In this paper,we establish the global existence of weak solutions with higher regularity to the com-pressible Navier-Stokes equations under the no-slip boundary conditions.Lions and Feireisl have established global weak solutions with finite energy under the Dirichlet boundary conditions by making use of effective viscous flux and oscillation defect measure,while Hoff has investigated global weak solutions with higher regularity when the domain is either whole space or half space with Navier-slip boundary conditions,yet the existence theory of global weak solutions with higher regularity under the Dirichlet boundary conditions remains unknown.In this paper,we prove that the system will admit at least one global weak solution with higher regularity as long as the initial energy is suitably small when the domain is a 2D solid disc.This is achieved by exploiting the structure of the exact Green function of the disc to decompose the effective viscous flux into three parts,which corresponds to the pressure term,the boundary term and the remaining term,respectively.In order to control the boundary term,one of the key observations is to use the geometry of the domain which successfully bounds the integral of the effective viscous flux on the boundary.
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