On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry
Xiangdi Huang, Weili Meng, Xueyao Zhang
TL;DR
The paper advances the global well-posedness theory for multi-dimensional viscous Saint-Venant and compressible Navier–Stokes systems under BD entropy by exploiting radial symmetry. It establishes global classical solutions with large initial data away from vacuum for subendpoint α and, in the endpoint α=1 case, in 2D including viscous shallow-water dynamics, along with large-time behavior toward equilibrium. It also develops a robust weak-solution theory that permits vacuum, using BD entropy-based a priori estimates and carefully constructed approximate systems to handle degeneracy and the origin. Collectively, the results extend BD-entropy methodologies to higher dimensions under radial symmetry and illuminate vacuum formation and vanishing phenomena in compressible flows with density-dependent viscosity.
Abstract
In 1871, Saint-Venant introduced the renowned shallow water equations. Since then, for the two-dimensional viscous or inviscid shallow water equations, the global existence of smooth solutions with arbitrarily large initial data has remained a challenging and long-standing open problem. In this paper, we provide an affirmative resolution to the viscous problem under the assumption of two-dimensional radial symmetry. Specifically, we establish the global existence of smooth solutions for the two-dimensional radially symmetric viscous shallow water equations with arbitrary smooth initial data. To achieve this goal, our approach relies crucially on overcoming two major obstacles: first, treating the viscous Saint-Venant system as the endpoint case of the BD entropy condition for the compressible Navier-Stokes equations; and second, addressing the critical embedding imposed by the spatial dimension, which currently holds only in two dimensions. However, the same result can be extended to three dimension for the compressible Navier-Stokes equations satisfying general BD entropy conditions excluding the endpoint case. Indeed, under the same symmtric framework, we also prove the global existence of smooth solutions for arbitrarily large initial data for both the two- and three-dimensional compressible Navier-Stokes equations subject to the BD entropy condition. It is particularly noteworthy that the aforementioned shallow water equations precisely correspond to the endpoint case of the compressible Navier-Stokes equations satisfying the BD entropy condition.