BCAM, Wednesday, July 30th, 2025, 17:00 - 18:00
Title: Analysis for hydrodynamic model of swarming
Nilasis Chaudhuri - University of Warsaw
In this talk, we consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and derive a corresponding energy-type inequality, in the spirit of the Bresch–Desjardins estimate. Furthermore, we identify dependencies between the communication weight and the interaction kernel, as well as between the pressure and the viscosity term, which allow this inequality to be uniform in time. This framework is then used to analyze the long-time asymptotics of solutions.
BCAM, Thursday, June 26th, 2025, 17:00 - 18:00
Title: Solutions of the divergence equation in Hardy and Lipschitz spaces.
Eugenia Cejas - Universidad de Buenos Aires
Given a bounded domain Ω and f with zero integral, the existence of a
vector field u vanishing on ∂Ω and satisfying div u = f has been widely studied because of its connection with many important problems.
It is known that for f ∈ Lp(Ω), 1 < p < ∞, there exists a solution u ∈ W_0^{1,p} (Ω).
It is also known that an analogous result does not hold for p = 1 or p = ∞.
The goal of this talk is to prove results for Hardy spaces when n/(n+1) < p ≤ 1, and in the other limiting case, for bounded mean oscillation (BMO) and Lipschitz spaces. This is joint work with Ricardo Durán.
BCAM, Thursday, June 12th, 2025, 17:00 - 18:00
Title: Some fully nonlinear problems arising in Physics
Martina Magliocca - Universidad de Sevilla
We will see some fourth-order problems arising in Physics which model different processes, such as the growth of crystal surfaces and wetting–dewetting processes.
Mathematically speaking, we will focus on global existence and regularity results for problems as:
u_t = F(t, x, ∇u, ..., Δ²u) in [0, T] × 𝕋^N
u(0, x) = u₀(x) in 𝕋^N
where 𝕋^N = [−π, π]^N is the N-dimensional torus and the initial data u₀ belong to Wiener spaces. The particular choices of F will describe the model in object.
These results are contained in:
https://doi.org/10.1016/j.nonrwa.2024.104137
and in a joint work with R. Granero Belinchón:
UPV/EHU, Thursday, June 5th, 2025, 12:00 - 13:00
Title: On the mass-critical inhomogeneous NLS equation
Luis Gustavo Farah - Universidade Federal de Minas Gerais
We consider the inhomogeneous nonlinear Schrödinger (INLS) equation
iu_t + ∆u + |x|^{−b}|u|^{4−2b/N}u = 0, x ∈ R^N
with N ≥ 1 and 0 < b < 1, which is a generalization of the classical nonlinear Schrödinger equation (NLS). Since the scaling invariant Sobolev index is zero, the equation is called mass-critical.
In this talk, we discuss some blow-up results in the non-radial setting, obtained in collaboration with Mykael Cardoso (UFPI-Brazil).
This work is partially supported by CNPq, CAPES and, FAPEMIG-Brazil.
UPV/EHU, Thursday, May 29th, 2025, 12:00 - 13:00
Jean-Bernard Bru - UPV/EHU
Title: Exchange Interactions and Cuprate Superconductivity
In this talk, we will explain the effect of quantum interactions exchanging different types of particles. We will consider a system made of two fermions and one boson, in order to study the effect of such an off-diagonal interaction term, having in mind the physics of cuprate superconductors. We will in particular show the existence of exponentially localized dressed bound fermion pairs. We will give particular attention to the regime of very large on-site (Hubbard) repulsions, because this situation can be relevant for cuprate superconductors.
BCAM, Thursday, May 22nd, 2025, 17:00 - 18:00
Title: Semigroups of composition operators on some new Banach spaces of analytic functions
Carlos Gómez Cabello - UPV/EHU
The theory of strongly continuous semigroups of bounded operators on Banach has proven fruitful in many areas of Analysis. A classical result related to this topic is Berkson and Porta's Theorem, establishing a one-to-one correspondence between continuous semigroups of analytic functions in the unit disc and strongly continuous semigroups of composition operators on Hardy spaces of the unit disc. In this talk, we shall discuss an analogue correspondence in the less known setting of Hardy spaces of Dirichlet series. Being the boundedness of composition operators in these spaces a delicate topic, we shall also present some results related to the existence of continuous semigroups of symbols in terms of its infinitesimal generator. In order to do so, we will introduce the most basic properties of Dirichlet series and of these new Hardy spaces.
