Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case

Article type: Research Article

Authors: Zhao, Liang^a | Xi, Shuai^{b; *}

Affiliations: [a] School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China | [b] College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P.R. China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.

Keywords: Convergence rate, first order balance laws, partial dissipation, isotropic parabolic system, stream function

DOI: 10.3233/ASY-211687

Journal: Asymptotic Analysis, vol. 124, no. 1-2, pp. 163-198, 2021