[Paper Review] Stability of solutions to nonlinear diffusion equations

Teemu Lukkari|arXiv (Cornell University)|Jun 12, 2012

Advanced Mathematical Modeling in Engineering12 references3 citations

TL;DR

This paper establishes the stability of solutions to nonlinear diffusion equations—specifically the porous medium and fast diffusion equations—under perturbations in the nonlinearity exponent $ m $, using elementary estimates based on weak solutions and compactness. It proves that solutions converge in $ L^{1+m} $ norm as $ m_i \to m $, even across degenerate ($ m>1 $) and singular ($ m<1 $) regimes, without relying on regularity theory or semigroup methods.

ABSTRACT

We prove stability results for nonlinear diffusion equations of the porous medium and fast diffusion types with respect to the nonlinearity power $m$: solutions with fixed data converge in a suitable sense to the solution of the limit problem with the same data as $m$ varies. Our arguments are elementary and based on a general principle. We use neither regularity theory nor nonlinear semigroups, and our approach applies to e.g. Dirichlet problems in bounded domains and Cauchy problems on the whole space.

Motivation & Objective

• To establish the stability of solutions to nonlinear diffusion equations with respect to variations in the nonlinearity power $ m $, particularly in the supercritical range $ m > m_c = (n-2)_+/n $.
• To provide a stability result that holds for both degenerate ($ m>1 $) and singular ($ m<1 $) cases, including the limiting case $ m=1 $ (heat equation), without requiring $ m \geq 1 $ or $ m \leq 1 $.
• To avoid reliance on advanced tools such as nonlinear semigroup theory, Hölder continuity, or Harnack inequalities, instead using only basic $ L^2 $-based estimates and compactness.
• To extend stability results to initial data that are positive measures of finite mass, such as the Barenblatt solution, which has a Dirac initial trace.
• To provide a direct, elementary proof of convergence in $ L^{1+m} $ norm for both Dirichlet problems on bounded domains and Cauchy problems on $ \mathbb{R}^n $.

Proposed method

• Uses the weak formulation of the equation $ \partial_t u - \Delta u^m = 0 $, where $ u^m $ and $ \nabla u^m $ are in $ L^2 $, enabling $ L^2 $-based energy estimates.
• Applies a compactness principle: locally uniformly bounded sequences of weak solutions admit pointwise a.e. convergent subsequences.
• Employs a test function $ \eta(x,t) = \int_t^T (u^m - u_i^{m_i}) \, ds $ to compare solutions $ u $ and $ u_i $ with different $ m $, leading to energy-type estimates.
• Uses the inequality $ |a-b|^{1+m} \leq c (a^m - b^m)(a-b) $ for $ a,b > 0 $, which holds for $ m > \frac{n-2}{n+2} $, to control the difference in solutions.
• Applies Hölder’s inequality and a pointwise estimate $ |u_i^{m_i} - u_i^m| \leq c(1 + u_i^{m_i+\varepsilon} + u_i^{m+\varepsilon})|m - m_i| $ to bound the difference in nonlinearities.
• Combines Sobolev embedding and energy estimates to obtain uniform $ L^1 $ bounds on $ u_i^{m_i(1+\varepsilon/m_i)(1+1/m)} $, ensuring integrability for small $ \varepsilon $.

Experimental results

Research questions

• RQ1 Does the solution to the nonlinear diffusion equation $ \partial_t u - \Delta u^m = 0 $ depend continuously on the exponent $ m $, especially across the degenerate and singular regimes?
• RQ2 Can stability of solutions be established without relying on nonlinear semigroup theory or regularity estimates such as Harnack inequalities?
• RQ3 Is the convergence of solutions valid when the initial data are positive measures (e.g., Dirac delta), as in the Barenblatt solution?
• RQ4 Can the convergence be quantified in terms of the difference $ |m - m_i| $, and what is the rate of convergence in $ L^{1+m} $ norm?
• RQ5 Does the stability result extend to the limit $ m \to 1 $, including the case where the limit equation is the classical heat equation?

Key findings

• Solutions to the porous medium and fast diffusion equations converge in $ L^{1+m} $ norm as $ m_i \to m $, with the convergence rate $ \|u - u_i\|_{L^{1+m}(\Omega_T)} \leq c |m - m_i|^{1/m} $ for large $ i $, under the condition $ m > \frac{n-2}{n+2} $.
• The stability result holds for both $ m > 1 $ (degenerate) and $ m < 1 $ (singular) cases, including the limiting case $ m = 1 $, which corresponds to the heat equation.
• The convergence is established for Dirichlet problems on bounded domains and Cauchy problems on $ \mathbb{R}^n $ with initial data being a positive measure of finite mass, such as the Barenblatt solution.
• The Barenblatt solution $ \mathcal{B}_m $ satisfies $ \mathcal{B}_{m_i} \to \mathcal{B}_m $ in $ L^p $ as $ m_i \to m $, a result not previously established in the literature.
• The proof avoids the use of nonlinear semigroup theory, Hölder regularity, or reverse Hölder inequalities, relying only on weak solutions and compactness.
• For $ m \geq 1 $, an alternative proof provides an estimate of the solution difference in terms of $ |m - m_i| $, but this approach is restricted to $ m \geq 1 $ and uses strong monotonicity.