Abstract
The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening, and eventual overturning of a wave. Using selfsimilar variables in two space dimensions and a power series expansion based on powers of $t_0t^{1/2}$, $t_0$ being the singularity time, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point. The shock spreads in the transversal direction as $t_0t^{1/2}$ and in the direction of propagation as $t_0t^{3/2}$, as also found in a onedimensional model problem.
Original language  English 

Pages (fromto)  208231 
Number of pages  24 
Journal  Journal of Fluid Mechanics 
Volume  820 
Early online date  5 May 2017 
DOIs  
Publication status  Published  Jun 2017 
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Profiles

Professor Jens G Eggers
 Cabot Institute for the Environment
 School of Mathematics  Professor of Applied Mathematics
 Fluids and materials
 Applied Mathematics
Person: Academic , Member