scholarly journals The Boltzmann Equation with Incoming Boundary Condition: Global Solutions and Navier--Stokes Limit

2019 ◽  
Vol 51 (3) ◽  
pp. 2504-2534
Author(s):  
Ning Jiang ◽  
Xu Zhang
Keyword(s):  
Boundary Condition ◽  
Boltzmann Equation ◽  
Global Solutions ◽  
Navier Stokes ◽  
The Boltzmann Equation

scholarly journals Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

2017 ◽  
Vol 27 (12) ◽  
pp. 2261-2296 ◽  
Author(s):  
Yan Guo ◽  
Shuangqian Liu
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Kinetic Equations ◽  
Navier Stokes ◽  
Viscous Heating ◽  
Text Setting ◽  
Hydrodynamic Approximation ◽  
Energy Conversation ◽  
The Boltzmann Equation

The incompressible Navier–Stokes–Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos–Levermore–Ukai–Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin.[Formula: see text] 3 (2008) 1–49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in [Formula: see text] setting in a periodic box. Based on an odd–even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an [Formula: see text] approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.

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