Navier‐Stokes Equations in Gas Dynamics: Green's Function, Singularity, and Well‐Posedness

2021 ◽  
Vol 75 (2) ◽  
pp. 223-348
Author(s):  
Tai‐Ping Liu ◽  
Shih‐Hsien Yu
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Gas Dynamics ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Navier Stokes Equations

scholarly journals GREEN'S FUNCTION AND LARGE TIME BEHAVIOR OF THE NAVIER–STOKES–MAXWELL SYSTEM

2012 ◽  
Vol 10 (02) ◽  
pp. 133-197 ◽  
Author(s):  
RENJUN DUAN
Keyword(s):  
Electromagnetic Field ◽  
Steady State ◽  
Green’S Function ◽  
Green's Function ◽  
Stokes Equations ◽  
Large Time Behavior ◽  
Navier Stokes ◽  
Constant Density ◽  
Time Behavior ◽  
Navier Stokes Equations

In this paper, we are concerned with the system of the compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The asymptotic stability of the steady state with the strictly positive constant density and the vanishing velocity and electromagnetic field is established under small initial perturbations in regular Sobolev space. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier–Stokes system. Moreover, the detailed analysis of the Green's function to the linearized system is made with applications to derive the rate of the solution converging to the steady state.

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

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