Inhomogeneous Incompressible Viscous Flows With Slowly Varying Initial Data

Abstract

Abstract. The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3-D inhomogeneous incompressible Navier-Stokes system in the whole space R 3 . This class of data is based on functions which vary slowly in one direction. The idea is that 2-D inhomogeneous Navier-Stokes system with large data is globally well-posedness and we construct the 3-D approximate solutions by the 2-D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2-D inhomogeneous Navier-Stokes system. We obtained the same optimal decay estimates as the solutions of 2-D homogeneous Navier-Stokes system.

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

“…On the other hand, Chemin and Gallagher [2] proved the global existence of smooth solution to three-dimensional incompressible Navier-Stokes equations (N S) with initial data which is slowly varying in one direction. This type of result was extended by Chemin and the second author [3] to the three-dimensional incompressible inhomogeneous Navier-Stokes equations and by Liu and the second author [17] for (N S) with unidirectional derivative of the initial velocity being sufficiently small in some critical functional space.…”

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Motivated by [2,3], we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.

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Large Time Behavior of the Navier-Stokes Flow

“…-If u 0 is a divergence-free vector field such that u 0 − w V 1,2 ≤ δ, then there is a unique solution u of the non-stationary Navier-Stokes equation (47) The work in [111,119,120]. analyzes the asymptotic behavior of infinite energy solutions to NS, under L 2 (R 3 ) perturbations.…”

“…See also the very recent paper by J.-Y. Chemin and P. Zhang [47], for an original construction of a class of "large", yet global smooth solutions to the system (70) and the corresponding decay results.…”

Global Solutions of 3-D Inhomogeneous Navier-Stokes System with Large Viscosity in One Variable

“…One may check [5] and the references therein concerning the well-posedness theory of the system (1.1) or (1.4). In particular, when ρ 0 ∈ L ∞ with a positive lower bound and initial velocity being sufficiently small in the critical Besove space,…”