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學術報告

報告題目：Global entropy-bounded solution to the heat conductive compressible (Navier-Stokes equations   (2019030)

報告人：李進開（華南師范大學 研究員）

報告時間：2019年05月31日（周五）下午14:00-15:00

報告地點：理學實驗樓312

報告摘要：The entropy is one of the fundamental physical states for compressible fluids. Due to the singularity of the logarithmic function at zero and the singularity of the entropy equation in the vacuum region, it is difficult to analyze mathematically the entropy of the ideal gases in the presence of vacuum. We will present in this talk that an ideal gas can retain its uniform boundedness of the entropy, up to any finite time, as long as the vacuum presents at the far field only and the density decays to vacuum sufficiently slowly at the far field. Precisely, for the Cauchy problem of the one-dimensional heat conductive compressible Navier-Stokes equations, in the presence of vacuum at the far field only, the local and global existence and uniqueness of strong solutions, and the uniform boundedness (up to any finite time) of the corresponding entropy have been established, provided that the initial density decays no faster than $O(\frac{1}{x^2})$ at the far field. By introducing the Jacobian between the Euler and Lagrangian coordinates to replace the density as one of the unknowns, we establish the global existence of strong solutions, in the presence of vacuum, and, thus, extend successfully the classic results in [1,2] from the non-vacuum case to the vacuum case. The main tools of proving the uniform boundedness of the entropy are some weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations, with the weights being singular at the far field, and the De Giorgi iteration technique applied to a certain class of degenerate parabolic equations in nonstandard ways. The De Giorgi iterations are carried out to different equations to obtain the lower and upper bounds of the entropy.

[1] Kazhikhov, A. V.: Cauchy problem for viscous gas equations, Siberian Math. J., 23 (1982),44-49.

[2] Kazhikhov, A. V.; Shelukhin, V. V.: Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.

個人簡介：李進開，男，博士，研究員，博士生導師。2013年博士畢業于香港中文大學數學研究所，導師為辛周平教授。2013年8月至2016年7月在以色列魏茨曼科學研究所從事博士后研究工作，合作導師為Edriss S. Titi教授，2016年8月至2018年7月在香港中文大學數學系任研究助理教授，2018年8月起在華南師范大學華南數學應用與交叉研究中心任研究員。主要研究方向為流體動力學偏微分方程組，具體包括大氣海洋動力學方程組、可壓縮Navier-Stokes方程組等，相關成果發表于CPAM, ARMA，CPDE, 及JFA等雜志，入選第14批國家重大人才工程項目入選者。