Global well-posedness for the compressible Euler–Korteweg equations with damping in L2-Lp critical Besov space and relaxation limit

In this paper, we investigate the Cauchy problem of compressible Euler–Korteweg equations with damping. The global well-posedness is established in $L^2$-$L^p$ critical Besov spaces. In our results, the existence theorem provides us with bounds that are independent of the relaxation parameter $\varepsilon$ and capillary coefficient $k$. As a consequence, we rigorously justify the relaxation limit and study the effect of the Korteweg-type dispersion on the relaxation limit. Specially, when $k\equiv 0$, our theorems reduce to the results in Crin-Barat and Danchin (2022) [28,29] on the Euler system with damping, and the smallness assumption for low-frequency initial data of velocity is weaker in some way.