A research team from South China University of Technology has made progress in understanding both the unconditional global existence and the vanishing viscosity limit of parabolic-elliptic coupled systems, with findings that extend existing research. The work, led by Prof. Changjiang Zhu and Dr. Qiaolong Zhu, is published in Acta Mathematica Scientia.
The study focuses on a parabolic-elliptic coupled system, which is a simplified model critical to understanding phenomena where fluid motion interacts with heat radiation. Such a model is considered with the viscosity. A natural question: how solutions of viscous systems behave as the viscosity coefficient grows infinitely small? To explore this, the researchers analyzed two types of mathematical problems. The first was the Cauchy problem, while the second was the initial-boundary value problem. For these two problems, they considered two conditions: initial data are close to a given wave with small wave strength, and initial data increase monotonically. Under these condition, the team solved this question and elaborated on the behavior of solutions when the viscosity is sufficiently small.
A major breakthrough of the research is the establishment of global existence of the parabolic-elliptic coupled system without any small condition of perturbation and wave strength, which is quite different from previous studies. Another lies in the derivation of explicit convergence rates. The team proved that as the viscosity coefficient tends to zero, solutions of the parabolic-elliptic system converge to those of the hyperbolic-elliptic system, which is a simplified model without viscosity in radiative hydrodynamics. These precise rates provide researchers with a critical tool for predicting how closely viscous models approximate inviscid behavior in different scenarios.
This work extends earlier research by offering precise convergence speeds and handling a wider range of conditions, providing deeper insight into how viscous flows transition to inviscid ones in mathematical models of radiation hydrodynamics.
See the article:
Vanishing viscosity limit of a parabolic-elliptic coupled system
https://doi.org/10.1007/s10473-025-0609-5
