| Abstract: |
| In this talk, we discuss the long-time behavior of one-dimensional compressible fluids with internal capillarity. For the barotropic Navier-Stokes-Korteweg system, we establish the time-asymptotic stability of a composite wave consisting of a rarefaction wave and a shifted viscous-dispersive shock. Furthermore, we extend this framework to the non-barotropic Navier-Stokes-Fourier-Korteweg system to prove the stability of composite waves involving a rarefaction wave, a viscous contact wave, and a shifted viscous-dispersive shock. Our analysis is based on the method of $a$-contraction with shifts, which combines weighted relative entropy estimates with dynamical shifts applied to the shock profiles. These results are based on joint works with Jeongho Kim and Moon-Jin Kang. |
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