arXiv subjects

Naotaka Ukai

Publications and source records attributed to Naotaka Ukai.

5 recordsLinked to original sources

arXiv2026

Large-Time Behavior of Pseudo-Parabolic Equations Associated with Generalized Total Variation Energies

In this paper, we study the well-posedness and large-time behavior of a pseudo-parabolic problem associated with a generalized total variation energy in the $BV$-framework. A main mathematical issue arises from the mismatch between the Sobolev regularity of solutions at finite times and the $BV$-structure of the corresponding steady-state problem. We prove the existence and uniqueness of solutions and investigate the relationship between their large-time behavior and solutions to the steady-state problem. Moreover, our results include the convergence of the solution trajectory to the unique steady-state solution under a typical setting arising in image processing.

math.AP↗

arXiv2026

Elliptic and Pseudo-Parabolic PDE System with Orientation-Adaptive Anisotropy

In this paper, we consider a coupled system of nonlinear elliptic and pseudo-parabolic PDEs arising in anisotropic monochrome image denoising with orientation-adaptation. The system is derived from the minimization process of a nonconvex energy functional. In particular, we focus on the problem of determining the initial data for the orientation variable. In previous studies, a natural procedure for determining such initial data has not been sufficiently clarified. To address this issue, we introduce a formulation in which the time derivative of the orientation variable is removed. This formulation enables the initial orientation data to be determined implicitly within a time-discrete scheme. On the other hand, this formulation weakens the conventional energy-dissipation structure and leads to new difficulties in constructing a stable variational time-evolution process. To overcome this issue, we develop an analysis based on a time-discretization method and establish the well-posedness of the proposed system, namely existence, uniqueness, and continuous dependence, as well as an energy-inequality. We also show that the proposed time-discrete scheme determines the initial orientation data consistently with the continuous model. These results provide a mathematical framework for the initial-orientation determination problem in orientation-adaptive anisotropic models.

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arXiv2026

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

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 $ - Δ\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.

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arXiv2025

Well-Posedness of Pseudo-Parabolic Gradient Systems with State-Dependent Dynamics

This paper develops a general mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics. The state dependence is induced by variable coefficient fields in the governing energy functional. Such coefficients arise naturally in scientific and technological models, including state-dependent mobilities in KWC-type grain boundary motion and variable orientation-adaptation operators in anisotropic image denoising. We establish two main results: the existence of energy-dissipating solutions, and the uniqueness and continuous dependence on initial data. The proposed framework yields a general well-posedness theory for a broad class of nonlinear evolutionary systems driven by state-dependent operators. As illustrative applications, we present an anisotropic image-denoising model and a new pseudo-parabolic KWC-type model for anisotropic grain boundary motion, and prove that both fit naturally within the abstract structure of $(\mathrm{S})_ν$.

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arXiv2024

A gradient system based on anisotropic monochrome image processing with orientation auto-adjustment

This paper is devoted to the mathematical analysis of a system of pseudo-parabolic partial differential equations governed by an energy functional, associated with anisotropic monochrome image processing. The energy functional is based on the one proposed by [Berkels et al. Cartoon extraction based on anisotropic image classification, SFB 611, 2006], which incorporates an orientation auto-adjustment mechanism, and our energy is a simplified version which reduces the order of derivatives to improve computational efficiency. The aim of this paper is to establish a stable minimization process that addresses some instability of the algorithm caused by the reduction of derivative order. As a part of the study, we here prove Main Theorems concerned with the well-posedness of the pseudo-parabolic system, and provide a mathematically rigorous guarantee for the stability of the image denoising process derived from our system.

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