High intensity (focused) ultrasound HIFU is used in numerous medical and industrial applications ranging from lithotripsy and thermotherapy via ultrasound cleaning and welding to sonochemistry. The relatively high amplitudes arising in these applications necessitate modeling of sound propagation via nonlinear wave equations and in this talk we will first of all dwell on this modeling aspect. Then in the main part of this lecture will deal with limiting cases of certain parameters, that are both of physical interest and mathematically challenging. The latter is due to the fact that these limits are singular in the sense that they change the qualitative behaviour of solutions. On a technical level, they require uniform bounds and therefore alternative energy estimates. We start with the classical Westervelt and Kuznetsov models and study the limit as the diffusivity of sound tends to zero – the parameter of a viscous damping term whose omission leads to a loss of regularity and global well-posedness as well as exponential decay. Secondly, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the already mentioned Westervelt and the Kuznetsov equation. We study the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. Making such a transition from third order to second order in time equations clearly necessitates compatibility conditions on the initial data. Finally, we provide a result on another higher order model in nonlinear acoustics, the Blackstock-Crighton-Brunnhuber-Jordan equation for vanishing thermal conductivity.

This is joint work with Vanja Nikolic, Radboud University, and Mechthild Thalhammer, University of Innsbruck.

Zoom meeting: Link