Global large solutions and optimal time-decay estimates to the Korteweg system

Abstract

We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

“…Lei et al 34 obtained the global well‐posedness for incompressible Navier‐Stokes equation in energy space with a class of large initial data which includes the Beltrami flow. Following the assumptions on the viscosity coefficients of Haspot, 18 Zhai and Li 35 recently proved the global solutions to system () without smallness condition imposed on the vertical component of the incompressible part of the velocity by using the weighted Chemin‐Lerner‐norm technique. Zhang 36 constructed a class of global large solutions to the compressible NSK system with constant viscosity coefficients in critical Besov spaces.…”

Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

“…In the present paper, the ρ * has been set to 1 without loss of generality. Furthermore, as recent contributions, we refer to the following papers: Zhang [39]; Zhai and Li [38]; Kawashima, Shibata, and Xu [15]; Charve, Danchin, and Xu [4].…”