Energy Dissipative Solution to a Nonlinear Parabolic Systems with Unknown Dependent Coefficients

TL;DR

This paper introduces an energy dissipative solution concept for nonlinear parabolic systems with unknown coefficients, unifying models in image processing and materials science.

Contribution

It establishes conditions for the existence of energy dissipative solutions, providing a new unified analytical framework for parabolic models.

Findings

Established conditions for existence of energy dissipative solutions.

Unified framework applicable to models in image processing and materials science.

Clarified the mathematical structure of parabolic systems with unknown coefficients.

Abstract

In this paper, we investigate a system of parabolic partial differential equations with unknown-dependent coefficients that integrates two models: an anisotropic orientation-adaptive denoising process in image processing and a phase-field model of grain-boundary motion in materials science. In recent years, several studies have attempted to develop a unified framework for treating these two research areas by considering pseudo-parabolic systems obtained through the introduction of the energy-dissipation operator $ - \Delta \partial_t $. However, the mathematical models for image processing and grain-boundary motion are originally formulated as parabolic systems. Therefore, establishing a unified analytical framework for such parabolic models remains an open problem. In this paper, we address this open problem by introducing a notion of solution that reproduces energy dissipation in parabolic systems, which we call an energy dissipative solution. As the main result, we clarify conditions that guarantee the existence of such solutions. The results of this paper establish a unified analytical framework for parabolic models, which has remained unresolved, and provide a solid theoretical foundation for advanced problems spanning both image processing and materials science.