Wave Propagation in Nonlinear Acoustics, Viscoelasticity, and Heat Transfer

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

This research investigates the qualitative behavior of solutions to nonlinear wave equations in nonlinear acoustics, viscoelasticity, and hyperbolic diffusion. The project studies existence, uniqueness, and long-time behavior of large-data solutions in four problem areas: (1) quasilinear wave equations (the Kuznetsov equation, together with its simplified version, the Westervelt equation) with degeneracy of the differential operator; (2) problems in nonlinear viscoelasticity; (3) the damped quasilinear wave equation as a model for unsteady heat conduction; and (4) development of a new non-local theory in solid mechanics, called peridynamics, an extension of classical continuum mechanics to allow effective modeling of fracture in materials. Among other methods, the PI proposes an original approach based on physical considerations to study existence of solutions for quasilinear hyperbolic equations. This approach enables one to show existence of solutions with arbitrarily large initial data starting from existence of solutions with small initial data. The ingredients needed (finite speed of propagation, uniqueness, energy identity) are available for many hyperbolic systems, hence the approach has potentially wide applicability. Nonlinear wave propagation phenomena appear in vibrations of elastic bodies, in theories of acoustic pressure, plasmas, and semiconductors, and in quantum mechanics. The investigation of nonlinear wave equations poses great difficulty since the interaction of waves does not follow the principle of superposition. Instead, the waves can generate new waves, blow up in finite time, or vanish at infinity. The project involves developing new mathematical tools to determine the factors that play a dominant role in these interactions and to predict the long time behavior of solutions. The problems under study model important phenomena in science and medicine: the lithotripsy model appears in shock wave propagation used in breaking up kidney stones; hyperbolic heat conduction is relevant in phase change transitions and superconductivity; and peridynamics is very promising in predicting fractures in material science. Progress in these areas will benefit both academic research and industrial applications. The PI has integrated her research efforts into the development of an interdisciplinary course, Math in the City, which attracts undergraduate students to study partial differential equations. Under this program, students work in collaboration with local businesses and research centers to develop and analyze mathematical models that use real data. The project also involves graduate students and a postdoctoral associate in the research.