Most of the results in this talk have been obtained in collaboration with Professor Manuel D. Contreras and Professor Luis Rodríguez Piazza.
UPV/EHU, Thursday, May 15th, 2025, 12:00 - 13:00
Title: Parabolic equation involving the fractional p-Laplacian in porous media.
Loic Constantin - UPV/EHU
In this talk we examine a class of nonlinear and nonlocal evolution equations involving the porous fractional p-Laplacian operator within a Sobolev space setting. Considering the equation over a smooth bounded domain, we will use a time discretization method paired with the accretive operator theory to prove the existence and uniqueness of weak-mild solutions for a nonlinear source term. We next explore the quantitative behavior of solutions: blow-up, extinction and stabilization.
UPV/EHU, Thursday, May 8th, 2025, 12:00 - 13:00
Title: Vorticity blowup in 2D compressible Euler equations
Giorgio Cialdea - Courant Institute New York
In this talk, I will discuss a finite-time vorticity blowup result for the 2D compressible Euler equations, with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by an axisymmetric implosion in which the swirl velocity enjoys full stability, as opposed to finite co-dimension stability. This is a joint work with Jiajie Chen, Steve Shkoller, and Vlad Vicol.
BCAM, Thursday, April 24th, 2025, 17:00 - 18:00
Title: The initial-to-final state inverse problem with time-independent potentials.
Thanasis Zacharopoulos - BCAM
In this talk we will explain why one can solve an initial boundary value problem for the Schrödinger equation with time-independent potential V(x), ignoring the exact Hamiltonian -Δ+V if enough data, regarding initial and final states, are available.
This is a joint work with M. Cañizares, P. Caro and I. Parissis.
UPV/EHU, Thursday, April 10th, 2025, 11:45 - 13:00
Title: Sharp blow-up stability of self-similar solutions for the modified Korteweg-de Vries equation
Simao Correia - Instituto Superior Técnico Lisboa
The evolution of vortex patches subject to Euler's equations can be described using the modified Korteweg-de Vries (mKdV) equation. The formation of singular geometric objects, such as corners or logarithmic spirals, corresponds to self-similar (i.e. scaling-invariant) solutions of (mKdV) at the blow-up time. Self-similar solutions present a number of critical features, from time and space decay to regularity. Moreover, at scaling-critical regularity, an instantaneous energy cascade is known to occur, blocking the derivation of a suitable well-posedness theory.
We will discuss the blow-up stability of self-similar solutions under arbitrarily large subcritical perturbations. This is a joint work with R. Côte (U. Strasbourg).
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Title: The role of the inhomogeneity on non-radial scattering for nonlinear Schrödinger equations
Luccas Campos - Universidade de Minas Gerais
The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds.
In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schrödinger (NLS) equation.
Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae.
This approach is versatile, as it was shown to work in the energy-subcritical setting for different nonlinearities, as well as for higher-order equations.
BCAM, Thursday, April 3rd, 2025, 17:00 - 18:15
1) Title: Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions
Felisia Angela Chiarello - Università degli Studi dell'Aquila
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in a periodic domain in one-space dimension with linear pressure term. The main result is the global existence of periodic entropy weak solutions, for periodic initial data having finite total variation and initial density bounded away from zero.
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2) Title: Critical non-linearity for some evolution equations with Fujita-type critical exponent
Giovanni Girardi - Università Politecnica delle Marche
We consider the Cauchy problem for a class of non-linear evolution equations in the form L u=F(u), where L is a linear partial differential operator with constant coefficients and F is a non-linear term.
