Relative entropy method for the relaxation limit of hydrodynamic models

Abstract

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.

Weak-Strong Uniqueness and High-Friction Limit for Euler-Riesz Systems

“…The first step to establish the desired theorem is what is called the relative energy inequality. The calculations involved are by now standard and similar results can be found, for example, in [2,9,18]. Nevertheless, regarding the present case, we provide the details for completeness.…”

“…In the present case, those integration by parts formulas do not apply. Moreover, the same range for the adiabatic exponent was obtained in [9] where the authors establish the high-friction limit of a Euler system with a bounded interaction kernel. The high-friction limit of a pressureless Euler-Riesz system towards an aggregation equation is studied in [10,11].…”

“…Since its origin, e.g. [14], this method has seen an extensive applicability to diffusive relaxation [2,3,9,17,18].…”

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

“…Essentially, the same method is used to prove the aforementioned weak-strong uniqueness when the relative entropy measures the distance between weak (measure-valued) and strong solutions. This strategy has been applied for several singular limits [3,21,22,25,57,60,61] and we also refer to the excellent review on weak-strong uniqueness [72].…”

The relaxation limit of bipolar fluid models

“…The relative energy method is used here to perform this limiting process for strong solutions of (1) in several space dimensions. This approach was successful for the relaxation limit in single-species fluid models [15,3], as well as for certain (weakly coupled through friction) multicomponent systems [9]. The relative energy method provides an efficient mathematical mechanism for stability analysis and establishing limiting processes; see [5] for early developments, [2,15,16] and references therein for applications to diffusive relaxation.…”