The general d-dimensional Riemann problem raises naturally the question of resolving the interaction of d planar shocks merging at a point. In gas dynamics, we may consider only standing shocks. This problem has received a satisfactory answer in dimension d = 2 (see [3, 4]). We investigate the 3-dimensional case. We restrict to the irrotational case, in order to keep the complexity of the solution within reasonable bounds. We show that a new kind of waves appears downstream, which we call a conical wave. When the equation of state is that of Chaplygin/von Kármán, we give a complete mathematical answer to this problem. This involves the existence and uniqueness of a complete minimal surface in a hyperbolic space, with prescribed asymptotics.
@article{CML_2011__3_3_543_0,
author = {Serre, Denis},
title = {Three-dimensional interaction of shocks in irrotational flows},
journal = {Confluentes Mathematici},
pages = {543--576},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {3},
year = {2011},
doi = {10.1142/S1793744211000394},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744211000394/}
}
TY - JOUR AU - Serre, Denis TI - Three-dimensional interaction of shocks in irrotational flows JO - Confluentes Mathematici PY - 2011 SP - 543 EP - 576 VL - 3 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744211000394/ DO - 10.1142/S1793744211000394 LA - en ID - CML_2011__3_3_543_0 ER -
%0 Journal Article %A Serre, Denis %T Three-dimensional interaction of shocks in irrotational flows %J Confluentes Mathematici %D 2011 %P 543-576 %V 3 %N 3 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744211000394/ %R 10.1142/S1793744211000394 %G en %F CML_2011__3_3_543_0
Serre, Denis. Three-dimensional interaction of shocks in irrotational flows. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 543-576. doi : 10.1142/S1793744211000394. http://www.numdam.org/articles/10.1142/S1793744211000394/
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