Global Strong Solution to Multi-Dimensional Compressible Quantum Navier-Stokes Equation in Bounded Domains with Large Initial Data
We consider the compressible quantum Navier-Stokes (QNS) system in a smooth bounded domain, supplemented with a curvature Navier-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the density. For arbitrarily large initial data away from vacuum, we establish the global existence and uniqueness of strong solutions when the adiabatic index \[ γ\in[1,\infty)\quad\text{for }N=2, \qquad γ\in\left[1,3\right)\quad\text{for }N=3. \] The argument relies on the effective velocity $v=u+\nabla\logρ$, which reveals the parabolic structure of the system. Compared with previous works in the periodic and Cauchy cases, additional difficulties arise from the treatment of boundary terms. We also use a bootstrap argument between the density integrability and the weighted integrability of the effective velocity to enlarge the admissible range of $γ$ from $\left[1,11/4\right)$ to $\left[1,3\right)$. Combining this bootstrap argument with a De Giorgi iteration for the density upper bound and a Nash-Moser type iteration for the positive lower bound, we derive a priori estimates on every finite time interval and extend the local strong solution globally in time. As far as we know, this is the first result on global strong solutions with large initial data for the initial-boundary value problem (IBVP) of the QNS system in bounded domains with compatibility boundary conditions.