Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms

Elisa Davoli; Greta Marino; Jan-Frederik Pietschmann

arxiv: 2408.07396 · v2 · pith:GEBK5H5Lnew · submitted 2024-08-14 · 🧮 math.AP

Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms

Elisa Davoli , Greta Marino , Jan-Frederik Pietschmann This is my paper

Pith reviewed 2026-05-23 21:58 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlocal Cahn-Hilliardcross-diffusionweak solutionsexistencelocal asymptoticsdegenerate mobilitygradient flowboundedness-by-entropy

0 comments

The pith

Global weak solutions exist for a nonlocal Cahn-Hilliard cross-diffusion system and converge to local limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves global-in-time existence of weak solutions for a multicomponent cross-diffusion system that includes nonlocal Cahn-Hilliard terms driven by a symmetric singular kernel together with degenerate mobility. The argument adapts the notion of weak solution to degeneracies by using the formal gradient-flow structure to pose an auxiliary variational problem, applies the boundedness-by-entropy method to secure uniform estimates on time-discrete approximations, and passes to the limit as the time step vanishes while handling the low-regularity nonlocal terms. The same construction yields convergence of the nonlocal solutions to their local counterparts. A reader would care because these systems model phase separation in mixtures, and the existence theory supplies a rigorous basis for comparing nonlocal and local descriptions before taking limits.

Core claim

The authors define a weak-solution notion suited to degeneracies and prove its global existence for the nonlocal system by constructing time-discrete solutions from minimization of an auxiliary energy that encodes the gradient-flow structure, deriving uniform bounds via an extension of the boundedness-by-entropy method, and passing to the continuous limit while treating the Cahn-Hilliard contributions separately. They further establish that solutions of this nonlocal class converge to solutions of the corresponding local Cahn-Hilliard equations.

What carries the argument

An auxiliary variational problem derived from the formal gradient-flow structure, to which the boundedness-by-entropy method is applied to obtain uniform estimates.

If this is right

Global weak solutions exist for the nonlocal system despite degenerate mobility and low regularity of the nonlocal terms.
Solutions of the nonlocal equations converge to solutions of the local system.
Time-discrete approximations obtained from the auxiliary variational problem converge to the continuous weak solution.
The boundedness-by-entropy method yields the necessary a-priori estimates when the gradient-flow structure is present.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The convergence result indicates that local models remain valid approximations when the nonlocality scale is small.
The same variational construction could be tested on other cross-diffusion systems that possess a formal gradient-flow structure but lack symmetry in the kernel.
Numerical schemes built for the local equations may serve as practical surrogates for the nonlocal system in the small-nonlocality regime.

Load-bearing premise

The symmetric singular kernel and the formal gradient-flow structure allow the boundedness-by-entropy method to produce uniform estimates on the auxiliary variational problem.

What would settle it

A concrete symmetric kernel or degenerate mobility for which the entropy estimates on the auxiliary problem cease to be uniform, so that the time-discrete minimizers fail to converge to a weak solution.

read the original abstract

We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to possible degeneracies and prove, as our first main result, its global-in-time existence. The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired nonlocal weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. Finally, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves global existence of weak solutions for a degenerate multicomponent cross-diffusion system with nonlocal Cahn-Hilliard terms via gradient-flow structure and boundedness-by-entropy, plus convergence to the local case.

