Formation of singularities in the Euler and Euler-Poisson equations

Abstract

We establish finite-time loss of smoothness for several variants of the one-dimensional Euler equations. We consider the Euler-Poisson equations in one dimension and the Euler and Euler-Poisson equations in three dimensions in the presence of spherical or cylindrical symmetry. We show that no matter how smooth the initial data are, solutions of these equations can be forced to lose smoothness in finite-time. By using an interesting variant of Lax's method for showing the finite-time loss of smoothness of solutions of pairs of hyperbolic conservation laws, we show the finite-time loss of smoothness of solutions of the one-dimensional Euler-Poisson equations. Using our method, we also find a sufficient condition for the existence of a globally smooth solution of the one-dimensional Euler-Poisson equations. For the other cases, we use a characteristic-based method. In all cases, we give local conditions for the finite-time loss of smoothness of solutions.