 Solution

Freeze a uniform state with $0<\alpha <1$, ${\rho }_i>0$, and seek a finite characteristic speed $c$. Denote its perturbation amplitudes by $a,v_1,v_2,\pi$. The two phase-mass equations and two momentum equations giveEliminating the amplitudes givesWith $s^2={\rho }_1(1-\alpha )/({\rho }_2\alpha )>0$, factor the quadratic over the complex numbers to obtainFor nonzero relative velocity the speeds are a nonreal conjugate pair. A Fourier perturbation $e^{ik(x-ct)}$ on the growing branch has magnitude $e^{|k||\mathrm{Im}c|t}$. Arbitrarily short waves therefore grow arbitrarily quickly, preventing a continuous dependence estimate in the usual finite-order Sobolev norms. This generic slip state is ill-posed as an initial value problem.

The pressure has no independent time-evolution equation; it enforces the incompressible volume constraint, so the request concerns the finite characteristic speeds rather than a freely propagating pressure mode. At $u_1=u_2$ the two speeds coincide and this particular nonreal-speed argument disappears; further degeneracy analysis is needed there, rather than calling coincident real speeds complex. The issue is complex characteristic speeds of a pressure-coupled two-fluid model.