The relaxation limit of bipolar fluid models

Abstract

<p style='text-indent:20px;'>This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid models, and it is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.</p>

The role of Riesz potentials in the weak-strong uniqueness for the Euler-Poisson system

“…where S(ρ) = −p(ρ)I − 1 2 |∇φ| 2 I + ∇φ ⊗ ∇φ is the stress tensor. Under this abstract setup one replicates the calculations done in [1,5] to derive the relative energy identity for this system. Let (ρ, ρū) with φ = K * ρ be another smooth solution of (3.2).…”

“…Next one presents the evolution of the relative total energy of the system. For a detailed exposition of the calculations involved refer to [1,5].…”

“…, and the second equality of (4.3). For a detailed presentation of the calculations involved refer to [1,5,7].…”

“…The relative energy method is an efficient tool for the analysis of stability of systems of conservation laws; see [4] for an early work on the stability of thermoelastic fluids, and [5] for more recent developments. This method has also been successful in establishing limiting processes; see [7] for the high-friction limit of single-species Euler flows towards gradient flows, and [1] for the relaxation limit of a two-species Euler-Poisson system with friction towards a bipolar drift-diffusion system.…”

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].…”