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| Nonlinear acoustics has a wide range of important applications, one of which is the medical use of high intensity focused ultrasound in lithotripsy. Here we face the problem of a physically unbounded domain which will be truncated for numerical computations. Absorbing boundary conditions are then used to avoid reflections on the artificial boundary of the computational domain. Ultrasound excitation can be modeled by Neumann boundary conditions on the rest of the boundary. \\
In the simulation of a focusing lens immersed in an acoustic medium the problem of coupling regions with different material parameters arises, and so the existence of spatially less regular solutions becomes significant. On the other hand, in the Westervelt equation, which is a classical model in nonlinear acoustics, degeneracy of the coefficient $1-2ku$ needs to be avoided. Adding a nonlinear strong damping term to the Westervelt equation enables us to obtain an $L_{\infty}$ estimate on $u$ in order to avoid degeneracy of the coefficient $1-2ku$, while refraining from estimates on $\Delta u$ (thus from too high regularity). \\
We will show local in time well-posedness of such a practically relevant problem for the Westervelt equation with nonlinear strong damping and Neumann as well as absorbing boundary conditions and give an outlook on acoustic-acoustic and elastic-acoustic coupling. |
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