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On a fourth order equation describing single-component film models

Martina Magliocca

TL;DR

This work analyzes a Mullins-type fourth-order PDE modeling controlled solid-state dewetting in a single-component film, with initial data in the Wiener space $A^0({\mathbb{T}^N})$ and a dimensionless reformulation for the fluctuation variable $v$. The authors establish global existence of a weak solution under small data and parameter constraints, together with exponential decay and baseline regularity; they further obtain enhanced regularity for data in $A^0\cap H^2$ under stronger smallness and a positivity condition on $1+G(v)$. The existence proof uses Faedo-Galerkin approximations to a truncated nonlinear flux $G_n$, uniform a priori estimates in Wiener Sobolev-type spaces, and compactness to pass to the limit, complemented by Fourier-based estimates. They also discuss open questions on uniqueness and propose fixed-point approaches to achieve analyticity and potential improvements on smallness requirements, linking to broader fourth-order PDE analysis in crystal growth and thin-film dynamics.

Abstract

We study existence results for a fourth order problem describing single-component film models assuming initial data in Wiener spaces.

On a fourth order equation describing single-component film models

TL;DR

This work analyzes a Mullins-type fourth-order PDE modeling controlled solid-state dewetting in a single-component film, with initial data in the Wiener space and a dimensionless reformulation for the fluctuation variable . The authors establish global existence of a weak solution under small data and parameter constraints, together with exponential decay and baseline regularity; they further obtain enhanced regularity for data in under stronger smallness and a positivity condition on . The existence proof uses Faedo-Galerkin approximations to a truncated nonlinear flux , uniform a priori estimates in Wiener Sobolev-type spaces, and compactness to pass to the limit, complemented by Fourier-based estimates. They also discuss open questions on uniqueness and propose fixed-point approaches to achieve analyticity and potential improvements on smallness requirements, linking to broader fourth-order PDE analysis in crystal growth and thin-film dynamics.

Abstract

We study existence results for a fourth order problem describing single-component film models assuming initial data in Wiener spaces.
Paper Structure (9 sections, 8 theorems, 117 equations, 1 table)

This paper contains 9 sections, 8 theorems, 117 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations and basic tools
  3. The dimensionless problem
  4. Main results
  5. Existence and regularity results with $v_0\in A^0({\mathbb{T}^N})$
  6. Regularity results with $v_0\in A^0({\mathbb{T}^N})\cap H^2({\mathbb{T}^N})$
  7. Final remarks
  8. On the uniqueness
  9. A fixed point technique

Key Result

Theorem 2.2

Assume that the parameters $c_1$, $c_2$ verify Let $v_0\in A^0({\mathbb{T}^N})$ such that Defined the values $\delta_i=\delta_i(\|v_0\|_{A^0})$ as we also require $\|v_0\|_{A^0}$ small enough in order to have Then, there exist at least one global weak solution to equation pb such that and for any $T>0$. Furthermore, we also have the continuity regularity and the exponential decay

Theorems & Definitions (19)

  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2: Existence and regularity results with $A^0$ data
  • Theorem 2.3: Regularity results with $A^0({\mathbb{T}^N})\cap H^2({\mathbb{T}^N})$ data
  • Remark 3.1: On the approximating problem
  • Proposition 3.2: A priori estimates in Wiener spaces
  • proof
  • Proposition 3.3: Compactness results
  • proof
  • Corollary 3.4: Further regularity results
  • ...and 9 more