Global classical solution to two-dimensional compressible Navier–Stokes equations with large data in [formula omitted

Abstract

In this paper, we prove the global well-posedness of the classical solution to the 2D Cauchy problem of the compressible Navier–Stokes equations with arbitrarily large initial data and non-vacuum far-fields when the shear viscosity μ is a positive constant and the bulk viscosity λ(ρ)=ρβ with β>43. Note that the initial data can be arbitrarily large with or without vacuum states. For the non-vacuum initial data, our global well-posedness result implies that the classical solution to the 2D Cauchy problem will not develop the vacuum states in any finite time. Moreover, the global well-posedness result still holds true when the initial data contains the vacuum states in a subset of R2 provided some compatibility conditions are satisfied. Some new weighted estimates for the density and the velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and velocity can propagate along with the flow, which are intrinsic to the two-dimensional Cauchy problem with the non-vacuum far-fields.