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

Harbir Antil, Daiki Mizuno, Ken Shirakawa, Naotaka Ukai

TL;DR

This work develops a general well-posedness theory for pseudo-parabolic gradient systems with state-dependent dynamics, proving existence of energy-dissipating solutions and uniqueness with continuous dependence on initial data. The analysis combines time-discretization, convex analysis, and Mosco/Γ-convergence to handle nonlinear, variable-coefficient operators. The framework is demonstrated through two applications: an anisotropic image-denoising model and a pseudo-parabolic anisotropic KWC-type grain-boundary model, both fitting into the abstract $( ext{S})_ u$ structure. The results advance nonlinear evolution theory with state-dependent dynamics and have implications for materials science and image processing.

Abstract

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 $(\mathrm{S})_ν$.

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

TL;DR

This work develops a general well-posedness theory for pseudo-parabolic gradient systems with state-dependent dynamics, proving existence of energy-dissipating solutions and uniqueness with continuous dependence on initial data. The analysis combines time-discretization, convex analysis, and Mosco/Γ-convergence to handle nonlinear, variable-coefficient operators. The framework is demonstrated through two applications: an anisotropic image-denoising model and a pseudo-parabolic anisotropic KWC-type grain-boundary model, both fitting into the abstract structure. The results advance nonlinear evolution theory with state-dependent dynamics and have implications for materials science and image processing.

Abstract

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 .

Paper Structure

This paper contains 10 sections, 6 theorems, 162 equations.

Table of Contents

  1. Preliminaries
  2. Main results
  3. Time-discretization scheme
  4. Proof of Main Theorem.
  5. Case: $\nu\downarrow0$
  6. Case: $\nu>0$ is a fixed positive constant
  7. Proof of Main Theorem 2.
  8. Applications
  9. Pseudo-parabolic image denoising process with anisotropy
  10. Pseudo-parabolic KWC type system with anisotropy

Key Result

Theorem 1

(Existence, uniqueness of solution with energy-dissipation) There exists a sufficiently small constant $\tau_0:=\tau_0(\nu,\|\nabla\gamma\|_{W^{1,\infty}})\in (0,1)$, possibly dependent on $\nu$ and $\gamma$, such that for any $\tau \in (0,\tau_0)$, $\mathrm{(AP)}^\tau_\nu$ admits a unique solution

Theorems & Definitions (23)

  • Example 1
  • Definition 1: Mosco-convergence: cf.MR0298508
  • Definition 2: $\Gamma$-convergence; cf.MR1201152
  • Remark 1
  • Example 2: Examples of Mosco-convergence
  • Definition 3
  • Definition 4
  • Theorem 1
  • Lemma 3.1
  • proof
  • ...and 13 more