For several different choices of L, many authors have investigated the existence of global (in time) solutions to this problem when F(u)=|u|^p is a power non-linearity, looking for a critical exponent p_c>1 such that global (in time) small data solutions exist in the supercritical case p>p_c, whereas no global (in time) weak solutions exist, under suitable sign assumptions on the data, in the subcritical case 1<p<p_c.
In this talk we consider a more general non-linear term F(u)=|u|^ph(|u|) where h is a modulus of continuity; for a large class of models, we provide an integral condition on h which allows to distinguish more precisely the region of existence of a global small data solution from that in which the problem admits no global weak solutions, refining the existing results about the critical exponents for power type non-linearities.
The talk is based on the results obtained in [1] and [2]
[1] Ebert, M. R., Girardi, G. and Reissig, M. Critical regularity of nonlinearities in semilinear classical damped wave equations. Math. Ann. (2020).
[2] Girardi, G. Critical non-linearity for some evolution equations with Fujita-type critical exponent. Nonlinear Differ. Equ. Appl. (2025).
BCAM, Thursday, March 27th, 2025, 17:00 - 18:00
Title: An optimization problem and point-evaluation in Paley–Wiener spaces
Sarah May Instanes (she/her) - NTNU Trondheim
We study the constant Cₚ defined as the smallest constant C such that |f(0)|^p ≤ C∥ f ∥ₚ^p holds for every function f in the Paley–Wiener space PWₚ. Brevig, Chirre, Ortega-Cerdà, and Seip have recently shown that Cₚ < p/2 for all p > 2. We improve this bound for 2 < p ≤ 5 by solving an optimization problem.
UPV/EHU, Thursday, March 20th, 2025, 12:00 - 13:00
Title: A pathological set regarding the propagation of almost sure properties of Gaussian measures
Pablo Merino - BCAM
Given the periodic and cubic wave equation in 3d, we deal with two questions:
• From a deterministic point of view, given p > 2 and σ ≥ 0 large enough, what can we say about the propagation of spatial derivative regularity, in terms of Wσ,p(T^3), for the linear and the nonlinear flow of the aforementioned equation?
• From a probabilistic point of view, given the loss of derivative regularity that we will encounter for the previous question, can we randomize the initial data in any way such that we do not see such regularity loss in Wσ,p(T^3), almost surely with respect certain meaningful measure?
I will talk about a complementary result to the quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of this equation, proved by T. S. Gunaratnam, T. Oh, N. Tzvetkov and H. Weber (2022).
Namely, we will discuss about the existence of dense sets of general Sobolev spaces Wσ,p(T^3), for large p and σ, which do not preserve the regularity σ throughout the aforementioned dynamics, as long as t ̸= 0. This is in sharp contrast with the propagation of almost sure properties of the Gaussian measure along the flow. This is a joint work with Nikolay Tzvetkov.
UPV/EHU, Thursday, March 13th, 2025, 12:00 - 13:00
Title: On certain dispersive equations and the behavior of their solutions
Rafael Granero Belinchón - Universidad de Cantabria
In many natural phenomena dispersive motion plays a key role. In particular, in this talk, we will introduce a number of nonlocal dispersive PDE describing the motion of certain two-dimensional fluid problems.
BCAM, Thursday, March 6th, 2025, 17:00 - 18:00
Title: Improved fractional Poincaré inequalities through interpolation
Irene Drelichman - Universidad Nacional de La Plata
In this talk, we will review the relation between classical and fractional Poincaré inequalities, discuss possible improvements, and show how the sharp dependence on the fractional parameter can be obtained through interpolation theory.
BCAM, Thursday, February 27th, 2025, 17:00 - 18:00
Title: Every classical inequality for the Fourier operator is trivial
Miquel Saucedo (he/him) - CRM Barcelona
In this talk we will see that if the Fourier operator is bounded between two classical spaces (for instance, rearrangement invariant spaces), then every operator which maps L^1 and L^2 to L^\infty and L^2, respectively, must also be bounded. We will also discuss why this means that there cannot be any "interesting" classical Fourier inequalities and give some applications. Joint work with Sergey Tikhonov.