read the letter

The core result here is a global-in-time existence theorem for weak solutions to this nonlocal degenerate system, built on a time-discrete approximation, an auxiliary variational problem, and careful passage to the limit that accounts for the low regularity of the nonlocal terms. They also show that solutions converge to the corresponding local Cahn-Hilliard cross-diffusion system. That combination is the actual new piece: prior work handled either local or non-degenerate cases, so extending the boundedness-by-entropy method to the symmetric singular kernel plus degeneracy is a genuine but incremental step inside the existing PDE toolkit. The symmetric kernel is used to close the entropy estimates, which is a reasonable choice and appears to work formally from the abstract. The convergence result is a clean bonus that strengthens the paper. The approach stays within established techniques rather than inventing new ones, so the technical lift comes from making the estimates close for this specific combination. The main soft spot is that the abstract leaves the details of the limit passage for the nonlocal terms somewhat opaque; without seeing the full estimates it is hard to judge whether any hidden regularity or approximation issues arise there. No load-bearing gaps are visible in the stated architecture, and the proof strategy does not look circular. This is a narrow but competent existence paper aimed at specialists in cross-diffusion and phase-separation models. Readers working on mathematical analysis of degenerate PDE systems will find the weak-solution notion and the convergence statement useful. It is solid enough on its own terms to merit a serious referee who can check the auxiliary problem and the nonlocal limit step in detail. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies a multicomponent cross-diffusion system with degenerate mobility and nonlocal Cahn-Hilliard terms driven by a symmetric singular kernel. It defines a suitable notion of weak solution, proves global-in-time existence via the formal gradient-flow structure combined with an extension of the boundedness-by-entropy method applied to an auxiliary variational problem (yielding uniform estimates for a time-discrete approximation), passes to the limit while treating the low-regularity nonlocal terms, and finally establishes convergence of the nonlocal solutions to their local counterparts.

Significance. If the central arguments hold, the work extends entropy-based techniques for degenerate cross-diffusion to a nonlocal singular-kernel setting and supplies a convergence result linking nonlocal and local models. These are useful contributions to the analysis of phase-separation systems, particularly for handling degeneracies and low-regularity terms without additional ad-hoc assumptions.

minor comments (3)

[Section 2] §2 (or wherever the kernel assumptions appear): the precise integrability and symmetry conditions on the singular kernel should be stated explicitly at the outset so that the reader can immediately verify they suffice for the entropy estimates.
[Definition of weak solution] Definition of weak solution (likely §3): the precise sense in which the nonlocal Cahn-Hilliard terms are tested against test functions of limited regularity should be spelled out, including any integration-by-parts or approximation arguments used to justify the formulation.
[Proof of existence] The auxiliary variational problem used in the boundedness-by-entropy step: a short remark on why the chosen regularization preserves the gradient-flow structure would help the reader follow the uniform-estimate argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were raised in the report, so there are no specific points requiring point-by-point responses or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes global existence of weak solutions for a nonlocal cross-diffusion system and convergence to local limits via a time-discrete approximation, boundedness-by-entropy estimates on an auxiliary variational problem, and passage to the limit using the formal gradient-flow structure. These steps constitute a standard constructive existence argument in degenerate PDE analysis; no equation or result is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness claim reduces to a self-citation chain. The derivation is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond standard functional-analysis background.

pith-pipeline@v0.9.0 · 5704 in / 1077 out tokens · 25353 ms · 2026-05-23T21:58:25.500388+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

K_ε(x,y) := ρε(|x−y|)/|x−y|^2 ... B_ε(v) = (K_ε∗1)v − K_ε∗v ... B_ε(v_ε) → Δv weakly

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

Abels and C

H. Abels and C. Hurm. Strong nonlocal-to-local converge nce of the Cahn-Hilliard equation and its operator. Journal of Diﬀerential Equations , 402:593–624, 2024

work page 2024
[2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient Flows in Metric Spaces and in the Space of Probabili ty Measures . Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser, 2. ed edition, 2008

work page 2008
[3]

Bakhta and V

A. Bakhta and V. Ehrlacher. Cross-diﬀusion systems with non-zero ﬂux and moving boundary conditions. ESAIM Math. Model. Numer. Anal. , 52(4):1385–1415, 2018

work page 2018
[4]

Benedetto, E

D. Benedetto, E. Caglioti, and M. Pulvirenti. A kinetic e quation for granular media. RAIRO Mod´ el Math. Anal. Num´ er., 31:615—-641, 1997

work page 1997
[5]

Berendsen, M

J. Berendsen, M. Burger, V. Ehrlacher, and J.-F. Pietsch mann. Uniqueness of strong solutions and weak–strong stability in a system of cross-diﬀusion equations. J. Evol. Equ. , Sep 2019

work page 2019
[6]

