Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions

TL;DR

This paper establishes local existence and well-posedness results for the Westervelt equation with nonlinear damping and various boundary conditions, relevant for high intensity focused ultrasound applications.

Contribution

It provides new local existence results for the Westervelt equation with nonlinear damping and variable coefficients, including boundary conditions, under small initial data.

Findings

Proved local in time existence of weak solutions.

Established well-posedness with spatially varying coefficients.

Applicable to high intensity focused ultrasound models.

Abstract

We investigate the Westervelt equation with several versions of nonlinear damping and lower order damping terms and Neumann as well as absorbing boundary conditions. We prove local in time existence of weak solutions under the assumption that the initial and boundary data are sufficiently small. Additionally, we prove local well-posedness in the case of spatially varying $L^{\infty}$ coefficients, a model relevant in high intensity focused ultrasound (HIFU) applications.