On dissipative turbulent solutions to the compressible anisotropic Navier-Stokes equations in unbounded domains
Ondřej Kreml, Šárka Nečasová, Tong Tang
TL;DR
The paper establishes global-in-time existence of dissipative turbulent solutions for the compressible Navier–Stokes system with an anisotropic viscous tensor on unbounded domains, under the pressure law $p(\rho)=\rho^\gamma$ with $\gamma>1$ and broad viscosity coefficients. It develops an invading-domain approximation combining artificial diffusion, Galerkin truncation, and compactness arguments to construct solutions on expanding bounded domains, yielding Reynolds and energy defect measures $(\mathfrak{R},\mathfrak{E})$ and a relative entropy framework. A key result is the relative entropy inequality, which, together with weak-strong uniqueness, shows that a strong solution coincides with any dissipative turbulent solution when the strong solution exists. The analysis extends prior bounded-domain results to unbounded geophysical contexts and demonstrates the robustness of the dissipative turbulent approach for anisotropic stresses, offering a rigorous foundation for geophysical fluid models in exterior domains.
Abstract
Inspired by Abbatiello, Feireisl and Novotný, we prove the global existence of dissipative turbulent solution for the compressible Navier-Stokes equations with anisotropic viscous stress tensor on unbounded domain. Our work complements the result of Bresch and Jabin, where the authors used the new compactness method to prove the existence of a weak solution to the same system in $\mathbb{T}^3$. By virtue of the concept of dissipative turbulent solutions, we are able to relax assumptions on the anisotropic tensor coefficients and the pressure law coefficient. We point out that we establish the existence result on a large class of unbounded domains, which is more conform to geophysical context. We also prove the weak-strong uniqueness property of acquired dissipative turbulent solutions.