Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions

Pages: 763 - 773, Issue Special, September 2011

 Abstract        Full Text (386.8K)              

Barbara Kaltenbacher - Institute of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria (email)
Irena Lasiecka - Kerchof Hall , P. O. Box 400137, University of Virginia, Charlottesville, VA 22904-4137, United States (email)

Abstract: In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We prove local in time well-posedness as well as global existence and exponential decay for a slightly modi ed model. A key step of the proof is an appropriate extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by the strongly damped wave equation.

Keywords:  nonlinear acoustics, Kuznetsov's equation, local and global well- posedness
Mathematics Subject Classification:  Primary: 35L72; Secondary: 35B40, 37N20

Received: July 2010;      Revised: January 2011;      Published: October 2011.