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BEGIN:VEVENT
SUMMARY:Alessio Figalli (ETH Zurich)
DTSTART:20210906T120000Z
DTEND:20210906T122500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/1
DESCRIPTION:Title: The singular set in the Stefan problem\nby Alessio Figalli (ETH Zu
rich) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between P
DEs\, Analysis and Geometry\n\n\nAbstract\nThe Stefan problem describes ph
ase transitions such as ice melting to water\, and it is among the most cl
assical free boundary problems. It is well known that the free boundary co
nsists of a smooth part (the regular part) and singular points. In this ta
lk\, I will describe a recent result with Ros-Oton and Serra\, where we an
alyze the singular set in the Stefan problem and prove a series of fine re
sults on its structure.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xavier Ros-Oton (ICREA and Universitat de Barcelona (Spain))
DTSTART:20210906T123000Z
DTEND:20210906T125500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/2
DESCRIPTION:Title: Stable cones in the thin one-phase free boundary problem\nby Xavie
r Ros-Oton (ICREA and Universitat de Barcelona (Spain)) as part of CMO-New
Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geome
try\n\n\nAbstract\nWe study homogeneous stable solutions to the thin (or f
ractional) one-phase free boundary problem. The problem of classifying sta
ble (or minimal) homogeneous solutions in dimensions $n\\geq3$ is complete
ly open. In this context\, axially symmetric solutions are expected to pla
y the same role as Simons’ cone in the classical theory of minimal surfa
ces\, but even in this simpler case the problem is open. The goal of this
talk is to present some new results in this direction.\nOn the one hand we
find\, for the first time\, the stability condition for the thin one-phas
e problem. Quite surprisingly\, this requires the use of "large solutions"
for the fractional Laplacian\, which blow up on the free boundary.\nOn th
e other hand\, using our new stability condition\, we show that any axiall
y symmetric homogeneous stable solution in dimensions \\(n<6\\) is one-dim
ensional\, independently of the parameter $s\\in(0\,1)$.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luciano Mari (Universtità di Torino (Italy))
DTSTART:20210906T130000Z
DTEND:20210906T132500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/3
DESCRIPTION:Title: Regularity for the prescribed Lorentzian mean curvature equation with
charges: the electrostatic Born-Infeld model\nby Luciano Mari (Univers
tità di Torino (Italy)) as part of CMO-New Trends in Nonlinear Diffusion:
a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nIn electrost
atic Born-Infeld theory\, the electrostatic potential $u_\\rho$ generated
by a charge distribution $\\rho$ on $\\mathbb{R}^m$ (typically\, a Radon m
easure) is required to minimize the action\n \\[\n \\int_{\\mathbb{R}^m} \
\Big( 1 - \\sqrt{1-|D\\psi|^2} \\Big) d x - \\langle \\rho\, \\psi \\rangl
e\n \\]\namong functions with a suitable decay at infinity and satisfying
$|D\\psi| \\le 1$. Formally\, the Euler-Lagrange equation $(\\mathcal{BI})
$ prescribes $\\rho$ as being the Lorentzian mean curvature of the graph o
f $u_\\rho$ in Minkowski spacetime $\\mathbb{L}^{m+1}$\; for instance\, if
$\\rho$ is a finite sum of Dirac deltas\, then the graph of $u_\\rho$ i
s a maximal spacelike hypersurface with singularities in $\\mathbb{L}^{m+1
}$. While the existence/uniqueness of $u_\\rho$ follows from standard vari
ational arguments\, finding sharp conditions on $\\rho$ to guarantee that
$u_\\rho$ solves $(\\mathcal{BI})$ is an open problem that has been addres
sed only in a few special cases. In this talk\, I will report on a recent
joint work with J. Byeon\, N. Ikoma and A. Malchiodi\, where we study the
solvability of $(\\mathcal{BI})$ and the regularity of $u_\\rho$ under mil
d conditions on $\\rho$. One of the main sources of difficulties is the po
ssible presence of light rays in the graph of $u_\\rho$\, which will be di
scussed in detail.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilles Carron (Universite de Nantes)
DTSTART:20210906T133000Z
DTEND:20210906T135500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/4
DESCRIPTION:Title: Yamabe flow on singular spaces\nby Gilles Carron (Universite de Na
ntes) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between P
DEs\, Analysis and Geometry\n\n\nAbstract\nIt is joint work with Boris Ver
tman (Oldenburg) and Jørgen Olsen Lye (Oldenburg). We study the convergen
ce of the normalized Yamabe flow with positive Yamabe constant on a class
of pseudo-manifolds that includes stratified spaces with iterated cone-edg
e metrics. We establish convergence under a low-energy condition. We also
prove a concentration-compactness dichotomy\, and investigate what the alt
