| Abstract: |
| In this talk, we study a parabolic gradient system that integrates two models: the free-energy functional for anisotropic-orientation-adaptive image processing; and the phase-field model of grain-boundary motion.
Through the analysis of this gradient system, we aim to construct a unified mathematical framework bridging the two research areas of image processing and materials science.
Recently, several attempts have been made to construct a unified framework for the pseudo-parabolic gradient system obtained by incorporating the energy-dissipation operator $-\Delta \partial_t$ into our system.
However, the mathematical models for image processing and grain boundary motion are originally formulated as parabolic gradient flows.
Therefore, establishing a unified analytical framework for these parabolic models still remains an open problem.
In this talk, we address this open problem.
Our main objective is to further advance the unification of this mathematical framework through the analysis of our system.
Building on the time-discretization method for the pseudo-parabolic system, we carry out the mathematical analysis of the corresponding parabolic system.
As a main result, we clarify conditions that ensure the well-posedness of the parabolic system with energy-dissipation.
Additionally, in the proof of the main result, we construct an energy-dissipating time-discrete scheme that reduces the cost associated with higher-order derivatives. |
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