Entropic structure and duality for multiple species cross-diffusion systems

Abstract

This paper deals with the existence of global weak solutions for a wide class of (multiple species) cross-diffusions systems. The existence is based on two different ingredients: an entropy estimate giving some gradient control and a duality estimate that gives naturally L 2 control. The heart of our proof is a semi-implicit scheme tailored for cross-diffusion systems firstly defined in [DLMT15] and a (nonlinear Aubin-Lions type) compactness result developped in [Mou16, ACM15] that turns the (potentially weak) gradient estimates into almost everywhere convergence. We apply our results to models having an entropy relying on the detailed balance condition exhibited by Chen et. al. in [CDJ16].

From Nonlocal to Classical Shigesada--Kawasaki--Teramoto Systems: Triangular Case with Bounded Coefficients

“…But the strength of the duality estimate is that it can be written as soon as the dual equation has one solution and it can hence be used as an a priori estimate for any solution of the Kolmogorov equation when the diffusion µ is only supposed integrable (and positively lower-bounded). This for instance was used in the context of cross-diffusion systems in the articles [15,16,27]. However, it is important to note that in all these references the duality lemma was either stated in an idealized setting (assuming z to be regular enough to justify all the manipulations) or a discretized one.…”

Improved duality estimates: time discrete case and applications to a class of cross-diffusion systems

“…We recall a general result on the construction procedure introduced in [12] and extended in [20]. We adapt lemma from [12] Lemma 15.…”

“…It has been established in [12,20] that for T > 0, we can extract a subsequence U τn that converges almost surely to U . Using then the L 2 (Q T ) standard bounds…”

“…For such systems, one can have additional estimates derived in the context of reaction-diffusion systems in [23], applied to cross-diffusion models in [4] for a specific case and in a more systematic way in [3,11,12,20]. This can be resumed in the fact that under general hypothesis we can deduce an a priori bounds on U L 2 (Q T ) (Q T standing for Ω × (0,T )).…”

About the entropic structure of detailed balanced multi-species cross-diffusion equations

“…nontriangular) local multi-species SKT system when one starts with a microscopic model (whether on a discrete set of positions or on a continuous set of positions, using a nonlocality which disappears in the limit). Note, however, that very recently, Moussa [21] manged to prove the convergence in the case of strictly triangular limiting local SKT model with bounded coefficients starting on a continuous set of positions with a nonlocality which disappears in the limit by using duality techniques (introduced for instance by Pierre and Schmitt in [13]).…”