Mixed Hyperbolic-Elliptic Systems in Industrial Problems

Abstract

In recent years there has been a large increase in the volume of literature pertaining to ‘mixed’ systems of partial differential equations. In order to fix the terminology which we shall use, consider a system of conservation laws $${w_t} + A{w_x} = S$$ (1) where A is an N × N matrix, w is a N-vector, S is an N-vector of source terms and subscripts denote differentiation. When the eigenvalues of A are non-zero, real and distinct, the system 1 is hyperbolic and many (though by no means all) of its mathematical properties are known (see, for example SMOLLER (1983)). In the strictest sense, the system 1 is said to be of ‘mixed type’ if any of the hyperbolicity conditions on the eigenvalues of A fail, so that for example two eigenvalues become equal, or an eigenvalue has a non-trivial zero. We shall be concerned mainly however with the much more serious case when some or all of the eigenvalues of A become complex, so that the problem is ‘elliptic in time’. In this case it is tempting to discard the problem completely, arguing that the innate ill-posedness of the system and need for boundary conditions at t = ∞ precludes any meaningful analytical or numerical results. The fact remains however that such systems occur with surprising regularity. Because of this, a number of questions can be posed: (a) Can such complex sound speeds ever occur in a correct mathematical model? (b) Even if complex eigenvalues are present, do they occur in regions of phase space which the model ever enters? (c) If complex eigenvalues can never occur in ‘correct’ models, is there ever any point in studying mixed problems for ‘physical’ systems? (d) If we do study such problems and ultimately attempt to find a numerical solution, what effects can we expect to arise from the ellipticity?