ernatives to convergence is.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (University of Chicago)
DTSTART:20210906T150000Z
DTEND:20210906T152500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/5
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without cutoff\nby
Luis Silvestre (University of Chicago) as part of CMO-New Trends in Nonli
near Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstrac
t\nWe study the regularization effect of the inhomogeneous Boltzmann equat
ion without cutoff. We obtain a priori estimates for all derivatives of th
e solution depending only on bounds of its hydrodynamic quantities: mass d
ensity\, energy density and entropy density. We use methods that originate
d in the study of nonlocal elliptic and parabolic equations: a weak Harnac
k inequality in the style of De Giorgi\, and a Schauder-type estimate.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yannick Sire (Johns Hopkins University)
DTSTART:20210906T153000Z
DTEND:20210906T155500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/6
DESCRIPTION:Title: KAM theory for ill-posed PDEs\nby Yannick Sire (Johns Hopkins Univ
ersity) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between
PDEs\, Analysis and Geometry\n\n\nAbstract\nI will review some results fo
r the construction of invariant tori in infinite dimensional systems model
ed on lattices and (some) PDEs\, with an emphasis on ill-posed PDEs arisin
g in fluids. I will in particular work out the details for the Boussinesq
equation and some other long-wave approximations of the water wave system.
\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Arnold (Technische Universitaet Wien)
DTSTART:20210906T160000Z
DTEND:20210906T162500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/7
DESCRIPTION:Title: Optimal non-symmetric Fokker-Planck equation for the convergence to a
given equilibrium\nby Anton Arnold (Technische Universitaet Wien) as p
art of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Anal
ysis and Geometry\n\n\nAbstract\nWe are concerned with finding Fokker-Plan
ck equations in whole space with the fastest exponential decay towards a g
iven equilibrium. For a prescribed\, anisotropic Gaussian we determine a n
on-symmetric Fokker-Planck equation with linear drift that shows the highe
st exponential decay rate for the convergence of its solutions towards equ
ilibrium. At the same time it has to allow for a decay estimate with a mul
tiplicative constant arbitrarily close to its infimum. This infimum is $1$
\, corresponding to the high-rotational limit in the Fokker-Planck drift.\
n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elvise Berchio (Politecnico di Torino)
DTSTART:20210906T163000Z
DTEND:20210906T165500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/8
DESCRIPTION:Title: Optimization of eigenvalues of partially hinged composite plates and r
elated theoretical issues\nby Elvise Berchio (Politecnico di Torino) a
s part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, A
nalysis and Geometry\n\n\nAbstract\nWe consider the spectrum of non-homoge
neous \npartially hinged plates having structural engineering \napplicatio
ns. A possible way to prevent instability \nphenomena is to optimize the f
requencies of certain \noscillating modes with respect to the density func
tion of \nthe plate\; we prove existence of optimal densities and we \ninv
estigate their qualitative properties. The analysis is \ncarried out by sh
owing fine properties of the involved \nfourth order operator\, such as th
e validity of the \npositivity preserving property.\n\nBased on a joint wo
rk with Alessio Falocchi.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel del Pino (University of Bath)
DTSTART:20210907T120000Z
DTEND:20210907T122500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/9
DESCRIPTION:Title: Dynamics of concentrated vorticities in 2d and 3d\nby Manuel del P
ino (University of Bath) as part of CMO-New Trends in Nonlinear Diffusion:
a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nA classical
problem that traces back to Helmholtz and Kirchhoff is the understanding o
f the dynamics of solutions to the Euler equations of an inviscid incompre
ssible fluid\, when the vorticity of the solution is initially concentrate
d near isolated points in 2d or vortex lines in 3d. We discuss some recent
results on the existence and asymptotic behaviour of these solutions. We
describe\, with precise asymptotics\, interacting vortices\, and travellin
g helices. We rigorously establish the law of motion of "leapfrogging vort
ex rings"\, originally conjectured by Helmholtz in 1858. This is joint wor
k with Juan Davila\, Monica Musso\, and Juncheng Wei.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatsuki Kawakami (Ryukoku University (Japan))