Berendsen, M

J. Berendsen, M. Burger, and J.-F. Pietschmann. On a cros s-diﬀusion model for multiple species with nonlocal interaction and size exclusion. Nonlinear Anal. , 159:10–39, 2017. Advances in Reaction-Cross-Diﬀusion Sy stems

work page 2017
[7]

Bourgain, H

J. Bourgain, H. Brezis, and P. Mironescu. Another look at sobolev spaces. Optimal control and partial diﬀerential equations, pages 439–455, 2001

work page 2001
[8]

H. Brezis. Functional analysis, Sobolev spaces and partial diﬀerenti al equations . Springer, New York, 2011

work page 2011
[9]

Burger, M

M. Burger, M. D. Francesco, J.-F. Pietschmann, and B. Sch lake. Nonlinear Cross Diﬀusion with Size Exclusion. SIAM J. Math. Anal. , 42(6):2842–2871, 2010

work page 2010
[10]

Burger, S

M. Burger, S. Hittmeir, H. Ranetbauer, and M.-T. W olfra m. Lane formation by side-stepping. SIAM Journal on Mathematical Analysis , 48(2):981–1005, 2016

work page 2016
[11]

Burger, B

M. Burger, B. Schlacke, and M.-T. W olfram. Nonlinear po isson-nerst-planck equations for ion ﬂux through conﬁned geometries. Nonlinearity, 25:961–990, 2012

work page 2012
[12]

Canc` es and B

C. Canc` es and B. Gaudeul. A convergent entropy diminis hing ﬁnite volume scheme for a cross-diﬀusion system. arXiv preprint arXiv:2001.11222 , 2020

work page arXiv 2001
[13]

Chen and A

L. Chen and A. J¨ ungel. Analysis of a multidimensional p arabolic population model with strong cross-diﬀusion. SIAM J. Math. Anal. , 36(1):301–322 (electronic), 2004

work page 2004
[14]

Chen and A

L. Chen and A. J¨ ungel. Analysis of a parabolic cross-di ﬀusion population model without self-diﬀusion. J. Diﬀerential Equations, 224(1):39–59, 2006

work page 2006
[15]

Cicalese, L

M. Cicalese, L. D. Luca, M. Novaga, and M. Ponsiglione. G round states of a two phase model with cross and self attractive interactions. SIAM J. Math. Anal. , pages 3412–3443, 2016

work page 2016
[16]

Davoli, C

E. Davoli, C. Gavioli, and L. Lombardini. Existence res ults for Cahn-Hilliard type systems driven by nonlocal integrodiﬀerential operators with singular kernels, 2024

work page 2024
[17]

Davoli, H

E. Davoli, H. Ranetbauer, L. Scarpa, and L. Trussardi. D egenerate nonlocal Cahn-Hilliard equations: W ell- posedness, regularity and local asymptotics. Ann. I. H. Poincar´ e- AN, 2019

work page 2019
[18]

Dreher and A

M. Dreher and A. J¨ ungel. Compact families of piecewise constant functions in lp(0, t; b). Nonlinear Anal., 75(6):3072– 3077, 2012

work page 2012
[19]

Ehrlacher, G

V. Ehrlacher, G. Marino, and J.-F. Pietschmann. Existe nce of weak solutions to a cross-diﬀusion system with Cahn-Hilliard type terms. Journal of Diﬀerential Equations , 286:578–623, 2021

work page 2021
[20]

Elbar and J

C. Elbar and J. Skrzeczkowski. Nonlocal-to-local conv ergence of the Cahn-Hilliard equation with degenerate mobi lity and the Flory-Huggins potential, 2024

work page 2024
[21]

Elliott and S

C. Elliott and S. Luckhaus. A generalized equation for p hase separation of a multi-component mixture with interfac ial free energy. preprint SFB 256 , 1991

work page 1991
[22]

Giacomin and J

G. Giacomin and J. Lebowitz. Phase segregation dynamic s in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys , 87(1):37–61, 1997

work page 1997
[23]

J¨ ungel

A. J¨ ungel. The boundedness-by-entropy method for cro ss-diﬀusion systems. Nonlinearity, 28(6):1963, 2015

work page 1963
[24]

