The dissipative structure of symmetric hyperbolic systems with symmetric diffusion or symmetric relaxation is completely characterized by Craftsmanship Condition (Umeda-Kawashima-Shizuta 1984). In that case the dissipative structure is expressed by the uniform dissipativity of the form $Re \lambda (i \xi)/(1+|\xi|^2)$. 

In this talk, we first consider symmetric hyperbolic systems with memory-type symmetric diffusion, where the memory kernel is given by $e^{-t}$. We observe that the systems are transformed into symmetric hyperbolic systems with symmetric relaxation. Moreover, Craftsmanship Condition for the original systems is preserved by this reduction. Therefore the dissipative structure of the reduced systems is expressed again by $Re \lambda (i \xi)/(1+|\xi|^2)$.  

Next we discuss symmetric hyperbolic systems with memory-type symmetric relaxation. In this case the systems are reduced to symmetric hyperbolic systems with non-symmetric relaxation, and Craftsmanship Condition for the original systems is not preserved by this reduction. Indeed, in this case we can show that the dissipative structure of the reduced systems is expressed by $Re \lambda (i \xi)/(1+|\xi|^2)^2$.

This dissipative structure, which was studied by Ueda-Duan-Kawashima 2012, is very weak in high frequency region and the corresponding decay estimate is of the regularity-loss type.

 

报告人介绍：川島秀一于1984年获得日本京都大学博士学位，1986年进入九州大学任教并在1994年成为数理学院教授。川島秀一教授致力于双曲方程组和流体力学方程的研究，在Arch. Rat. Mech. Anal.、SIAM J. Math. Anal.、J. Diff. Equations等知名数学杂志上发表论文100余篇。