Global Existence of Classical Solutions to the Relativistic Quantum Hydrodynamic System with Small Initial Data

Abstract

We establish global existence, decay, and scattering for sufficiently small, smooth, and localized perturbations of a constant non-vacuum equilibrium of a relativistic quantum hydrodynamic system in three space dimensions. In logarithmic-amplitude and phase variables, the equations form a semilinear system of coupled wave equations. The skew-symmetric coupling between the time derivatives cancels in the energy identity, yielding a natural derivative energy. The linearized system has two dispersion branches. At low frequencies, the slow branch exhibits Schr\"odinger-type dispersion, while the fast branch has a spectral gap; at high frequencies, both branches are wave-like. The main nonlinear difficulty arises from quadratic interactions with nontrivial time and space-time resonances. We show that the symbol of every active quadratic interaction contains the corresponding interaction phase as an exact factor. This structural cancellation removes the resonant denominator and allows us to eliminate the quadratic terms by a nonsingular normal-form transformation. Combining this transformation with dispersive and energy estimates, we obtain uniform-in-time bounds for the derivative energy and $\langle t\rangle^{-3/2}$ decay of the first derivatives in $L^\infty$. We further prove that the nonlinear solution scatters to a solution of the linearized system in the high-order derivative energy norm.