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Sara Daneri

Publications and source records attributed to Sara Daneri.

18 recordsLinked to original sources
arXiv2024

A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions

We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. We consider a general class of nonlocal variational problems in dimension $d\geq 2$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae). The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory. Among others, we detect a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. The power of decay of the considered kernels at infinity is $p\geq d+3$ and it is related to pattern formation in synthetic antiferromagnets.

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arXiv2022

Deterministic particle approximation of aggregation diffusion equations with nonlinear mobility

We consider a class of aggregation-diffusion equations on unbounded one dimensional domains with Lipschitz nonincreasing mobility function. We show strong $L^1$-convergence of a suitable deterministic particle approximation to weak solutions of a class aggregation-diffusion PDEs (coinciding with the classical ones in the no vacuum regions) for any bounded initial data of finite energy. In order to prove well-posedness and convergence of the scheme with no BV or no vacuum assumptions and overcome the issues posed in this setting by the presence of a mobility function, we improve and strengthen the techniques introduced in arXiv:2012.01966(2).

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arXiv2021

Periodic striped configurations in the large volume limit

We show striped pattern formation in the large volume limit for a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension previously considered Goldman-Runa and Daneri-Runa and in Giuliani-Lieb-Lebowitz and Giuliani-Seiringer in the discrete setting. In such a model the relative strength between the short range attractive term favouring pure phases and the long range repulsive term favouring oscillations is modulated by a parameter $τ$. For $τ<0$ minimizers are trivial uniform states. It is conjectured that $\forall\,d\geq2$ there exists $0<\barτ\ll1$ such that for all $0<τ\leq\barτ$ and for all $L>0$ minimizers are striped/lamellar patterns. In Daneri-Runa arXiv:1702.07334 the authors prove the above for $L=2kh^*_τ$, where $k\in\N$ and $h^*_τ$ is the optimal period of stripes for a given $0<τ\leq\barτ$. The purpose of this paper is to show the validity of the conjecture for generic $L$.

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arXiv2021

Exact periodic stripes for a local/nonlocal minimization problem with volume constraint

We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection positive power law kernel. Breaking of symmetry with respect to coordinate permutations and pattern formation for functionals in this class have been shown in~\cite{gr,dr_arma} and previously by~\cite{gs_cmp} in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~\cite{dr_arma} to be given by periodic stripes of volume density $1/2$ in any cube having optimal period size, also in the large volume limit. In this paper we study the minimization problem with arbitrarily prescribed volume constraint $α\in(0,1)$. We show that, in the large volume limit, minimizers are periodic stripes of volume density $α$, namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density $α$ in each period. Results of this type in the one-dimensional setting, where no symmetry breaking occurs, have been previously obtained in \cite{muller1993singular, alberti2001new,ren2003energy,chen2005periodicity,giuliani2009modulated}.

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arXiv2021

One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension

In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica-Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in \cite{GR} and in \cite{DR}. In the discrete setting it has been previously studied in \cite{GLL, GLS, GS}. The model contains two parameters: $τ$ and $\varepsilon$. The parameter $τ$ represents the relative strength of the local term with respect to the nonlocal one, while the parameter $\varepsilon$ describes the transition scale in the Modica-Mortola type term. If $τ< 0$ one has that the only minimizers of the functional are constant functions with values in $\{0,1\}$. In any dimension $d\geq1$ for small but positive $τ$ and $\varepsilon$, it is conjectured that the minimizers are non-constant one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension~$d >1$.

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arXiv2021

One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension

In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in Daneri-Kerschbaum-Runa arXiv:1907.06419. The parameters of the model are denoted by $τ$ and $\varepsilon$: the parameter $τ$ represents the relative strength of the local term with respect to the nonlocal one, while the parameter $\varepsilon$ describes the transition scale in the Modica-Mortola type term. Restricting to a periodic box of size $L$, with $L$ multiple of the period of the minimal one-dimensional minimizers, in Daneri-Kerschbaum-Runa arXiv:1907.06419 the authors prove that in any dimension $d\geq1$ and for small but positive $τ$ and $\varepsilon$ (eventually depending on $L$), the minimizers are non-constant one-dimensional periodic functions. In this paper we prove that periodicity and one-dimensionality of minimizers occurs also in the zero temperature analogue of the thermodynamic limit, namely as $L\to+\infty$.

