Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weakly elliptic damping gives sharp decay

Lassi Paunonen, Nicolas Vanspranghe, Ruoyu P. T. Wang

Abstract

We prove that weakly elliptic damping gives sharp energy decay for the abstract damped wave semigroup, where the damping is not in the functional calculus. In this case, there is no overdamping. We show applications in linearised water waves and Kelvin--Voigt damping.

Weakly elliptic damping gives sharp decay

Abstract

We prove that weakly elliptic damping gives sharp energy decay for the abstract damped wave semigroup, where the damping is not in the functional calculus. In this case, there is no overdamping. We show applications in linearised water waves and Kelvin--Voigt damping.
Paper Structure (6 sections, 11 theorems, 71 equations)

This paper contains 6 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Introduction
  4. Main results
  5. Acknowledgement
  6. Proof

Key Result

Corollary 1.1

Assume $(a(x))^{-1}\in L^{p}(M)$ for $p\in (d,\infty)$. Then there is $C>0$ that uniformly in $t>0$ and $u$ satisfying eq:5. For $p\in (1,d]$, we have uniformly in $t>0$ and $u$ solving eq:5 with initial data in $W^{1,2}(M)\times W^{\frac{1}{2},2}(M)$.

Theorems & Definitions (31)

  • Corollary 1.1
  • Definition 1.2: $m$-ellipticity
  • Definition 1.3: $m$-boundedness
  • Remark 1.4
  • Example 1.5: Linearised water waves
  • Theorem 1: Weak ellipticity gives stability
  • Theorem 2: Weak boundedness gives sharpness
  • Corollary 1.6: Weak ellipticity gives decay
  • Remark 1.7: Strong monotonicity
  • Example 1.8: Linearised water waves, bounded from below
  • ...and 21 more