Global well-posedness and large-time behavior of classical solutions to the 3D Navier-Stokes system with changed viscosities

Abstract

We establish the global existence of classical solutions to the Cauchy problem for the Navier-Stokes equations with viscosities depending on density in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state. Moreover, we obtain some large-time behavior and decay rate estimates of the solutions.

Blow‐up criterion for compressible Navier–Stokes equations with degenerate viscosities

“…In order to overcome these difficulties, we will study from a mathematical view point. Inspired by Bresch et al 12 and Guo and Song, 25 we found that the boundedness of the temporal integral of the super‐norm in space of the deformation tensor guarantees estimate on $$$ \nabla \rho /\rho $$$ and the velocity $$$ u $$$. In the process of the proof, we can see that the estimates on gradient of the velocity $$$ u $$$ can be derived from $$$ {\left\Vert \nabla \rho /\rho \right\Vert}_{L^{\infty}\left(0,T;{L}^2\right)} $$$; see Lemma 3.3 and Lemma 3.4.…”

Global Existence of Finite Energy Weak Solutions of Quantum Navier–Stokes Equations

“…To the best of our knowledge, the Cauchy Problem for (1.1) with κ ≥ 0 and far-field behavior (1.2) has not been previously investigated in literature in the class of weak solutions for d = 2, 3. Local strong solutions have been constructed in [20,30,28] for (1.1) with ν > 0 and κ = 0. For d = 1, ν > 0 and κ = 0, existence and uniqueness of global strong solutions with (1.2) has been shown in [34,21].…”