DEVELOPMENT OF SINGULARITIES IN THE RELATIVISTIC EULER EQUATIONS

摘要

The purpose of this paper is to study the phenomenon of singularity formation in large data problems for C1 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 x 2 hyperbolic systems, the break-down of C1 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 C1 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 C1 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.