Papers by Marco Morandotti
Energetic Relaxation to Structured Deformations, 2023
Milan Journal of Mathematics
The well-posedness of a multi-population dynamical system with an entropy regularization and its ... more The well-posedness of a multi-population dynamical system with an entropy regularization and its convergence to a suitable mean-field approximation are proved, under a general set of assumptions. Under further assumptions on the evolution of the labels, the case of different time scales between the agents’ locations and labels dynamics is considered. The limit system couples a mean-field-type evolution in the space of positions and an instantaneous optimization of the payoff functional in the space of labels.
Journal of Elasticity
whose broad and deep scientific contributions and leadership never cease to evoke admiration and ... more whose broad and deep scientific contributions and leadership never cease to evoke admiration and to provide inspiration.
Meccanica
A swimmer embedded on an inertialess fluid must perform a non-reciprocal motion to swim forward. ... more A swimmer embedded on an inertialess fluid must perform a non-reciprocal motion to swim forward. The archetypal demonstration of this unique motion-constraint was introduced by Purcell with the so-called “scallop theorem”. Scallop here is a minimal mathematical model of a swimmer composed by two arms connected via a hinge whose periodic motion (of opening and closing its arms) is not sufficient to achieve net displacement. Any source of asymmetry in the motion or in the forces/torques experienced by such a scallop will break the time-reversibility imposed by the Stokes linearity and lead to subsequent propulsion of the scallop. However, little is known about the controllability of time-reversible scalloping systems. Here, we consider two individually non-controllable scallops swimming together. Under a suitable geometric assumption on the configuration of the system, it is proved that controllability can be achieved as a consequence of their hydrodynamic interaction. A detailed anal...
Zeitschrift für angewandte Mathematik und Physik
An energy for first-order structured deformations in the context of periodic homogenization is ob... more An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral-type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process underlying the theory of homogenization.
The existence of minimizers of the Canham--Helfrich functional in the setting of generalized Gaus... more The existence of minimizers of the Canham--Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham--Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss graphs, then lower semicontinuity and compactness are proved under a suitable condition on the bending constants ensuring coerciveness; the minimization follows by the direct methods of the Calculus of Variations. Remarks on the regularity of minimizers and on the behavior of the functional in case there is lack of coerciveness are presented.
We introduce and study a mean-field model for a system of spatially distributed players interacti... more We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represented by an evolving probability measure Σ on an infinite dimensional state space (pairs (x, σ) of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number N of the players goes to infinity, and the initial discrete distribution of positions and strat...
An advective Cahn-Hilliard model motivated by thin film formation is studied in this paper. The o... more An advective Cahn-Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn-Hilliard equation. CONTENTS
MRS Advances, 2017
In this note we discuss two aspects of screw dislocations dynamics: their behavior near the bound... more In this note we discuss two aspects of screw dislocations dynamics: their behavior near the boundary and a way to confine them inside the material. In the former case, we obtain analytical results on the estimates of collision times (one dislocation with the boundary and two dislocations with opposite Burgers vectors with each other); numerical evidence is also provided. In the latter, we obtain analytical results stating that, under imposing a certain type of boundary conditions, it is energetically favorable for dislocations to remain confined inside the domain.
Journal of Biological Dynamics, 2012
We extend the existence, uniqueness, and regularity result obtained in [4] for the case of a self... more We extend the existence, uniqueness, and regularity result obtained in [4] for the case of a self-propelled swimmer in a viscous fluid to the case of a self-propelled swimmer immersed in a particulate medium, modeled as a Brinkman fluid.
Energetic Relaxation to Structured Deformations, 2023
We describe multiscale geometrical changes via structured deformations ( g , G ) and the non-loca... more We describe multiscale geometrical changes via structured deformations ( g , G ) and the non-local energetic response at a point x via a function Ψ of the weighted averages of the jumps [ u n ] ( y ) of microlevel deformations u n at points y within a distance r of x. The deformations u n are chosen so that lim n → ∞ u n = g and lim n → ∞ ∇ u n = G . We provide conditions on Ψ under which the upscaling “ n → ∞ ” results in a macroscale energy that depends through Ψ on (1) the jumps [ g ] of g and the “disarrangement field” ∇ g − G , (2) the “horizon”r, and (3) the weighting function α r for microlevel averaging of [ u n ] ( y ) . We also study the upscaling “ n → ∞ ” followed by spatial localization “ r → 0 ” and show that this succession of processes results in a purely local macroscale energy I ( g , G ) that depends through Ψ upon the jumps [ g ] of g and the “disarrangement field” ∇ g − G alone. In special settings, such macroscale energies I ( g , G ) have been shown to support...
SIAM Journal on Mathematical Analysis
The mean-field analysis of a multi-population agent-based model is performed. The model couples a... more The mean-field analysis of a multi-population agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markow-type jump process on the probability that each agent has of belonging to a given population. A general functional analytic framework for the well-posedness of the problem is established, and some concrete applications are presented, both in the case of discrete and continuous set of labels. In the particular case of a leader-follower dynamics, the existence and approximation results recently obtained in [2] are recovered and generalized as a byproduct of the abstract approach proposed.
We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a f... more We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.
arXiv: Optimization and Control, 2019
The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the ... more The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimisation of the power expended are addressed and a qualitative description of the optimal strategies is provided.
A mean-field selective optimal control problem of multipopulation dynamics via transient leadersh... more A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. This way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via Γ -convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some nume...
arXiv: Analysis of PDEs, 2017
This note collects some results on the behaviour of screw dislocation in an elastic medium. By us... more This note collects some results on the behaviour of screw dislocation in an elastic medium. By using a semi-discrete model, we are able to investigate two specific aspects of the dynamics, namely (i) the interaction with free boundaries and collision events and (ii) the confinement inside the domain when a suitable Dirichlet-type boundary condition is imposed. In the first case, we analytically prove that free boundaries attract dislocations and we provide an expression for the Peach--Koehler force on a dislocation near the boundary. Moreover, we use this to prove an upper bound on the collision time of a dislocation with the boundary, provided certain geometric conditions are satisfied. An upper bound on the collision time for two dislocations with opposite Burgers vectors hitting each other is also obtained. In the second case, we turn to domains whose boundaries are subject to an external stress. In this situation, we prove that dislocations find it energetically favourable to st...
arXiv: Optimization and Control, 2019
An integral representation result is obtained for the asymptotics of energies including both loca... more An integral representation result is obtained for the asymptotics of energies including both local and non-local terms, in the context of structured deformations. Starting from an initial energy featuring a local bulk and interfacial contribution and a non-local measure of the jump discontinuities, an iterated limiting procedure is performed. First, the initial energy is relaxed to structured deformation, and then the measure of non-locality is sent to zero, with the effect of obtaining an explicit local energy in which the non-linear contribution of submacroscopic slips and separations is accounted for. Two terms, different in nature, emerge in the bulk part of the final energy: one coming from the initial bulk energy and one arising from the non-local contribution to the initial energy. This structure turns out to be particularly useful for studying mechanical phenomena such as yielding and hysteresis. Moreover, in the class of invertible structured deformations, applications to c...
arXiv: Analysis of PDEs, 2016
A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied ... more A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $n\to\infty$, which is treated in [17].
arXiv: Analysis of PDEs, 2014
We present a variational framework for studying screw dislocations subject to antiplane shear. Us... more We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by Cermelli and Gurtin, methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-K\"ohler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in a forthcoming work.
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Papers by Marco Morandotti