J¨ ungel, S

A. J¨ ungel, S. Portisch, and A. Zurek. Nonlocal cross-d iﬀusion systems for multi-species populations and network s. Nonlinear Analysis, 219:112800, 2022

work page 2022
[25]

J¨ ungel, S

A. J¨ ungel, S. Portisch, and A. Zurek. A convergent ﬁnit e-volume scheme for nonlocal cross-diﬀusion systems for multi-species populations. ESAIM: Mathematical Modelling and Numerical Analysis , 58:759–792, 2024

work page 2024
[26]

Klinkert

T. Klinkert. Comprehension and optimisation of the co-evaporation depo sition of Cu (In, Ga) Se2 absorber layers for very high eﬃciency thin ﬁlm solar cells . PhD thesis, Universit´ e Pierre et Marie Curie-Paris VI, 20 15

work page
[27]

K¨ ufner

K. K¨ ufner. Invariant regions for quasilinear reactio n-diﬀusion systems and applications to a two population mod el. NoDEA-Nonlinear Diﬀ. , 3:421–444, 1996. CROSS-DIFFUSION EQUATION WITH NONLOCAL CAHN-HILLIARD TER MS 23

work page 1996
[28]

Lepoutre, M

T. Lepoutre, M. Pierre, and G. Rolland. Global Well-Pos edness of a Conservative Relaxed Cross Diﬀusion System. SIAM J. Math. Anal. , 44(3):1674–1693, 2012

work page 2012
[29]

Melchionna, H

S. Melchionna, H. Ranetbauer, L. Scarpa, and L. Trussar di. From nonlocal to local Cahn-Hilliard equation. Adv. Math. Sci. Appl. , 28(2):197–211, 2019

work page 2019
[30]

Mogilner and L

A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Biol. , 38:534—-570, 2012

work page 2012
[31]

Painter and T

K. Painter and T. Hillen. Volume-ﬁlling and quorum-sen sing in models for chemosensitive movement. Can. Appl. Math. Q. , 10(4):501 – 543, 2002

work page 2002
[32]

K. J. Painter. Continuous models for cell migration in t issues and applications to cell sorting via diﬀerential che mo- taxis. Bulletin of Mathematical Biology , 71(5):1117–1147, 2009

work page 2009
[33]

A. C. Ponce. An estimate in the spirit of Poincar´ e’s ine quality. Journal of the European Mathematical Society , 6(1):1–15, 2004

work page 2004
[34]

A. C. Ponce. A new approach to Sobolev spaces and connect ions to Γ-convergence. Calc. Var. Partial Diﬀerential Equations, 19(3):229–255, 2004

work page 2004
[35]

E. W einan. Dynamics of vortex liquids in ginzburg-land au theories with applications to superconductivity. Physical Review B , 50:1126—-1135, 2012

work page 2012
[36]

W enisch, R

R. W enisch, R. H¨ ubner, F. Munnik, S. Melkhanova, S. Gem ming, G. Abrasonis, and M. Krause. Nickel-enhanced graphitic ordering of carbon ad-atoms during physical vapo r deposition. Carbon, 100:656 – 663, 2016. (E. Davoli) Tu Wien, Institute for Analysis and Scientific Computing, Wi edner Hauptstraße 8-10, 1040 Wien, Austria Email address : elisa.davoli@...

work page 2016

[1] [1]

Abels and C

H. Abels and C. Hurm. Strong nonlocal-to-local converge nce of the Cahn-Hilliard equation and its operator. Journal of Diﬀerential Equations , 402:593–624, 2024

work page 2024

[2] [2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient Flows in Metric Spaces and in the Space of Probabili ty Measures . Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser, 2. ed edition, 2008

work page 2008

[3] [3]

Bakhta and V

A. Bakhta and V. Ehrlacher. Cross-diﬀusion systems with non-zero ﬂux and moving boundary conditions. ESAIM Math. Model. Numer. Anal. , 52(4):1385–1415, 2018

work page 2018

[4] [4]

