scholarly journals Time Decay Estimate with Diffusion Wave Property and Smoothing Effect for Solutions to the Compressible Navier-Stokes-Korteweg System

2021 ◽  
Vol 64 (2) ◽  
pp. 163-187
Author(s):  
Takayuki Kobayashi ◽  
Kazuyuki Tsuda
Keyword(s):  
Decay Estimate ◽  
Navier Stokes ◽  
Smoothing Effect ◽  
Time Decay ◽  
Diffusion Wave ◽  
Wave Property

scholarly journals Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 683
Author(s):  
Takayuki Kobayashi ◽  
Masashi Misawa ◽  
Kazuyuki Tsuda
Keyword(s):  
Hardy Space ◽  
Potential Flow ◽  
Decay Estimate ◽  
Initial Conditions ◽  
Bounded Mean Oscillation ◽  
Navier Stokes ◽  
Diffusion Wave ◽  
Mean Oscillation ◽  
Asymptotic Profile ◽  
Flow Part

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

Sign in / Sign up

Export Citation Format

Share Document