Propagation of support and singularity formation for a class of 2D quasilinear hyperbolic systems

Abstract

In this paper we consider a class of quasilinear, non-strictly hyperbolic 2 x 2 systems in two space dimensions. Our main result is finite speed of propagation of the support of smooth solutions for these systems. As a consequence, we establish nonexistence of global smooth solutions for a class of sufficiently large, smooth initial data. The nonexistence result applies to systems in conservation form, which satisfy a convexity condition on the fluxes. We apply the nonexistence result to a prototype example, obtaining an upper bound on the lifespan of smooth solutions with small amplitude initial data. We exhibit explicit smooth solutions for this example, obtaining the same upper bound on the lifespan and illustrating loss of smoothness through blow-up and through shock formation. Consider a quasilinear, hyperbolic system of differential equations of the form: [GRAPHICS] where U(x, y, t) = (u(x, y, t), v(x, y, t)) is the state vector and A(.), B(.) are smooth, 2 x 2 matrix-valued functions of the state vector. Hyperbolicity means that the matrix C(xi, U) = xi(1)A(U) + xi(2)B(U) has real eigenvalues for any U in state space and for any xi = (xi(1), xi(2)) is an element of S-1. A partially aligned state U-0 for the system above is a state such that A(U-0) and B(U-0) have a common eigenvector. A partially aligned system is one for which all states are partially aligned. This class of systems was introduced by the authors in [8], along with some basic properties and a classification of different kinds of partial alignment. In this article we will use the results and terminology from [8]. An earlier version of the results contained here was announced in [10].57222924