Wave dynamics of linear hyperbolic relaxation systems (Q2804483)

The authors study a linear hyperbolic relaxation system NEWLINE\[NEWLINE \partial_t V+ A \partial_x V =\frac{1}{\varepsilon}RV, \qquad x\in\mathbb{R}, V\in\mathbb{R}^N NEWLINE\]NEWLINE where \(R\) is a \(N\times N\)-matrix of rank \(1\) with one negative eigenvalue. Associated with the relaxation system one considers the homogeneous system NEWLINE\[NEWLINE \partial_t V+ A\partial_x V=0 NEWLINE\]NEWLINE and the equilibrium system NEWLINE\[NEWLINE \partial_t v+ B\partial_x v=0 NEWLINE\]NEWLINE where \(v\in \mathbb{R}^{N-1}\) describes the system on the equilibrium manifold \(M=\{V;\; RV=0\}\). It is shown that for each Fourier component the characteristic polynomial for the linear relaxation system is a convex combination of the characteristic polynomials for the homogeneous and for the equilibrium system. Using results on the roots of complex polynomials this leads to the equivalence of linear stability and the subcharacteristic condition. Similarly, using different properties of complex polynomials the authors give a new proof of the fact that the characteristic velocities of a stable linear relaxation systems are bounded by those of the associated homogeneous system.