Benedetto, E

D. Benedetto, E. Caglioti, and M. Pulvirenti. A kinetic e quation for granular media. RAIRO Mod´ el Math. Anal. Num´ er., 31:615—-641, 1997

work page 1997

[5] [5]

Berendsen, M

J. Berendsen, M. Burger, V. Ehrlacher, and J.-F. Pietsch mann. Uniqueness of strong solutions and weak–strong stability in a system of cross-diﬀusion equations. J. Evol. Equ. , Sep 2019

work page 2019

[6] [6]

Berendsen, M

J. Berendsen, M. Burger, and J.-F. Pietschmann. On a cros s-diﬀusion model for multiple species with nonlocal interaction and size exclusion. Nonlinear Anal. , 159:10–39, 2017. Advances in Reaction-Cross-Diﬀusion Sy stems

work page 2017

[7] [7]

Bourgain, H

J. Bourgain, H. Brezis, and P. Mironescu. Another look at sobolev spaces. Optimal control and partial diﬀerential equations, pages 439–455, 2001

work page 2001

[8] [8]

H. Brezis. Functional analysis, Sobolev spaces and partial diﬀerenti al equations . Springer, New York, 2011

work page 2011

[9] [9]

Burger, M

M. Burger, M. D. Francesco, J.-F. Pietschmann, and B. Sch lake. Nonlinear Cross Diﬀusion with Size Exclusion. SIAM J. Math. Anal. , 42(6):2842–2871, 2010

work page 2010

[10] [10]

Burger, S

M. Burger, S. Hittmeir, H. Ranetbauer, and M.-T. W olfra m. Lane formation by side-stepping. SIAM Journal on Mathematical Analysis , 48(2):981–1005, 2016

work page 2016

[11] [11]

Burger, B

M. Burger, B. Schlacke, and M.-T. W olfram. Nonlinear po isson-nerst-planck equations for ion ﬂux through conﬁned geometries. Nonlinearity, 25:961–990, 2012

work page 2012

[12] [12]

Canc` es and B

C. Canc` es and B. Gaudeul. A convergent entropy diminis hing ﬁnite volume scheme for a cross-diﬀusion system. arXiv preprint arXiv:2001.11222 , 2020

work page arXiv 2001

[13] [13]

Chen and A

L. Chen and A. J¨ ungel. Analysis of a multidimensional p arabolic population model with strong cross-diﬀusion. SIAM J. Math. Anal. , 36(1):301–322 (electronic), 2004

work page 2004

[14] [14]

Chen and A

L. Chen and A. J¨ ungel. Analysis of a parabolic cross-di ﬀusion population model without self-diﬀusion. J. Diﬀerential Equations, 224(1):39–59, 2006

work page 2006

[15] [15]

Cicalese, L

M. Cicalese, L. D. Luca, M. Novaga, and M. Ponsiglione. G round states of a two phase model with cross and self attractive interactions. SIAM J. Math. Anal. , pages 3412–3443, 2016

work page 2016

[16] [16]

Davoli, C

E. Davoli, C. Gavioli, and L. Lombardini. Existence res ults for Cahn-Hilliard type systems driven by nonlocal integrodiﬀerential operators with singular kernels, 2024

work page 2024

[17] [17]

Davoli, H

E. Davoli, H. Ranetbauer, L. Scarpa, and L. Trussardi. D egenerate nonlocal Cahn-Hilliard equations: W ell- posedness, regularity and local asymptotics. Ann. I. H. Poincar´ e- AN, 2019

work page 2019

[18] [18]

Dreher and A

M. Dreher and A. J¨ ungel. Compact families of piecewise constant functions in lp(0, t; b). Nonlinear Anal., 75(6):3072– 3077, 2012

work page 2012

[19] [19]

Ehrlacher, G

V. Ehrlacher, G. Marino, and J.-F. Pietschmann. Existe nce of weak solutions to a cross-diﬀusion system with Cahn-Hilliard type terms. Journal of Diﬀerential Equations , 286:578–623, 2021

work page 2021

[20] [20]

Elbar and J

C. Elbar and J. Skrzeczkowski. Nonlocal-to-local conv ergence of the Cahn-Hilliard equation with degenerate mobi lity and the Flory-Huggins potential, 2024

work page 2024

[21] [21]

