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Trace regularity of solutions to the Navier equations

Jerin Tasnim Farin, Giusy Mazzone

TL;DR

The paper addresses the trace regularity of the boundary stress vector $P(u)\cdot n$ for the time-dependent Navier equations in a bounded 3D domain under mixed Dirichlet/Neumann boundary conditions. It develops a global-in-time framework using weak, transposition, and strong solutions and derives new trace estimates for $P(u)\cdot n$ on $\Gamma_0$ that go beyond standard energy methods, drawing an analogy with hidden trace regularity for scalar waves. A key finding is that the boundary stress trace is one degree less smooth than the prescribed boundary data, with explicit $T$-dependent bounds for weak formulations and $T$-independent bounds for strong solutions. These results extend the hidden-trace regularity paradigm (as in LLT-86) from scalar wave equations to the Navier system and have potential implications for coupled problems such as fluid-structure interaction, where sharp boundary regularity aids in proving existence and stability of solutions.

Abstract

We present results on the trace regularity of the stress vector on the boundary of an elastic solid satisfying the time-dependent, displacement-traction problem for the Navier equations of linear elasticity in a bounded domain of $\mathbb{R}^3$. Specifically, the solid's displacement is subject to Dirichlet- and Neumann-type conditions on different portions of its boundary and possibly non-zero body forces and initial data. Our regularity results are reminiscent of the so-called "hidden trace regularity" results for solutions to the scalar wave equation obtained in [12].

Trace regularity of solutions to the Navier equations

TL;DR

The paper addresses the trace regularity of the boundary stress vector for the time-dependent Navier equations in a bounded 3D domain under mixed Dirichlet/Neumann boundary conditions. It develops a global-in-time framework using weak, transposition, and strong solutions and derives new trace estimates for on that go beyond standard energy methods, drawing an analogy with hidden trace regularity for scalar waves. A key finding is that the boundary stress trace is one degree less smooth than the prescribed boundary data, with explicit -dependent bounds for weak formulations and -independent bounds for strong solutions. These results extend the hidden-trace regularity paradigm (as in LLT-86) from scalar wave equations to the Navier system and have potential implications for coupled problems such as fluid-structure interaction, where sharp boundary regularity aids in proving existence and stability of solutions.

Abstract

We present results on the trace regularity of the stress vector on the boundary of an elastic solid satisfying the time-dependent, displacement-traction problem for the Navier equations of linear elasticity in a bounded domain of . Specifically, the solid's displacement is subject to Dirichlet- and Neumann-type conditions on different portions of its boundary and possibly non-zero body forces and initial data. Our regularity results are reminiscent of the so-called "hidden trace regularity" results for solutions to the scalar wave equation obtained in [12].
Paper Structure (5 sections, 19 theorems, 179 equations)

This paper contains 5 sections, 19 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Notation and basic functional setting
  3. Main results
  4. Useful equalities
  5. Regularity properties of Hilbert spaced-valued functions

Key Result

Theorem 2.1

LM-1 Let $s>1/2$ and denote with $k$ the largest integer satisfying $k<s-1/2$. Let $\mathcal{D}\subset\mathbb{R}^d$ be a bounded domain with smooth boundary $\Gamma$. Then, the trace operator is linear, continuous, surjective with a continuous linear right inverse.

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 1
  • proof
  • Remark 2
  • Definition 1: Solutions in the sense of transposition
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 29 more