In a previous post I mentioned the recent work of Demetrios Christodoulou on shock waves. On 08.12 I heard a talk by Christodoulou on this subject and since then he has explained some of the most important points of this work to me in more detail. Here I will present a little of what I learned. These results concern solutions of the relativistic Euler equations in Minkowski space. According to Christodoulou analogous results could be obtained in the non-relativistic case but no details of this have been published. The initial data is given on a hyperplane and is assumed to coincide outside a compact set with a constant state where the fluid is at rest with constant density. On the compact set the data is close to the constant state in a suitable sense.
The object of study is the maximal smooth solution evolving from the given data and its future boundary. This boundary will be non-empty exactly in the case when a shock is formed. Necessary and sufficient conditions are given for there to be a shock. Most of the results concern a fluid which is isentropic and irrotational. The two conditions are intimately connected and cannot be assumed independently of each other. These results do have consequences for the general case since a sufficiently large region exists where the extra conditions are satisfied. Here I will concentrate on the isentropic and irrotational case. A central point is that while the solution becomes singular at the future boundary it is actually smooth up to and including the boundary with respect to a non-standard differential structure. Key computations are done in coordinates which define this different kind of smoothness. One of these coordinates is constant on sound cones of the solution being considered. The condions for shock formation depend very much on the sign of the quantity at the constant state, where is the equation of state of the fluid. This is the same as the sign of the quantity introduced in the book. If this quantity actually vanishes on the constant state then there are no shocks. For an irrotational and isotropic flow the evolution equations of the fluid can be written as a quasilinear wave equation for a scalar function . In the case that is identically zero for a given equation of state this coincides with a geometric equation which is obtained as follows. Suppose that a timelike hypersurface in five-dimensional Minkowski space can be expressed as the graph of a function on . Suppose in other words that the hypersurface is of the form . Then the condition that this hypersurface has vanishing mean curvature is equivalent to the equation of motion for this particular type of fluid. The fluid is related to the Chaplygin gas which has been studied in cosmology in recent years. It has equation of state for a constant and satisfies . This type of fluid originally came up in aerodynamics in the early years of the twentieth century. Sergei Chaplygin, after whom this fluid is named, seems to have been quite a prominent figure since the town he grew up in is now named after him, as is a crater on the moon. For fluids under normal physical conditions but, as pointed out in the book, there are physical situations where the opposite sign occurs.