Elliott and S

C. Elliott and S. Luckhaus. A generalized equation for p hase separation of a multi-component mixture with interfac ial free energy. preprint SFB 256 , 1991

work page 1991

[22] [22]

Giacomin and J

G. Giacomin and J. Lebowitz. Phase segregation dynamic s in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys , 87(1):37–61, 1997

work page 1997

[23] [23]

J¨ ungel

A. J¨ ungel. The boundedness-by-entropy method for cro ss-diﬀusion systems. Nonlinearity, 28(6):1963, 2015

work page 1963

[24] [24]

J¨ ungel, S

A. J¨ ungel, S. Portisch, and A. Zurek. Nonlocal cross-d iﬀusion systems for multi-species populations and network s. Nonlinear Analysis, 219:112800, 2022

work page 2022

[25] [25]

J¨ ungel, S

A. J¨ ungel, S. Portisch, and A. Zurek. A convergent ﬁnit e-volume scheme for nonlocal cross-diﬀusion systems for multi-species populations. ESAIM: Mathematical Modelling and Numerical Analysis , 58:759–792, 2024

work page 2024

[26] [26]

Klinkert

T. Klinkert. Comprehension and optimisation of the co-evaporation depo sition of Cu (In, Ga) Se2 absorber layers for very high eﬃciency thin ﬁlm solar cells . PhD thesis, Universit´ e Pierre et Marie Curie-Paris VI, 20 15

work page

[27] [27]

K¨ ufner

K. K¨ ufner. Invariant regions for quasilinear reactio n-diﬀusion systems and applications to a two population mod el. NoDEA-Nonlinear Diﬀ. , 3:421–444, 1996. CROSS-DIFFUSION EQUATION WITH NONLOCAL CAHN-HILLIARD TER MS 23

work page 1996

[28] [28]

Lepoutre, M

T. Lepoutre, M. Pierre, and G. Rolland. Global Well-Pos edness of a Conservative Relaxed Cross Diﬀusion System. SIAM J. Math. Anal. , 44(3):1674–1693, 2012

work page 2012

[29] [29]

Melchionna, H

S. Melchionna, H. Ranetbauer, L. Scarpa, and L. Trussar di. From nonlocal to local Cahn-Hilliard equation. Adv. Math. Sci. Appl. , 28(2):197–211, 2019

work page 2019

[30] [30]

Mogilner and L

A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Biol. , 38:534—-570, 2012

work page 2012

[31] [31]

Painter and T

K. Painter and T. Hillen. Volume-ﬁlling and quorum-sen sing in models for chemosensitive movement. Can. Appl. Math. Q. , 10(4):501 – 543, 2002

work page 2002

[32] [32]

K. J. Painter. Continuous models for cell migration in t issues and applications to cell sorting via diﬀerential che mo- taxis. Bulletin of Mathematical Biology , 71(5):1117–1147, 2009

work page 2009

[33] [33]

A. C. Ponce. An estimate in the spirit of Poincar´ e’s ine quality. Journal of the European Mathematical Society , 6(1):1–15, 2004

work page 2004

[34] [34]

A. C. Ponce. A new approach to Sobolev spaces and connect ions to Γ-convergence. Calc. Var. Partial Diﬀerential Equations, 19(3):229–255, 2004

work page 2004

[35] [35]

E. W einan. Dynamics of vortex liquids in ginzburg-land au theories with applications to superconductivity. Physical Review B , 50:1126—-1135, 2012

work page 2012

[36] [36]

W enisch, R

R. W enisch, R. H¨ ubner, F. Munnik, S. Melkhanova, S. Gem ming, G. Abrasonis, and M. Krause. Nickel-enhanced graphitic ordering of carbon ad-atoms during physical vapo r deposition. Carbon, 100:656 – 663, 2016. (E. Davoli) Tu Wien, Institute for Analysis and Scientific Computing, Wi edner Hauptstraße 8-10, 1040 Wien, Austria Email address : elisa.davoli@...

work page 2016