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 ut + (u22)x = σx,σt + uσx = k2ux, where k is a real number and the unknowns u(x,t) and σ(x,t) 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 δ shock wave are established. For initial conditions containing a Dirac measure, a δ shock wave solution is also presented.

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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
Authors
C. O. R. Sarrico
CMAFCIO
Universidade de Lisboa
Campo Grande
Lisboa
Portugal