THE STRUCTURE OF THE SET OF HYPERBOLIC SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

摘要

In this dissertation we consider several questions about the structure of the set, H, of hyperbolic systems in the space of all homogeneous systems of N, m('th) order, constant coefficient, partial differential equations in n + 1 independent variables, --t, x(,1), ... ,x(,n)-- for which the plane t = 0 is non-characteristic. We then use these results to prove that the Cauchy initial value problem is well-posed for a set of systems with variable coefficients.;We determine the interior of H in two cases. First we show that, for first order systems, the interior of H is just the set of strictly hyperbolic systems. For systems of an arbitrary number of equations of equations of arbitrary order in 4 independent variables, we show that the interior of H is the interior of the closure of the strictly hyperbolic systems.;Consider the space of systems of an arbitrary number of first order equations in 4 independent variables. We look at a subset of this space, consisting of the least degenerate type of hyperbolic, but not strictly hyperbolic systems. The roots of the characteristic equation of such a system are at most double and are simple except in a finite number of isolated directions. Roots which coalesce must split to first order, and the system must always be diagonalizable. We denote this subset of H by (DELTA).;We show that all hyperbolic systems near an element of (DELTA) are also in (DELTA). Moreover, we show that all the algebraic properties of the roots of the characteristic equation of an element of (DELTA) are preserved under all hyperbolic perturbations of that system. Since the elements of (DELTA) are not strictly hyperbolic, the co-dimension of H near all elements of (DELTA) is strictly positive. Although a first order perturbation of a system in (DELTA) is not, in general, hyperbolic, we show that all lower order perturbations of that system are hyperbolic. This means that all systems in (DELTA) are strongly hyperbolic.;To show that our theory is non-trivial, we produce a system of 7 equations that is in (DELTA). We then show that, in a neighborhood of this system, the co-dimension of H is 4 and that H is not affine.;Consider systems with variable coefficients such that all systems derived from it by freezing coefficients are strongly hyperbolic. If such a system is smoothly symmetrizable, the Cauchy initial value problem for that system is well-posed. We will show by example that smooth symmetrizability is not a necessary condition for well-posedness. If a system is such that its coefficients are smooth and such that all systems derived from it by freezing coefficients are in (DELTA), we show that it is smoothly symmetrizable.