Multiplication of distributions and a nonlinear model in elastodynamics

C. O. R. Sarrico

Vol. 294 (2018), No. 1, 195–212

DOI: 10.2140/pjm.2018.294.195

Abstract

We consider the system $u_t+{\left(u^2∕2\right)}_x={\sigma }_x,{\sigma }_t+u{\sigma }_x=k^2{u}_x$, where $k$ is a real number and the unknowns $u\left(x,t\right)$ and $\sigma \left(x,t\right)$ belong to convenient spaces of distributions. For this simplified model from elastodynamics, a rigorous solution concept defined in the setting of a distributional product is used. The explicit solution of a Riemann problem and the possible emergence of a $\delta$ shock wave are established. For initial conditions containing a Dirac measure, a ${\delta }^{\prime }$ shock wave solution is also presented.

Keywords

products of distributions, nonconservative systems, nonstrictly hyperbolic systems, strictly hyperbolic systems, $\delta$ waves, $\delta^\prime$ waves, Riemann problem.

Mathematical Subject Classification 2010

Primary: 35D99, 35L67, 46F10

Milestones

Received: 1 March 2017

Accepted: 26 October 2017

Published: 5 January 2018