Asymptotic Convergence of Hyperbolic Relaxation Systems
Organizers
Speaker
Zeyu Jin
Time
Thursday, November 27, 2025 3:15 PM - 4:15 PM
Venue
Online
Online
Zoom 928 682 9093
(BIMSA)
Abstract
Hyperbolic relaxation systems are fundamental in modeling non-equilibrium processes across physics, ranging from kinetic theory and traffic flows to plasmas and reacting gases. The central mathematical challenge in this field lies in rigorously justifying the relaxation limit: establishing that the solution to the full non-equilibrium system converges to the solution of the reduced equilibrium system as the relaxation time tends to zero.
Historically, the asymptotic analysis of these problems has been successfully developed. Classical results, such as those by Chen-Levermore-Liu and W.-A. Yong, established a robust framework relying on entropy dissipation or specific structural stability conditions. While these structural conditions are incredibly powerful for ensuring stability, recent investigations suggest they are sufficient rather than strictly necessary—even within purely dissipative settings. The limitations of the classical dissipative framework become even more apparent when addressing highly oscillatory systems, such as rotating fluids or magnetized plasmas.
In this talk, I will present our recent work establishing an optimal convergence theory in linear regime. We identify uniform boundedness of the solution operator as the minimal, necessary, and sufficient stability condition for asymptotic convergence. This result can be viewed as a "Lax Equivalence Theorem" for hyperbolic relaxation approximations: stability is sufficient to imply convergence. I will also discuss the key analytical tools used to treat the high-frequency oscillations, including the Mori-Zwanzig projection formalism and a generalized Riemann-Lebesgue lemma.
Historically, the asymptotic analysis of these problems has been successfully developed. Classical results, such as those by Chen-Levermore-Liu and W.-A. Yong, established a robust framework relying on entropy dissipation or specific structural stability conditions. While these structural conditions are incredibly powerful for ensuring stability, recent investigations suggest they are sufficient rather than strictly necessary—even within purely dissipative settings. The limitations of the classical dissipative framework become even more apparent when addressing highly oscillatory systems, such as rotating fluids or magnetized plasmas.
In this talk, I will present our recent work establishing an optimal convergence theory in linear regime. We identify uniform boundedness of the solution operator as the minimal, necessary, and sufficient stability condition for asymptotic convergence. This result can be viewed as a "Lax Equivalence Theorem" for hyperbolic relaxation approximations: stability is sufficient to imply convergence. I will also discuss the key analytical tools used to treat the high-frequency oscillations, including the Mori-Zwanzig projection formalism and a generalized Riemann-Lebesgue lemma.