Shock waves, part 2

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 $\rho f''(\rho)+2f'(\rho)$ at the constant state, where $p=f(\rho)$ is the equation of state of the fluid. This is the same as the sign of the quantity $H$ 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 $\phi$ . In the case that $H$ 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 $\phi$ on ${\bf R}^4$ . Suppose in other words that the hypersurface is of the form $(x_0,x_1,x_2,x_3,\phi(x_0,x_1,x_2,x_3))$ . 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 $p=-A\rho^{-1}$ for a constant $A$ and satisfies $H=0$ . 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 $\rho f''(\rho)+2f'(\rho)>0$ but, as pointed out in the book, there are physical situations where the opposite sign occurs.

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2 Responses to “Shock waves, part 2”

Uwe Brauer Says:
January 13, 2009 at 6:36 pm | Reply
Hello,

right now I see the document with some
formula does not parse in yellow!

Besides this, it looks extremely interesting what you are talking about.

“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.”

could you outline what this non-standard differential structure is supposed to be?

thanks

Uwe Brauer
hydrobates Says:
January 13, 2009 at 8:33 pm | Reply
I do not know what is the problem with parsing. When I view the post it looks OK.

As far as the non-standard differential structure is concerned perhaps the following is helpful. One idea is to follow a family of sound cones starting on the initial surface and watch their density in space. This density blows up at the shock. Now this sounds strange since it seems to depend so much on which family you started with on the initial surface. If I understood Christodoulou correctly the answer is: it does not matter – whichever choice you make the density of hypersurfaces blows up at the same place. He seemed to find this very remarkable. If you tried to define a coordinate which is constant on these sound cones then it would become singular when the shock is reached. The variables defining the fluid also become singular. The statement is now that if you change to a suitable new coordinate system where one of the coordinates is constant on these sound cones then something amazing happens. Of course the family of sound cones remains smooth in the new coordinate system, it was built that way, but the fluid variables are also smooth when expressed in these new coordinates. The statement about the differential structure is just a way of saying that in a coordinate-free way.

Hydrobates