UPV/EHU, Thursday, February 20th, 2025, 12:00 - 13:00
Title: The Cauchy-Dirichlet Problem for singular nonlocal diffusions on bounded domains
Mikel Ispizua - UPV/EHU
We will discuss about homogeneous Cauchy-Dirichlet Problem for a nonlocal fast diffusion equation posed on a bounded Euclidean domain. The prototype equation is the "Fractional Fast Diffusion Equation", when L is one of the three possible Dirichlet Fractional Laplacians on Ω. The main results shall provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far; sharp smoothing estimates; extinction rates and regularity of solutions.
UPV/EHU, Thursday, February 13th, 2025, 12:00 - 13:00
Title: Dynamic Refinement of Pressure Decomposition in Navier-Stokes Equations
Pedro Gabriel Fernandez Dalgo (he/him) - BCAM
We will discuss how the local decomposition of pressure in the Navier-Stokes equations can be dynamically refined to prove that a relevant critical energy of a suitable Leray-type solution inside a backward paraboloid, regardless of its aperture, is controlled near the vertex by a critical behavior confined to a neighborhood of the paraboloid’s boundary. This neighborhood excludes the interior near the vertex and remains separated from the temporal profile of the vertex, except at the vertex itself.
UPV/EHU, Thursday, February 6th, 2025, 12:00 - 13:00
Title: Unique continuation for the heat operator with potentials in weak spaces
Sanghyuk Lee - Seoul National University
This talk concerns the strong unique continuation property for the differential inequality
|(∂/∂t + Δ)u(x, t)| ≤ V(x, t) |u(x, t)|
where the potential $V$ belongs to suitable function spaces. In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has remained open since the earlier works of Escauriaza and Escauriaza--Vega. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces, which in turn are built on the spectral projection estimates for the Hermite operator.
BCAM, Thursday, January 30th, 2025, 17:00 - 18:00
Title: Phase retrieval and Fourier uniqueness problems
UPV/EHU, Thursday, January 23rd, 2025, 12:00 - 13:00
Title: Nonlinear fluids and Lipschitz truncations
We provide existence of very weak solutions and a-priori estimates for steady flows of non-Newtonian fluids of Stokes and Navier-Stokes type when the right-hand sides are not in the natural existence class. This includes stress laws that depend non-linearly on the shear rate of the fluid like power-law fluids. To obtain the a-priori estimates we make use of a refined solenoidal Lipschitz truncation that preserves zero boundary values.
We provide also estimates in (Muckenhoupt) weighted spaces which permit us to regain a duality pairing, which than can be used to prove existence of solutions. Joint work with Sebastian Schwarzacher (Uppsala and Prague).
UPV/EHU, Thursday, January 9th, 2025, 12:00 - 13:00
Title: On overview on nonlinear Schrödinger systems
In this talk I will present some recent existence results on nonlinear Schrödinger type systems in a weak fully attractive or repulsive regime in presence of an external trapping potential with subcritical ora critical growth
UPV/EHU, Thursday, December 12th, 2024, 12:00 - 13:00
Title: A novel method for the derivation of a universal expansion for the free energy of a Bose gas
The Lee-Huang-Yang formula provides a universal expression for the first two terms in the dilute regime expansion of the energy density of a Bose gas in the thermodynamic limit. In the model considered, the particles interact through a pairwise, spherically symmetric, repulsive potential. This universality lies in its dependence solely on the scattering length of the interaction potential, irrespective of the potential’s specific shape. Introducing a new method that combines the renormalization of the potential with a Neumann localization, we establish a lower bound for the free energy density at low temperatures for a broad class of singular potentials, including the case of hardcore interactions. The derived expression incorporates the Lee-Huang-Yang terms along with an additional thermal contribution arising from the excitation spectrum.
From a joint work with S. Fournais, T. Girardot, L. Junge, L. Morin, and A. Triay.
