In this work, which is mainly divided into two parts, the emphasis is on the equations of nonlinear acoustics as integral parts of models for medical applications of ultrasound.
The first part treats the question of well-posedness for two equations of nonlinear acoustics, the Westervelt equation and the Kuznetsov equation. The formulation of the equations is taken from engineering books, where starting from the basic equations of thermodynamics, the continuity equation and the Euler equation, the acoustic wave equations (linear as well as nonlinear) are derived. Then the appropriate formulation is taken, namely the one in terms of acoustics pressure, put in the context of an initial-boundary value problem on the domain in R3 of suitable smoothness and studied for the existence and uniqueness of solutions. Of course, some assumptions on the initial and boundary data are necessary in order to obtain the results. In addition, we first prove well-posedness on the time interval [0,T] for T taken small enough and then extend the result to the case of arbitrarily chosen positive T. The proof is based on Banach’s Fixed-Point Theorem.
In the second part we use the theory for the equations of nonlinear acoustics to develop models for different types of lithotripters. In those models we need to describe propagation of ultrasound through different media and therefore the equations of nonlinear acoustics are used. In case of an electromagnetic lithotripter, where the focusing mechanism relies upon an acoustic lens, we analyse in detail the shape optimization problem for the lens. Specifically, we consider the well-posedness of this optimization problem, perform a shape sensitivity analysis and investigate numerical aspects of it, as well. The second example of using the well-posedness results from the first part in an optimization context comes from another type of lithotripter, namely the one whose operation is based on the piezoelectric principle. Since this device has a self-focusing property, the objective here is to achieve favorable focusing by controlling an input signal which comes from one part of the boundary. So optimal boundary controllability for the aforementioned equations is investigated, existence of an optimal control is established and the first order necessary optimality conditions are discussed.