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New \(\delta\)-shock waves in the \(p\)-system: a distributional product approach. (English) Zbl 1446.74150

Summary: This article studies a Riemann problem for the so-called “\(p\)-system”\(u_t - v_x = 0, v_t - [\sigma (u)]_x = 0\), which rules one-dimensional isentropic thermoelastic media. Such study is made using a product of distributions that allows us to extend both the classical solution concept and a weak solution concept. By considering \(\sigma\) as an entire function that takes real values on the real axis, this product also extends for certain distributions \(u\) the meaning of \(\sigma (u)\). Under certain conditions, this Riemann problem has solutions that are \(\delta\)-shock waves. Furthermore, those \(\delta\)-shock waves satisfy the so-called generalized Rankine-Hugoniot conditions.

MSC:

74J40 Shocks and related discontinuities in solid mechanics
74F05 Thermal effects in solid mechanics
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