BCAM, Thursday, December 5th, 2024, 17:00 - 18:00
Title: Quantitative characterizations of weights and parabolic boundary value problems
Weights (non-negative locally integrable functions) satisfying a reverse H\"older condition are important in the study of harmonic measure and boundary value problems for elliptic and parabolic partial differential equations. In this talk, I will discuss a quantitative version of a Carleson measure characterization of reverse Hölder weights (originally found by Fefferman, Kenig and Pipher in the 90s) and its application to elliptic measure. In addition, I will explain extensions and modifications of the results needed in the analogous theory for parabolic measures. This is based on joint work with Simon Bortz and Moritz Egert.
UPV/EHU, Thursday, November 28th, 2024, 12:00 - 13:00
Title: Interaction of liquid crystals with a rigid body
This talk addresses the interaction problem of liquid crystals with a rigid body. The physical motivation for such problems is the presence of so-called liquid crystal colloids formed by dispersion of colloidal particles in the liquid crystal host medium, where a colloidal particle is viewed as a rigid body. In the first part of the talk, we investigate the interaction problem involving a simplified Ericksen-Leslie model. We verify that the director condition |d| = 1 is preserved in the interaction problem. After transforming the moving boundary problem to a fixed domain, we establish the local strong well-posedness by showing maximal L^p-L^q-regularity of the linearized problem. Moreover, we prove global strong well-posedness close to constant equilibria, where we perform a splitting argument of the director into its mean value zero and average part to overcome the lack of invertibility.
The second part of the talk is dedicated to the study of the interaction problem of a general Beris-Edwards Q-tensor model. In contrast to Ericksen-Leslie models, which are vector models, Q-tensor models build on symmetric, traceless 3×3-matrices Q to describe the biaxial alignment of molecules. In order to tackle the resulting quasilinear mixed-order problem with moving boundary, we first transform it to a fixed domain and then establish maximal L^p-regularity in an anisotropic ground space of the form L^2×H^1 by means of a “monolithic” approach. The proofs of the local strong well-posedness for large data and the global strong well-posedness for small data are completed by suitable nonlinear estimates.
The talk is based on joint work with Tim Binz, Matthias Hieber and Arnab Roy.
BCAM, Thursday, November 21st, 2024, 17:00 - 18:00
Title: Self-improving Poincaré-Sobolev type functionals in product spaces.
We give a geometric condition which ensures that (q, p)-Poincaré-Sobolev inequalities are implied from generalized (1,1)-Poincaré inequalities related to $L^1$ norms in the context of product spaces. We provide several (1,1)-Poincaré type inequalities adapted to different geometries and then show that our self-improving method can be applied to obtain special interesting Poincaré-Sobolev estimates.
The results are based on joint work with E. Cejas (UNL, Argentina), C. Pérez (BCAM) and, E. Rela (UBA, Argentina).
UPV/EHU, Thursday, November 14th, 2024, 12:00 - 13:00
Title: On the Landis conjecture
In this talk, I will present partial affirmative answer to the Landis conjecture in all dimensions. The conjecture concerns the fastest speed at which a solution of a Schrödinger equation in $\mathbb{R}^N$ can decrease at infinity. Such results are referred as unique continuation at infinity. We provide a sharp decay criterion that ensures when a solution of a nonnegative Schrödinger equation in $\mathbb{R}^N$ with a potential V ≤ 1 is trivial. Our approach relies on the application of Liouville comparison principles and criticality theory.
Based on a joint work with Yehuda Pinchover.
BCAM, Thursday, November 7th, 2024, 17:00 - 18:00
Title: Existence of the extremals for the high order Hardy-Sobolev-Maz'ya inequalities
We establish the existence of the extremal function for the high order critical Hardy-Sobolev-Maz'ya (HSM) inequalities on the upper half space. The loss of translation invariance and rearrangement technique on the upper half space, together with the presence of the Hardy-singularity around the boundary of the upper half space for the HSM inequality cause much challenge to the study of the extremal problem. Instead of directly considering HSM inequality, we establish the existence of the extremal for its equivalent version: Poincare-Sobolev inequality on the hyperbolic space. We overcome the loss of compactness for minimizing sequence of Poincare-Sobolev inequality using Helgason-Fourier analysis, Riesz's rearrangement inequality and Lions concentration-compactness principle on the hyperbolic space. As an application, we also obtain the existence and symmetry of the positive solution for the high order Brezis-Nirenberg equation on the entire hyperbolic space.
