On the signed porous medium flow
Edoardo Mainini
Networks & Heterogeneous Media 2012, 7(3): 525-541 doi: 10.3934/nhm.2012.7.525
We prove that the signed porous medium equation can be regarded as limit of an optimal transport variational scheme, therefore extending the classical result for positive solutions of [13] and showing that an optimal transport approach is suited even for treating signed densities.
keywords: signed transport. Porous media equation changing sign-solutions gradient flow optimal transport
Carbon-nanotube geometries: Analytical and numerical results
Edoardo Mainini Hideki Murakawa Paolo Piovano Ulisse Stefanelli
Discrete & Continuous Dynamical Systems - S 2017, 10(1): 141-160 doi: 10.3934/dcdss.2017008

We investigate carbon-nanotubes under the perspective ofgeometry optimization. Nanotube geometries are assumed to correspondto atomic configurations whichlocally minimize Tersoff-type interactionenergies. In the specific cases of so-called zigzag and armchairtopologies, candidate optimal configurations are analytically identifiedand their local minimality is numerically checked. Inparticular, these optimal configurations do not correspond neither tothe classical Rolled-up model [5] nor to themore recent polyhedral model [3]. Eventually, theelastic response of the structure under uniaxial testing is numericallyinvestigated and the validity of the Cauchy-Born rule is confirmed.

keywords: Carbon nanotubes Tersoff energy variational perspective new geometrical model stability Cauchy-Born rule
Uniqueness for Keller-Segel-type chemotaxis models
José Antonio Carrillo Stefano Lisini Edoardo Mainini
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1319-1338 doi: 10.3934/dcds.2014.34.1319
We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.
keywords: Chemotaxis displacement convexity. Wasserstein distance Gradient flows Keller-Segel model

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