#### Vol. 294, No. 1, 2018

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Multiplication of distributions and a nonlinear model in elastodynamics

### C. O. R. Sarrico

Vol. 294 (2018), No. 1, 195–212
##### 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