Development of singularities in the relativistic Euler equations

Abstract:

The purpose of this paper is to study the phenomenon of singularity formation in large data problems for $C^1$ solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by [P. D. Lax [J. Math. Phys. 5 (1964), pp. 611–613] shows that, for $2\times 2$ hyperbolic systems, the break-down of $C^1$ solutions occurs in finite time if initial data contain any compression in some truly non-linear characteristic field under some additional conditions, which include genuine non-linearity and the strict positivity of the difference between two corresponding eigenvalues. These harsh structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density $\rho$ does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound of $\rho$. To this end, based on introducing several key artificial quantities and some elaborate analysis on the difference of the two Riemann invariants, we characterized the decay of mass-energy density lower bound in time, and ultimately made some concrete progress. On the one hand, for the $C^1$ solutions with large data and possible far field vacuum to the isentropic flow, we verified the theory obtained by P. D. Lax in 1964. On the other hand, for the $C^1$ solutions with large data and strictly positive initial mass-energy density to the non-isentropic flow, we exhibit a numerical value $N$, thought of as representing the strength of an initial compression, above which all initial data lead to a finite-time singularity formation. These singularities manifest as a blow-up in the gradient of certain Riemann invariants associated with corresponding systems.