Vanja Nikolic (Universität Klagenfurt, A) - On some mathematical aspects of nonlinear acoustics: well-posedness, interface coupling and shape optimization

Due to the large number of applications of high intensity focused ultrasound in medicine and industry, models of nonlinear acoustics and their rigorous mathematical treatment have recently gained increasing interest. Most commonly used model in nonlinear acoustics is the Westervelt equation - a quasilinear potentially degenerate wave equation. As a way of avoiding degeneracy, Westervelt's equation is considered with a strong nonlinear damping term of the q-Laplace type. Using this model also allows to show well-posedness for an acoustic-acoustic coupling problem which arises in the context of medical applications of ultrasound in treatment of kidney stones.

In this talk, we will address the question of local in time well-posedness for the Westervelt equation with nonlinear damping and different boundary conditions, as well as for the coupled problem. Another important question is optimal focusing of the ultrasound in order to achieve the desired acoustic pressure at a kidney stone without harming  the surrounding tissue. We will present first order shape sensitivity analysis for the optimal focusing problem by employing the general approach of Ito, Kunisch and Peichl, 2008.