DTSTART:20210907T123000Z
DTEND:20210907T125500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/10
DESCRIPTION:Title: The large diffusion limit for the heat equation with a dynamical boun
dary condition\nby Tatsuki Kawakami (Ryukoku University (Japan)) as pa
rt of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analy
sis and Geometry\n\n\nAbstract\nWe study the heat equation in a half-space
or the exterior of the unit ball with a dynamical boundary condition. In
this talk\, we construct a global-in-time solution of this problem and sho
w that\, if the diffusion coefficient tends to infinity\, then the solutio
ns converge (in a suitable sense) to solutions of the Laplace equation wit
h the same dynamical boundary condition. Furthermore\, we give the optimal
rate of convergence.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikaela Iacobelli (ETH-Zurich)
DTSTART:20210907T130000Z
DTEND:20210907T132500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/11
DESCRIPTION:Title: Singular limits for Vlasov equations via kinetic-type Wasserstein dis
tances\nby Mikaela Iacobelli (ETH-Zurich) as part of CMO-New Trends in
Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nA
bstract\nThe Vlasov-Poisson system with massless electrons (VPME) is widel
y used in plasma physics to model the evolution of ions in a plasma. It di
ffers from the classical Vlasov-Poisson system in that the Poisson couplin
g has an exponential nonlinearity that creates several mathematical diffic
ulties. In this talk\, we will discuss the well-posedness of VPME\, the st
ability of solutions\, and its behaviour under singular limits. Then\, we
will introduce a new class of Wasserstein-type distances specifically desi
gned to tackle stability questions for kinetic equations. As we shall see\
, these distances allow us to improve classical stability estimates by Loe
per and Dobrushin and to obtain\, as a consequence\, improved rates in qua
si-neutral limits.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Edoardo Mainini (Università di Genova (Italy))
DTSTART:20210907T133000Z
DTEND:20210907T135500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/12
DESCRIPTION:Title: Fractional PDEs and steady states for aggregation-diffusion models\nby Edoardo Mainini (Università di Genova (Italy)) as part of CMO-New T
rends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometr
y\n\n\nAbstract\nAggregation-diffusion models describe the motion of inter
acting agents towards states of overall balance between diffusion effects
and mutual attraction. The Newtonian and the Riesz interaction potentials
provide relevant examples of aggregation modeling with long range effects.
They give rise to local and nonlocal PDEs for the characterization of sta
tionary states: we will focus on existence\, uniqueness and regularity pro
perties of radial entire solutions to the equilibrium equations. This is a
joint work with H. Chan\, M.D.M. González\, Y. Huang and B. Volzone.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Loss (Georgia Institute of Technology)
DTSTART:20210907T150000Z
DTEND:20210907T152500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/13
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Michael Loss (Geor
gia Institute of Technology) as part of CMO-New Trends in Nonlinear Diffus
ion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nI presen
t some results concerning the size of magnetic fields that support zero mo
des for the three dimensional Dirac equation and related problems for spin
or equations. The critical quantity\, is the $3/2$ norm of the magnetic fi
eld $B$. The point is that the spinor structure enters the analysis in a c
rucial way. This is joint work with Rupert Frank at Caltech.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Gualdani (University of Texas Austin)
DTSTART:20210907T153000Z
DTEND:20210907T155500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/14
DESCRIPTION:by Maria Gualdani (University of Texas Austin) as part of CMO-
New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Ge
ometry\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Schmeiser (University of Vienna)
DTSTART:20210907T160000Z
DTEND:20210907T162500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/15
DESCRIPTION:Title: A kinetic model for myxobacteria with binary reversal and alignment i
nteraction and with Brownian forcing\nby Christian Schmeiser (Universi
ty of Vienna) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge b
etween PDEs\, Analysis and Geometry\n\n\nAbstract\nThe competition between
directional dispersal caused\nby Brownian forcing and tendency towards co
ncentration caused\nby alignment is studied. Main results are the stabilit
y of uniform\nstates for dominating Brownian forcing (proven by hypocoerci
vity\nmethods) as well as the existence of nontrivial steady states (shown
