The breakdown of solutions of quasilinear first order systems of partial differential equations (Q1093818)

Beginning with a one-dimensional conservation law for motivation, the authors lead up to a discussion and exploitation of matrix Riccati equations in proving blow-up in \(C^ 1\) of solutions to certain quasilinear equations. The partial differential equations take the form either of \[ (1)\quad \partial u/\partial t+\sum^{n}_{i=1}f_ i(x,t,u)\partial u/\partial x_ i+g(x,t,u,\nabla u)=0, \] \(x\in {\mathbb{R}}^ n\), \(u\in {\mathbb{R}}^ n\), or of two similar coupled systems. Defining the total derivative \(D/Dt=\partial /\partial t+\sum^{n}_{i=1}f_ i\partial /\partial x_ i,\) (1) implies \((2)\quad Du/Dt+g=0.\) If \(J=\nabla u\) then J satisfies an equation of the form \[ (3)\quad DJ/Dt+J\cdot \nabla_ uf\cdot J\quad +\quad other\quad terms = 0, \] where \(\nabla_ uf\) has entries \(\partial f_ i/\partial u_ j\). If one evaluates the coefficient terms in (3) along characteristics, then (3) is a matrix Riccati system of ODEs for J. A particular result is the following. Suppose \(f_ i=f_ i(t,u)\) and \(g=0\). Then if \((J.\nabla_ uf)|_{t=0}\) has a negative eigenvalue, the solution J(t) to (3) becomes unbounded in finite time.