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报告题目Global entropy-bounded solution to the heat conductive compressible (Navier-Stokes equations   (2019030)

报告人:李进开(华南师范大学 研究员)



报告摘要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.

个人简介:李进开,男,博士,研究员,博士生导师。2013年博士毕业于香港中文大学数学研究所,导师为辛周平教授。20138月至20167月在以色列魏茨曼科学研究所从事博士后研究工作,合作导师为Edriss S. Titi教授,20168月至20187月在香港中文大学数学系任研究助理教授,20188月起在华南师范大学华南数学应用与交叉研究中心任研究员。主要研究方向为流体动力学偏微分方程组,具体包括大气海洋动力学方程组、可压缩Navier-Stokes方程组等,相关成果发表于CPAM, ARMACPDE, JFA等杂志,入选第14批国家青年千人。



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