Relativistic hydrodynamics of quark-gluon plasma and stability of scaling solutions

The scaling solutions of the relativistic hydrodynamics are expected to play an important role in describing the expansion stage of a quark-gluon plasma which may be formed in nucleus-nucleus collisions at high energies. After summarizing some general properties of the scaling solutions, we study in detail their stability against small perturbations. In some typical cases of the two-dimensional scaling solution it is found that (i) the scaling solution is stable if the Reynolds number R defined in terms of the viscosity coefficients is larger than a critical value Rc (=1), (ii) it is also stable for a long-wavelength perturbation if R is small enough, and (iii) it becomes unstable when R approaches Rc from below. It is also shown that these results are related to the time dependence of the Reynolds number, the entropy density, and the temperature, and the point R=Rc corresponds to a critical instant when the heating due to the dissipative processes balances with the cooling due to the expansion of the fluid. The stability of the scaling solution of the quark-gluon plasma is examined for typical ranges of the relevant parameters