A Note on Aubin-Lions-Dubinskiĭ Lemmas

In this work we study a degenerate pseudo-parabolic system with cross diffusion describing the evolution of the densities of an unsaturated twophase flow mixture with dynamic capillary pressure in porous medium with saturation-dependent relaxation parameter and hypocoercive diffusion operator modeling cross diffusion. The equations are derived in a thermodynamically correct way from mass conservation laws. Global-in-time existence of weak solutions to the system in a bounded domain with equilibrium boundary conditions is shown. The main tools of the analysis are an entropy inequality and a crucial apriori bound which allows for controlling the degeneracy.

show abstract

“…Assume that Assumptions (H1)-(H5) hold. Then there exists a global-in-time weak solution S : Ω × (0, ∞) → D to (1)- (4).…”

Section: Resultsmentioning

confidence: 99%

Global existence for a two-phase flow model with cross-diffusion

Daus¹,

Milišić²,

Zamponi³

2020

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

show abstract

“…We also mention the results obtained in [3,7,23], where generalizations of the Aubin-Lions-Simon lemma in various types of nonlinear settings were obtained, and the work in [11] where a version of Aubin-Lions-Simon result was obtained in the context of finite element spaces.…”

Section: Literature Reviewmentioning

confidence: 98%

A generalization of the Aubin–Lions–Simon compactness lemma for problems on moving domains

Muha

Čanić

2019

Journal of Differential Equations

View full text Add to dashboard Cite

This work addresses an extension of the Aubin-Lions-Simon compactness result to generalized Bochner spaces L 2 (0, T ; H(t)), where H(t) is a family of Hilbert spaces, parameterized by t. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin-Lions-Simon compactness result holds. Concrete examples of the application of the compactness theorem are presented, including a classical problem for the incompressible, Navier-Stokes equations defined on a given non-cylindrical domain, and a class of fluid-structure interaction problems for the incompressible, Navier-Stokes equations, coupled to the elastodynamics of a Koiter shell. The compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear, moving boundary problems, using Rothe's method. *

show abstract

“…This implies that ∂ t u k is bounded in L s (0, T ; W 1,s (Ω)) ′ uniformly with respect to k. In order to obtain compactness for the sequence √ u k , we need a modified version of Aubin-Lions Lemma. In particular, if a > 1 we apply 20 Theorem 1 of [32] (with Φ(·) = | · |, so that | · | ∈ W 1,1 (R) and meas({ | · | > δ}) → 0 in L 1 as δ → 0), otherwise we apply Theorem 3 of [24] (with m = 1 2 ). We deduce that the sequence √ u k is compact in L 2 (Ω T ).…”

Section: Passage To the Limit And Proof Of Theoremmentioning

confidence: 99%

The role of a strong confining potential in a nonlinear Fokker–Planck equation

2020

View full text Add to dashboard Cite

We show that solutions of nonlinear nonlocal Fokker-Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for L 2 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.

show abstract

A Note on Aubin-Lions-Dubinskiĭ Lemmas

Cited by 74 publications

References 19 publications

Global existence for a two-phase flow model with cross-diffusion

Global existence for a two-phase flow model with cross-diffusion

A generalization of the Aubin–Lions–Simon compactness lemma for problems on moving domains

The role of a strong confining potential in a nonlinear Fokker–Planck equation

Contact Info

Product

Resources

About