) = |u|^{q-1}u + u $ with $ q > 0 $.

Experimental results

Research questions

• RQ1 Can Brézis's theory for subdifferential operators be applied directly to quasilinear parabolic equations on unbounded domains?
• RQ2 Does the direct application of Brézis's theory yield existence and uniqueness of solutions for porous media-type equations on unbounded domains?
• RQ3 Can the same framework be extended to Cahn–Hilliard-type systems with $ \varepsilon $-regularization on unbounded domains?
• RQ4 What is the convergence rate of the $ \varepsilon $-regularized solution to the original problem in the unbounded domain setting?
• RQ5 How can apriori estimates be derived directly from the subdifferential structure without compactness methods?

Key findings

• Existence and uniqueness of solutions to the porous media-type equation $ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g $ are established on unbounded domains $ \Omega \subset \mathbb{R}^N $.
• The same result is extended to the Cahn–Hilliard-type system with $ \varepsilon $-regularization, even when $ \Omega $ is unbounded.
• An error estimate of order $ \varepsilon^{1/2} $ is proven between the solution of the approximate system $ (P)_\varepsilon $ and the original problem $ (P) $, matching Colli and Fukao's result in the bounded case.
• The initial data $ u_{0\varepsilon} $ satisfies $ |u_{0\varepsilon} - u_0|_{H^{-1}(\Omega)} \leq \varepsilon^{1/2}|u_0|_{L^2(\Omega)} $, ensuring convergence in the energy space.
• The method avoids compactness arguments, thus overcoming the limitation of previous approaches that required bounded domains.
• The results hold for $ \beta(u) = |u|^{q-1}u + u $ with $ q > 0 $, including both porous media ($ q > 1 $) and fast diffusion ($ 0 < q < 1 $) cases.