Quasi-static limit for a hyperbolic conservation law

Marchesani, Stefano; Olla, Stefano; Xu, Lu

doi:10.1007/s00030-021-00716-5

“…The existence and uniqueness of the quasi-static limit ρ ε → ρ as ε → 0 is proven in [8]. Moreover, the ρ = ρ(x, t) satisfies the variational conditions ([9, 13, 1, 8]):…”

Section: Asep With Open Boundariesmentioning

confidence: 98%

Quasi-static limit for the asymmetric simple exclusion

De Masi,

Marchesani,

Olla

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the one-dimensional asymmetric simple exclusion process on the lattice {1, . . . , N } with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale N −a with a > 0. We prove that at the time scale N 1+a the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy-entropy flux pairs and a coupling argument.

show abstract

“…is then defined as the weak-limit, for ε → 0 + of ρ ε . The existence and uniqueness of this limit is proven in [10] as explained below. Define the critical line…”

Section: Definition 31 a Boundary Lax Entropy-entropy Flux Pair Formentioning

confidence: 99%

Quasi-static limit for the asymmetric simple exclusion

Masi

¹

,

Marchesani

²

,

Olla

³

et al. 2022

Probab. Theory Relat. Fields

Self Cite

6

0

View full text Add to dashboard Cite

We study the one-dimensional asymmetric simple exclusion process on the lattice {1, . . . , N } with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale N −a with a > 0. We prove that at the time scale N 1+a the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy-entropy flux pairs and a coupling argument.

show abstract

Scalar conservation law in a bounded domain with strong source at boundary

Xu

2024

Nonlinear Differ. Equ. Appl.

1

0

View full text Add to dashboard Cite

We consider a scalar conservation law with source in a bounded open interval $$\Omega \subseteq \mathbb R$$ Ω ⊆ R . The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function $$\varrho $$ ϱ with an intensity function $$V:\Omega \rightarrow \mathbb R_+$$ V : Ω → R + that grows to infinity at $$\partial \Omega $$ ∂ Ω . We define the entropy solution $$u \in L^\infty $$ u ∈ L ∞ and prove the uniqueness. When V is integrable, u satisfies the boundary conditions introduced by F. Otto (C. R. Acad. Sci. Paris, 322(1):729–734, 1996), which allows the solution to attain values at $$\partial \Omega $$ ∂ Ω different from the given boundary data. When the integral of V blows up, u satisfies an energy estimate and presents essential continuity at $$\partial \Omega $$ ∂ Ω in a weak sense.

show abstract

Quasi-static limit for a hyperbolic conservation law

Cited by 6 publications

References 11 publications

Quasi-static limit for the asymmetric simple exclusion

Quasi-static limit for the asymmetric simple exclusion

Quasi-static limit for the asymmetric simple exclusion

Scalar conservation law in a bounded domain with strong source at boundary

Contact Info

Product

Resources

About