Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation

TL;DR

This paper establishes well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation, a model for nonlinear acoustic waves in viscous fluids, using operator semigroups and energy estimates.

Contribution

It introduces a new analysis framework for this complex nonlinear PDE, proving well-posedness and decay properties for the first time.

Findings

Proved well-posedness of the Blackstock-Crighton-Kuznetsov equation.

Established exponential decay of solutions over time.

Used operator semigroup theory and energy estimates in the analysis.

Abstract

The present work provides well-posedness and exponential decay results for the Blackstock-Crighton-Kuznetsov equation arising in the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. First, we treat the associated linear equation by means of operator semigroups. Moreover, we derive energy estimates which we will use in a fixed-point argument in order to obtain well-posedness of the Blackstock-Crighton-Kuznetsov equation. Using a classical barrier argument we prove exponential decay of solutions.