BCAM, Thursday, October 31st, 2024, 17:00 - 18:00
Title: Quantitative propagation of smallness in the plane and 1D spectral estimates
We investigate the connection between the propagation of smallness in two dimensions and one-dimensional spectral estimates. The propagation of smallness in the plane obtained by Y. Zhu, reveals how the value of solutions in a small region extends to a larger domain. By revisiting Zhu’s proof, we obtain a quantitative version that includes an explicit dependence on key parameters. This refinement enables us to establish spectral inequalities for one-dimensional Schrödinger operators.
UPV/EHU, Thursday, October 24th, 2024, 12:00 - 13:00
Title: The critical exponent for k-evolution equations with double dispersion
UPV/EHU, Thursday, October 24th, 2024, 11:00 - 12:00
Title: Strong ill-posedness in $L^{\infty} $ of the 2D Boussinesq equations
In this talk, I will present a recent work in which the strong ill-posedness of the two-dimensional Boussinesq system is proven. I will show explicit examples of initial data with vorticity and density gradient in $L^{\infty}(\mathbb{R}^2)$ for which the horizontal density gradient has a strong norm inflation in infinitesimal time. This is a joint work with Roberta Bianchini (CNR) and Lars Eric Hientzsch (Bielefeld University)
UPV/EHU, Thursday, October 17th, 2024, 12:00 - 13:00
Title: Existence and regularity results for the sub-Riemannian Navier-Stokes system
In this talk, I will introduce a new class of models called sub-Riemannian Navier-Stokes systems that model an incompressible fluid whose velocity field admits an anisotropy resulting from a geometric structure. We show the global existence of finite energy (weak) solutions for any initial data belonging to $L^2$. Then, we will establish the well-posedness in a suitable Sobolev-type critical space that takes into account the structure of the system and the underlying geometry. Finally, I will describe the smoothing effects through which we found the anisotropic structure of the equations.
The analysis strongly involves the sub-Riemannian geometric structure of stratified Lie groups and tools from non-commutative harmonic analysis.
UPV/EHU, Thursday, October 10th, 2024, 12:00 - 13:00
Title: A lattice approach to singular integrals
In this talk we discuss the problem of characterizing the boundedness of singular integrals in Banach lattices such as weighted Morrey and variable Lebesgue spaces. Moreover, we apply this lattice viewpoint to the setting of matrix weights to obtain new characterizations and Rubio de Francia extrapolation results for matrix Muckenhoupt weights.
BCAM, Thursday, October 3rd, 2024, 17:00 - 18:00
Title: On the rigorous derivation of the Wave Kinetic Equation from the periodic cubic NLS
In a system formed of many objects (particles, molecules, electrons, ...), instead of computing the evolution of each individual object described by elementary laws of physics, a kinetic equation predicts the distribution of such objects. The most famous example is likely Boltzmann’s equation, which describes the evolution of the density of gas molecules, each of which moves according to Newton’s laws. An analogue theory exists for a wave system described by a dispersive PDE, whose “average” behavior is described by a wave kinetic equation (WKE). The classic derivation of these kinetic equations is a heuristic limiting procedure, but it is a major pending task is to make it rigorous.
This talk will be a very informal introduction to the derivation of the Wave Kinetic Equation. I will discuss its heuristic derivation from the periodic cubic NLS and the main challenges to make it rigorous, give the most recent results, and show how the problem has very relevant probabilistic, combinatorial and number theoretic components.
BCAM, Thursday, September 19th, 2024, 17:00 - 18:00
Title: Mathematical Analysis of Fluid-structure interaction models
In this talk, we will explain what a fluid-structure interaction problem is. We describe the interesting problems in this area. In particular, we focus on the following problem: we consider the motion of several small rigid bodies immersed in a viscous incompressible fluid contained in a domain. We show that the fluid flow is not influenced by the presence of the infinitely many bodies in the asymptotic limit. The result depends solely on the geometry of the bodies and is independent of their mass densities.