Optimal Decay Rate of the Compressible Quantum Navier-Stokes Equations

Abstract

For quantum fluids governed by the compressible quantum Navier-Stokes equations in $\mathbb{R}^3$ with viscosity and heat conduction, we prove the optimal $L^p − L^q$ decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.