Some Analytical Solutions to the Compressible Euler-Korteweg Equations

When \(\gamma =\frac{3}{2}\), we construct some analytical solutions to the compressible Euler-Korteweg equations, where \(\gamma \) is the adiabatic exponent. For the one-dimensional case, we provide a self-similar analytical solution for the vacuum free boundary problem on a finite interval, the vacuum free boundary problem on a half line and the Cauchy problem, respectively. From the constructed solutions, we find that the free boundary for the vacuum free boundary problem on a finite interval expands out linearly in time, this is same to the case when the capillarity force is absent. But for the vacuum free boundary problem on a half line and the Cauchy problem, we find that the capillarity effect plays a crucial role in preventing the smooth solutions from blowing up. We also extend these results to the N-dimensional radially symmetric case and the three-dimensional cylindrically symmetric case, respectively.