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The Time-Periodic Cahn-Hilliard-Gurtin System on the Half Space as a Mixed-Order System with General Boundary Conditions

Guillaume Neuttiens, Jonas Sauer

TL;DR

The paper develops a rigorous $L^2$-theory for the time-periodic CHG system in the half-space and reveals that classical Lopatinskiĭ–Shapiro conditions are insufficient for mixed-order systems. It introduces Newton-polygon-based potential spaces and a factorization of the CHG determinant to separate interior invertibility from boundary solvability, and proves maximal regularity and well-posedness results in the time-periodic setting. A novel complementing boundary condition is defined via an extended boundary matrix, ensuring solvability with general boundary data and providing trace theorems on Newton-polygon spaces. The results yield an isomorphism between natural solution and data spaces and establish precise boundary-trace criteria, with implications for boundary-value problems for mixed-order vector-valued systems. The methods offer a flexible framework for broader classes of time-periodic, boundary-coupled, mixed-order PDEs in unbounded domains.

Abstract

A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinskiĭ-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinskiĭ-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.

The Time-Periodic Cahn-Hilliard-Gurtin System on the Half Space as a Mixed-Order System with General Boundary Conditions

TL;DR

The paper develops a rigorous -theory for the time-periodic CHG system in the half-space and reveals that classical Lopatinskiĭ–Shapiro conditions are insufficient for mixed-order systems. It introduces Newton-polygon-based potential spaces and a factorization of the CHG determinant to separate interior invertibility from boundary solvability, and proves maximal regularity and well-posedness results in the time-periodic setting. A novel complementing boundary condition is defined via an extended boundary matrix, ensuring solvability with general boundary data and providing trace theorems on Newton-polygon spaces. The results yield an isomorphism between natural solution and data spaces and establish precise boundary-trace criteria, with implications for boundary-value problems for mixed-order vector-valued systems. The methods offer a flexible framework for broader classes of time-periodic, boundary-coupled, mixed-order PDEs in unbounded domains.

Abstract

A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinskiĭ-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinskiĭ-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.
Paper Structure (17 sections, 34 theorems, 166 equations, 3 figures)

This paper contains 17 sections, 34 theorems, 166 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Newton Polygons and Order Functions
  4. Mixed-Order Systems
  5. Time-Periodic Setting and Purely Oscillatory Function Spaces
  6. Fourier Transform on $G$
  7. Purely Oscillatory Spaces
  8. Time-Periodic Function Spaces Associated to the Newton Polygon
  9. The Whole Space Problem
  10. Maximal Regularity of the CHG Determinant
  11. Mixed-Order Systems and Maximal Regularity for the CHG System
  12. The Half Space Problem with General Boundary Conditions
  13. Quotient Spaces
  14. Trace Theory
  15. Operators Preserving Upper and Lower Support
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p\in (1,\infty)$. Consider the solution space $\mathbb{E}:=\mathbb{E}_1\times \mathbb{E}_2$, the data space $\mathbb{F}:=\mathbb{F}_1\times\mathbb{F}_2$, and the trace space $\mathbb{G}:=\mathbb{G}_1\times\mathbb{G}_2$, given by For all $f=(f_1, f_2)\in \mathbb{F}$ and $g=(g_1,g_2)\in \mathbb{G}$, system eqn: CHG_system_1 admits a unique solution $u=(u_1, u_2)\in\mathbb{E}$, and it holds wh

Figures (3)

  • Figure 1: The Newton polygon of $D$, where $r_0=r_3=s_0=s_1=0$, $r_1=4$, $r_2=2$, and $s_2=s_3=1$.
  • Figure 2: The Newton polygons associated to the trace spaces of $\mu_D$.
  • Figure 3: The Newton polygons associated to certain spaces considered in the proof of Theorem \ref{['thm: main_2']}. Observe that $T_0(\mathcal{N}_{D^-})$ is a proper subset of $T_2(\mathcal{N}_D)$. In particular, we can choose $g\in H^{T_0(\mu_{D^-})}_\perp(G^{n-1})$ such that $g\notin H^{T_2(\mu_D)}_\perp(G^{n-1})$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • ...and 76 more