## DE Seminar / Applied Math Colloquium: Dr. Xin Liu

#### Texas A&M University

Tuesday, January 22, 2019 · 1 PM - 2 PM

**Title:**Solutions with bounded entropy to compressible Navier-Stokes equations

**Abstract:**Compressible ﬂuids (or gas) can be modeled by hydrodynamic equations. The second law of thermodynamics states that for a closed system, the total entropy is non-decreasing. Our goal is to ﬁnd solutions to the hydrodynamic equations with bounded entropy. Demonstrated by Makino, Ukai, Kawashima, Xin, Yan, etc., the associated Cauchy problems with smooth solutions will blow up in ﬁnite time with the appearance of vacuum states for either inviscid or viscous ﬂows. These results motivate us to study the free boundary problems. As a ﬁrst step, 1. we establish the equilibrium for a model of radiation gaseous stars, which serves as an example of stationary ﬂows with bounded entropy; 2. we show local-in-time existence of evolving ﬂows with bounded entropy; 3. we also construct a class of global-in-time solutions with bounded entropy. These works contain part of the authors’ PhD thesis with Prof. Zhouping Xin as our advisor.

Joint work with Yuan Yuan, South China Normal University