On one-dimensional models for hydrodynamics

To date it has not been possible to prove whether or not the three-dimensional incompressible Euler equations develop singular behaviour in finite time. Some possible singular scenarios, for instance shock-waves, are very important from a physical point of view, since they imply a connection between the macroscopic and the microscopic scale. Therefore, the appearance of this type of singularity or a similar one might be interpreted as a possible explanation of the transition to turbulence. In order to clarify the question, some one-dimensional models for ideal incompressible hydrodynamics have been introduced and analysed, and it was proven that shock-waves appear in finite time within this type of flow. In this work we question the validity of these models and analyse the physical meaning that the occurrence of a singularity in an incompressible flow, if it happens, may have. (c) 2006 Elsevier B.V. All rights reserved.