Archive for the ‘hydrodynamics’ Category

Shock waves, part 3

April 6, 2010

In a previous post I wrote about shock waves in fluids, including the case that they are described by the Einstein-Euler equations for a self-gravitating fluid in general relativity. I mentioned there a result of Fredrik Ståhl and myself proving that smooth solutions of the Einstein-Euler system can lose regularity in the course of their time evolution. This was done in the framework of spacetimes with plane symmetry. Here I want to describe some complementary results which were recently obtained by Philippe LeFloch and myself. These new results concern the existence of global weak solutions in situations where shocks may be present. This work is done under the assumption of Gowdy symmetry, which is weaker than plane symmetry. It allows the presence of gravitational waves, which plane symmetry does not. It uses time coordinates different from the constant mean curvature (CMC) coordinate used in the work with Ståhl. This difference in the time coordinates makes it difficult to relate the results of the two papers directly. It would be interesting to adapt the results of either of these papers to the time coordinates of the other.

In the paper with LeFloch we use coordinates (areal, conformal) which have previously been used in analysing analogous problems for vacuum spacetimes or spacetimes where the matter content is described by collisionless kinetic theory. A big difference is the weak regularity. One effect of this is that while in the given context it has been possible to prove global existence theorems for the initial value problem, nothing is known about the uniqueness of the solutions in terms of initial data. It should, however, be noted that in the corresponding analytical framework uniqueness is not even known for a one-dimensional non-relativistic fluid without gravity. Another new element introduced by the use of weak solutions is that it is only possible to evolve in one time direction. This model is not reversible, a fact implemented mathematically by the imposition of entropy inequalities. One of the results obtained concerns a forever expanding cosmological model. The other one concerns a contracting model which ends in a singularity. The second is not a global existence result in the conventional sense but it can be thought of as saying that the solution can be extended until certain specific things happen (a big crunch singularity).

To finish this post I want to indicate the type of regularity of the solutions obtained. I only state this roughly – more precise information can be found in the paper. The energy density and momentum density of the fluid is integrable in space, with the L^1 norms locally bounded in time. The quantities parametrizing the spacetime metric have first order derivatives which are square integrable in space. These conditions allow for jump discontinuities in the energy density which is what comes up in shock waves. It also allows singularities of Dirac \delta type in the metric, corresponding to what are often called impulsive gravitational waves.

Shock waves, part 2

December 14, 2008

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.

Shock waves

June 13, 2008

A partial differential equation is the requirement that partial derivatives up to order k 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 k 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 k 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.


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