A partial differential equation is the requirement that partial derivatives up to order of some unknown function satisfy a certain algebraic relation. An evident prerequisite for this to make sense is that the derivatives of the function up to order exist. It turns out, however, that this apparently obvious statement is not true. In many cases it is possible to reformulate a PDE so as to make it meaningful to talk about solutions which have less derivatives than appear to be necessary at first sight. This leads to the concept of weak (or generalized) solutions, in contrast to classical solutions which have the derivatives needed for the straightforward interpretation of the equation.

In hydrodynamics, the study of fluids, weak solutions play an important role due to the phenomenon of shock waves. Solutions of the Euler equations which describe a fluid while neglecting viscosity have the tendency to develop discontinuities in the basic fluid variables (e.g. the density) even when starting from a perfectly smooth configuration. If the solution is to be used for modelling a physical situation beyond the time where a discontinuity appears it is necessary to use weak solutions. It turns out that weak solutions of the Euler equations can be used effectively in hydrodynamics and there are powerful methods for handling them numerically. At the same time it is very difficult to prove rigorous mathematical results about solutions of the Euler equations in the presence of shocks. Most of the theorems available in the literature concern situations which are reduced to an effective problem in one space dimension by means of a symmetry assumption (plane symmetry). Recently a notable exception to this appeared in the form of a book ‘The formation of shocks in 3-dimensional fluids‘ by Demetrios Christodoulou which treats the dynamics of solutions of the Euler equations up to the moment of shock formation without requiring symmetry assumptions. See also the recent review article of Christodoulou for background to this work and a concise history of mathematical developments in hydrodynamics (Bull. Amer. Math. Soc. 44, 581).

If viscosity is included in the description of fluids then the Euler equations are replaced by the Navier-Stokes equations. There is reason to suspect that in this case shock waves are smoothed out and a smooth initial configuration remains smooth in the course of the evolution, for all time. There are simple examples where this can be seen but there is still no global regularity result for Navier-Stokes (and no counterexample). The Clay Foundation has offered a prize of one million dollars for the solution of this problem in either direction. The fact that the prize has not yet been collected is a sign of the difficulty of the problem. For a discussion of this question and its broader mathematical significance I recommend the excellent account of Tao.

What effect does gravity have on the formation of shock waves? It is reasonable to suppose that the answer to this question is ‘almost none’. This intuition applies not only in Newtonian physics but also to the case of a fully relativistic description in the context of general relativity. The gravitational field curves space but in many situations the curvature produced should not affect the qualitative nature of the process of shock formation. A couple of years ago Fredrik Ståhl and I set out to confirm this idea rigorously. The project has been delayed by other things but now we have finished a manuscript and put it on the arXiv (Shock Waves in Plane Symmetric Spacetimes). The result is that there are plane symmetric solutions of the relativistic Euler equations coupled to the Einstein equations of general relativity which lose smoothness in an arbitrarily short time, depending on the initial size of the spatial derivatives of the fluid variables. The fact is used that the mechanism of breakdown bears a sufficiently close resemblance to that in a non-relativistic fluid without gravity. In this one-dimensional setting there are left-moving and right-moving waves in the fluid and gravity leads to a coupling between them. A central idea of the proof of our result is to use changes of variable so as to attain a sufficient amount of decoupling of the degrees of freedom corresponding to the two types of waves.

August 24, 2012 at 8:59 pm |

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