A Nonlinear Conservation Law with Memory.
Abstract
In this paper we study a history-boundary value problem for a nonlinear conservation with fading memory in one space dimension. The motivation for studying this problem is an earlier work by C. M. Dafermos and the author concerning the motion of a nonlinear, one-dimensional viscoelastic body. Using a variant of an energy method applied to the viscoelastic problem it is shown that under physically reasonable assumptions the nonlinear conservation law has a unique, classical solution (global in time), provided the data are sufficiently smooth and 'small' in a suitable norm; moreover, the solution and its first order derivatives decay to zero as t goes to infinity. The proof illustrates the versatility of the energy method combined with frequency domain techniques for Volterra operators. A preliminary analysis based on current work of R. Malek-Madani and the author is presented concerning the development of singularities in smooth solutions of the conservation law (in finite time) for sufficiently 'large' smooth data; under special assumptions it is shown that such singularities necessarily develop. The hope is to apply such a procedure to the viscoelastic problem. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1981
- Accession Number
- ADA103882
Entities
People
- J. A. Nohel
Organizations
- University of Wisconsin–Madison