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arXiv2020

Deterministic particle approximation of aggregation-diffusion equations on unbounded domains

We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle approximations with piecewise constant densities introduced in~\cite{Di Francesco-Rosini} starting from general bounded initial densities converge strongly in $L^1$ to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded domains and for arbitrary nonnegative bounded initial densities, thus extending the results in \cite{Gosse-Toscani, Matthes-Osberger, Mathes-Soellner} (in which a no-vacuum condition is required) and giving an alternative approach to \cite{Carrillo-Craig-Patacchini} in the one-dimensional case, including also subquadratic and superquadratic diffusions.

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arXiv2020

Non-uniqueness for the Euler equations up to Onsager's critical exponent

In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an $L^2$-dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical $1/3$. This improves previous results on non-uniqueness obtained by Daneri in arXiv:1302.0988 and by Daneri and Szekelyhidi Jr. in arXiv:1603.09714 and generalizes the result obtained by Buckmaster, De Lellis, Szekelyhidi Jr. and Vicol in arXiv:1701.08678.

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arXiv2019

Pattern formation for a local/nonlocal interaction functional arising in colloidal systems

In this paper we study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the $1$-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential. This model is physically interesting as it is the $Γ$-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal systems \cite{BBCH,CCA,IR,GCLW}. Following a strategy introduced in~\cite{DR} we prove that in a suitable regime minimizers are periodic stripes, in any space dimension.

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arXiv2018

Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension

We study the functional considered in~\cite{2011PhRvB..84f4205G,2014CMaPh.tmp..127G,GiuSeirGS} and a continuous version of it, analogous to the one considered in~\cite{GR}. The functionals consist of a perimeter term and a non-local term which are in competition. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in~\cite{GiuSeirGS}.

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arXiv2017

Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations

In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Székelyhidi], we prove that the set of Hölder $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible Hölder $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.

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arXiv2016

Towards a stationary Monge-Kantorovich dynamics: the Physarum Polycephalum experience

In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.

math.NA↗
arXiv2014

On Sudakov's type decomposition of transference plans with norm costs

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $|\cdot|_{D^*}$ \[ \min \bigg\{\int |\mathtt T(x) - x|_{D^*} dμ(x), \ \mathtt T : \mathbb R^d \to \mathbb R^d, \ ν= \mathtt T_\# μ\bigg\}, \] with $μ$, $ν$ probability measures in $\mathbb R^d$ and $μ$ absolutely continuous w.r.t. $\mathcal L^d$. The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in $Z_\mathfrak a\times\mathbb R^d$, where $\{Z_\mathfrak a\}_{\mathfrak a\in\mathfrak A} \subset \mathbb R^d$ are disjoint regions such that the construction of an optimal map $\mathtt T_\mathfrak a : Z_\mathfrak a \to \mathbb R^d$ is simpler than in the original problem, and then to obtain $\mathtt T$ by piecing together the maps $\mathtt T_\mathfrak a$. In this paper we show how the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set $Z_\mathfrak a$ and then in $\mathbb R^d$. The strategy is sufficiently powerful to be applied to other optimal transportation problems.

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arXiv2013

Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent $θ<\frac{1}{16}$. Using the techniques introduced in \cite{DS12} and \cite{DS12H}, we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.

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arXiv2011

A planar bi-Lipschitz extension Theorem

We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\widetilde L \leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \cite{Tukia}), using a completely different argument, but without any estimate on the constant $\widetilde L$. In particular, the function $v$ can be taken either smooth or (countably) piecewise affine.

math.FA↗
arXiv2010

Lecture Notes on Gradient Flows and Optimal Transport

We present a short overview on the strongest variational formulation for gradient flows of geodesically $λ$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

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arXiv2008

Eulerian calculus for the displacement convexity in the Wasserstein distance

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.

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