Well-posedness of a first-order formulation for fractionally damped nonlinear acoustics

Abstract

In this work, we study the well-posedness of a quasilinear first-order-in-time system with memory arising in nonlinear acoustics. The model features a memory kernel describing the fractional damping and covers, as special cases, first-order formulations of Kuznetsov- and Westervelt-type equations. For completely monotone convolution kernels, we prove that the Westervelt-type system admits unique local-in-time solutions in bounded domains for space dimensions $d\leq3$, under homogeneous Dirichlet boundary conditions, suitable regularity, and smallness assumptions. The analysis is based on energy estimates exploiting novel nonlinear coercivity properties of the memory terms. Under additional assumptions on the resolvent of the kernel, we show that the smallness conditions can be significantly relaxed. For finite sums of exponential kernels our results generalize to the Kuznetsov-type system. Additionally, we show that the inviscid case (absence of memory kernel) in \(\R^d\) can be treated by standard hyperbolic arguments in the case of the Kuznetsov-type system.