On higher regularity for the Westervelt equation with strong nonlinear damping

TL;DR

This paper establishes higher interior regularity results for the Westervelt equation with nonlinear damping and analyzes an interface coupling problem relevant to medical ultrasound applications, demonstrating piecewise regularity under certain conditions.

Contribution

It provides new regularity results for the Westervelt equation with nonlinear damping and studies a coupled interface problem with implications for medical ultrasound modeling.

Findings

Higher interior regularity for the Westervelt equation with nonlinear damping.

Piecewise H^2 regularity in space for the coupled problem under bounded gradient conditions.

Relevance to numerical methods and optimization in medical ultrasound applications.

Abstract

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term of the $q$-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g., in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solution to the coupled problem exhibits piecewise $H^2$ regularity in space, provided that the gradient of the acoustic pressure is essentially bounded in space and time on the whole domain. This result is of importance in numerical approximations of the present problem, as well as in gradient based algorithms for finding the optimal shape of the focusing acoustic lens in lithotripsy.