Non-resonant Solutions in Hyperbolic-Parabolic Systems with Periodic Forcing

Aday Celik

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Aday Celik, Non-resonant Solutions in Hyperbolic-Parabolic Systems with Periodic Forcing (2020), Logos Verlag, Berlin, ISBN: 9783832586584

Beschreibung / Abstract

This thesis is a mathematical investigation of damping effects in hyperbolic systems. In the first part two models from nonlinear acoustics are studied. Existence of time-periodic solutions to the Blackstock-Crighton equation and the Kuznetsov equation are established for time-periodic data sufficiently restricted in size. This leads to the conclusion that the dissipative effects in these models are sufficient to avoid resonance.

In the second part the interaction of a viscous fluid with an elastic structure is studied. A periodic cell structure filled with a viscous fluid interacting with a deformable boundary of the cell is considered under time-periodic forcing. The motion of the fluid is governed by the Navier-Stokes equations and the deformable boundary is governed by the plate equation. It is shown that the damping mechanism induced by the viscous fluid is sufficient to avoid resonance in the elastic structure.

Inhaltsverzeichnis

  • BEGINN
  • 1 Introduction
  • 1.1 A Historical Background of Nonlinear Acoustics
  • 1.2 A Historical Background of Fluid-Structure Interaction
  • 1.3 Nonlinear Acoustics with Periodic Forcing
  • 1.4 Fluid-Structure Interaction with Periodic Forcing
  • 2 Preliminaries
  • 2.1 General Notation
  • 2.2 Topology and di erentiable structure
  • 2.3 Fourier Transform and Multiplier Theory
  • 2.4 Sobolev and Bessel Potential Spaces
  • 2.5 Interpolation
  • 2.6 Embedding and Trace properties of Time-Periodic Sobolev spaces
  • 2.7 Mathematical Tools from Fluid Mechanics
  • 3 Nonlinear Acoustics
  • 3.1 Models
  • 3.2 The Damped Wave Equation
  • 3.3 The Kuznetsov Equation
  • 3.4 The Blackstock-Crighton Equation
  • 4 Fluid-Structure Interaction
  • 4.1 Viscous Fluid Flow on an Elastic Plate
  • 4.2 Reformulation in a Reference Con guration
  • 4.3 Uniqueness
  • 4.4 The Time-Periodic Stokes Equations
  • 4.5 The Coupled Resolvent Problem
  • 4.6 A priori Estimates
  • 4.7 The Stationary Linear System
  • 4.8 The Linear Fluid-Structure Problem
  • 4.9 The Nonlinear Problem
  • Appendix
  • A.1 Some Bounded Functions
  • A.2 Fourier Multiplier

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