The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Shu Wang,Yongxin Wang

doi:10.3390/math8071195

Abstract

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

Highlights

In this paper, we consider the Cauchy problem for the three-dimensional (3D) incompressibleNavier–Stokes (ν > 0) and Euler (ν = 0) equations ut + u · ∇u + ∇ P = ν∆u, x ∈ R, t>0, divu = 0, x ∈ R, t>0, u(0, x) = u (x), x ∈ R3 (1)and the initial boundary value problem for the 3D incompressible Navier–Stokes (ν > 0) equations in the bounded domain ut + u · ∇u + ∇ P = ν∆u, x ∈ Ω, t > 0, divu = 0, x ∈ Ω, t > 0, (2)u = 0, x ∈ ∂Ω, t > 0, u(0, x) = u0 (x), x ∈ Ω respectively
Incompressible Euler and quasi-geostrophic equations, we investigate the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions to the three-dimensional incompressible flows by studying non-axisymmetric global smooth solution with respect to any coordinate axis xi, i = 1, 2, 3 in the Cartesian coordinate system in R3 and looking for the domain to guarantee the global wellposedness for the initial and boundary value problem for the 3D incompressible Navier–Stokes and Euler system based on a class of variant spherical coordinates
Theorems 1–4 imply that the form of the velocity u and the geometry structure of the domain

Summary

Introduction

We consider the Cauchy problem for the three-dimensional (3D) incompressible. Ut + u · ∇u + ∇ P = ν∆u, x ∈ R , t>0, divu = 0, x ∈ R , t>0, u(0, x) = u (x), x ∈ R3. The initial boundary value problem for the 3D incompressible Navier–Stokes (ν > 0) equations in the bounded domain ut + u · ∇u + ∇ P = ν∆u, x ∈ Ω, t > 0, divu = 0, x ∈ Ω, t > 0,. X = ( x1 , x2 , x3 ), the unknown u = (u1 (t, x), u2 (t, x), u3 (t, x)) T , denotes the fluid velocity vector field, P = P(t, x) is the scalar pressure, ν ≥ 0 is the viscosity coefficient. U0 is a given initial velocity with div u0 = 0.

Background

A Class of Variant Spherical Coordinates

Research Model

Preliminaries

Main Results

Conclusions