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Poisson systems , Comptes Rendus Mathematique , vol. Lieb , Loss, Analysis. Second edition , Graduate Studies in Mathematics , vol. Lions , The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1.

César R. de Oliveira

Lions and B. Perthame , Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system , Inventiones Mathematicae , vol. French [Isoperimetric inequalities and applications to physics] Travaux en Cours , Progress] Hermann , Marchioro and M. Pulvirenti , Some considerations on the nonlinear stability of stationary planar Euler flows , Communications in Mathematical Physics , vol. Pfaffelmoser , Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data , Journal of Differential Equations , vol.

Schaeffer , Global existence of smooth solutions to the vlasov poisson system in three dimensions , Communications in Partial Differential Equations , vol. Sygnet, G. Forets, M. Lachieze-rey, and R. Pellat , Stability of gravitational systems and gravothermal catastrophe in astrophysics , The Astrophysical Journal , vol.

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Talenti , Elliptic equations and rearrangements , Ann. Scuola Norm. Pisa Cl. We also propose the presence of mutual interactions among the three layers during formation of the 3D structures of the medulla columns. We first study the global asymptotic stability of thehomogeneous steady state.

We then focus our attention on the stationary system, and prove the existenceof nontrivial solutions branching from the homogeneous steady state, through possibly infinitely manybifurcations, under appropriate assumptions on the interaction potential.

We also provide sufficientconditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase theseresults by applying them to several examples of interaction potentials such as the noisy Kuramoto modelfor synchronisation, the Keller—Segel model for bacterial chemotaxis, and the noisy Hegselmann—Kraussemodel for opinion dynamics. We propose a new splitting scheme for general reaction—taxis—diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena.

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The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables. We discuss several continuum cell-cell adhesion models based on the underlying microscopic assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level.

The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. Cell Biol. We also review some of the applications of these models in other areas of tissue growth in developmental biology. We finally explore the resulting qualitative behavior due to cell-cell repulsion.

This result is one of the very few examples wherethe minimiser of a nonlocal anisotropic energy is explicitly computed. Finally, we showcasethese results by applying them to several examples of interaction potentials such as the noisy Kuramotomodel for synchronisation, the Keller—Segel model for bacterial chemotaxis, and the noisy Hegselmann—Krausse model for opinion dynamics. We show that solutions of nonlinear nonlocal Fokker—Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space.

Two different approaches are analyzed, making crucial use of uniform estimates for L2 energy functionals and free energy or entropy functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials.

The free energy approach extends to the case degenerate diffusion.

César R. de Oliveira

As a counterpoint to classical stochastic particle methods for diffusion, we developa deterministic particle method for linear and nonlinear diffusion. At first glance, deterministicparticle methods are incompatible with diffusive partial differential equations since initial data givenby sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocityfield that ensures particles do remain particles and apply this to develop a numerical blob methodfor a range of diffusive partial differential equations of Wasserstein gradient flow type, includingthe heat equation, the porous medium equation, the Fokker—Planck equation, and the Keller—Segelequation and its variants.

Our choice of regularization is guided by the Wasserstein gradient flowstructure, and the corresponding energy has a novel form, combining aspects of the well-knowninteraction and potential energies. In the presence of a confining drift or interaction potential,we prove that minimizers of the regularized energy exist and, as the regularization is removed,converge to the minimizers of the unregularized energy.

We then restrict our attention to nonlineardiffusion of porous medium type with at least quadratic exponent. Under sufficient regularityassumptions, we prove that gradient flows of the regularized porous medium energies converge tosolutions of the porous medium equation. As a corollary, we obtain convergence of our numericalblob method.

We analyze stability of conservative solutions of the Cauchy problem on the line for the integrated Hunter—Saxton HS equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular.

The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric. In this work we focus on the construction of numerical schemes for the approximation of stochastic mean-field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos gPC expansion in the random space.

In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported. We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance.

We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension. Following the paradigm set by attraction-repulsion-alignment schemes, a myriad of individual-based models have been proposed to calculate the evolution of abstract agents.

While the emergent features of many agent systems have been described astonishingly well with force-based models, this is not the case for pedestrians. Many of the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely that a purely mechanical model will succeed. As a response to the mechanistic literature, we will consider a model for pedestrian dynamics that attempts to reproduce the rational behaviour of individual agents through the means of anticipation. Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage.

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We will discuss the validity of this game theoretical-based model in regimes with varying degrees of congestion, ultimately presenting a correction to the mechanistic model in order to achieve realistic high-density dynamics. Motile organisms often use finite spatial perception of their surroundings to navigate and search their habitats. Yet standard models of search are usually based on purely local sensory information. To model how a finite perceptual horizon affects ecological search, we propose a framework for optimal navigation that combines concepts from random walks and optimal control theory.

On local connectivity for the Julia set of rational maps: Newton's famous example Annals of Mathematics.

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Equivalence of summatory conditions along sequences for bounded holomorphic functions Complex Variables. Schematic homotopy types and non-abelian Hodge theory Compositio Mathematica , Vol. Landsberg, J. On the infinitesimal rigidity of homogeneous varieties Compositio Mathematica , Vol.

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