On higher regularity for the Westervelt equation with strong nonlinear damping

Abstract

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term of 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 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 a focusing acoustic lens in lithotripsy.

Keywords:

• Nonlinear acoustics
• interface coupling
• Westervelt’s equation
• q-Laplacian

AMS Subject Classifications:

• 35L05
• 35L20

Acknowledgements

The authors would like to thank the reviewer for the valuable comments which led to an improvement of the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was support by the Austrian Science Fund (FWF) under [grant number P24970] is gratefully acknowledged as well as the support of the Karl Popper Kolleg “Modeling-Simulation-Optimization”, which is funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF).