Geodesic
On the acceleration of shock waves and concentration of energy
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 162-169

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By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures. 
@article{TM_2013_281_a12,
     author = {A. N. Golubyatnikov},
     title = {On the acceleration of shock waves and concentration of energy},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {162--169},
     publisher = {mathdoc},
     volume = {281},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_281_a12/}
}
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A. N. Golubyatnikov. On the acceleration of shock waves and concentration of energy. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 162-169. http://geodesic.mathdoc.fr/item/TM_2013_281_a12/