Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations

Shu Wang; Shu Wang

doi:10.3934/cam.2025023

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 582-605. doi: 10.3934/cam.2025023

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Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations

Shu Wang ^,

School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, People's Republic of China

Received: 10 July 2024 Revised: 19 January 2025 Accepted: 15 April 2025 Published: 11 June 2025
76D05, 76D03, 35Q30, 35Q35, 74F10

The global well-posedness theory and viscosity vanishing limit of the initial-boundary value problem on two/three-dimensional (2D/3D) incompressible Navier-Stokes (NS) equations and/or Boussinesq equations with nonlinear boundary conditions are studied. The global existence of weak solution to the initial boundary value problem for 2D/3D incompressible NS equation with one kind of boundary of pressure-velocity's relation and the global existence and uniqueness of the smooth solution to the corresponding problem in 2D case for large smooth initial data are proven. The viscosity vanishing limit of the corresponding initial-boundary value problem for 2D/3D incompressible NS equations in the bounded domain is also established. And the corresponding results are extended to the 2D/3D incompressible Boussinesq equations.
Citation: Shu Wang. Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations[J]. Communications in Analysis and Mechanics, 2025, 17(2): 582-605. doi: 10.3934/cam.2025023

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Abstract

The global well-posedness theory and viscosity vanishing limit of the initial-boundary value problem on two/three-dimensional (2D/3D) incompressible Navier-Stokes (NS) equations and/or Boussinesq equations with nonlinear boundary conditions are studied. The global existence of weak solution to the initial boundary value problem for 2D/3D incompressible NS equation with one kind of boundary of pressure-velocity's relation and the global existence and uniqueness of the smooth solution to the corresponding problem in 2D case for large smooth initial data are proven. The viscosity vanishing limit of the corresponding initial-boundary value problem for 2D/3D incompressible NS equations in the bounded domain is also established. And the corresponding results are extended to the 2D/3D incompressible Boussinesq equations.

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