| Abstract: |
| We study the initial boundary value problem of the 3D non--conservative compressible two--fluid model with common pressure (P^+=P^-) in a bounded domain with no--slip boundary conditions. The global existence and uniqueness of classical solution are established when the initial data is near its equilibrium in H^4(\Omega) by delicate energy methods. By a product, the exponential convergence rates of the pressure and velocities in H^3(\Omega) are obtained. To overcome the difficulties arising from boundary effects, on the one hand, we separate the energy estimates for the spatial derivatives into that over the region away from the boundary and near the boundary by using cutoff functions and localizations of \Omega. On the other hand, by exploiting the dissipation structure of the system, we employ regularity theory of the stationary Stokes equations and elliptic equations to get higher--order spatial derivatives of pressure and velocities, which is very different from the Cauchy problem in [Wu--Yao--Zhang, Math. Ann., 2024] where the effective viscous flux played an important role in their analysis. |
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