\nby a bifurcation approach).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Craig (University of California Santa Barbara)
DTSTART:20210907T163000Z
DTEND:20210907T165500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/16
DESCRIPTION:Title: A blob method for nonlinear diffusion and applications to sampling an
d two layer neural networks\nby Katy Craig (University of California S
anta Barbara) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge b
etween PDEs\, Analysis and Geometry\n\n\nAbstract\nGiven a desired target
distribution and an initial guess of that distribution\, composed of finit
ely many samples\, what is the best way to evolve the locations of the sam
ples so that they accurately represent the desired distribution? A classic
al solution to this problem is to allow the samples to evolve according to
Langevin dynamics\, the stochastic particle method corresponding to the F
okker-Planck equation. In today’s talk\, I will contrast this classical
approach with a deterministic particle method corresponding to the porous
medium equation. This method corresponds exactly to the mean-field dynamic
s of training a two layer neural network for a radial basis function activ
ation function. We prove that\, as the number of samples increases and the
variance of the radial basis function goes to zero\, the particle method
converges to a bounded entropy solution of the porous medium equation. As
a consequence\, we obtain both a novel method for sampling probability dis
tributions as well as insight into the training dynamics of two layer neur
al networks in the mean field regime.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuhiro Ishige (The University of Tokyo)
DTSTART:20210908T120000Z
DTEND:20210908T122500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/17
DESCRIPTION:Title: Power concavity and Dirichlet heat flow\nby Kazuhiro Ishige (The
University of Tokyo) as part of CMO-New Trends in Nonlinear Diffusion: a B
ridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nWe show that log
-concavity is the weakest power concavity preserved by the Dirichlet heat
flow in $N$-dimensional convex domains\, where $N\\ge 2$. Jointly with wha
t we already know\, i.e. that log-concavity is the strongest power concavi
ty preserved by the Dirichlet heat flow\, we see that log-concavity is ind
eed the only power concavity preserved by the Dirichlet heat flow.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Goro Akagi (Tohoku University)
DTSTART:20210908T123000Z
DTEND:20210908T125500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/18
DESCRIPTION:Title: Rates of convergence to non-degenerate asymptotic profiles for fast d
iffusion equations via an energy metho\nby Goro Akagi (Tohoku Universi
ty) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDE
s\, Analysis and Geometry\n\n\nAbstract\nThis talk is concerned with a qua
ntitative analysis of\nasymptotic behavior of solutions to the Cauchy-Diri
chlet problem\nfor the fast diffusion equation posed on bounded domains wi
th\nSobolev subcritical exponents. More precisely\, rates of convergence\n
to non-degenerate asymptotic profiles will be discussed via an energy meth
od.\nThe rate of convergence for positive profiles was recently discussed\
nbased on an entropy method by Bonforte and Figalli (2021\, CPAM).\nAn alt
ernative proof will also be provided to their result.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabio Punzo (Politecnico di Milano (Italy))
DTSTART:20210908T130000Z
DTEND:20210908T132500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/19
DESCRIPTION:Title: Global existence for a class of nonlinear reaction-diffusion equation
s on Riemannian manifolds: an approach via Sobolev and Poincaré inequalit
ies\nby Fabio Punzo (Politecnico di Milano (Italy)) as part of CMO-New
Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geome
try\n\n\nAbstract\nWe discuss existence of global-in-time solutions to the
porous medium equation with a reaction term on Riemannian manifolds\, whe
re Sobolev and Poincaré inequalities are assumed to hold. Smoothing estim
ates are also established. The results have been recently obtained jointly
with Gabriele Grillo and Giulia Meglioli (Politecnico di Milano).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Quirós (Universidad Autónoma de Madrid (Spain))
DTSTART:20210908T133000Z
DTEND:20210908T135500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/20
DESCRIPTION:Title: Large-time behaviour in nonlocal heat equations with memory\nby F
ernando Quirós (Universidad Autónoma de Madrid (Spain)) as part of CMO-N
ew Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geo
metry\n\n\nAbstract\nIn this talk we will review several recent results\,
in collaboration with Carmen Cortázar (PUC\, Chile) and Noemí Wolanski (
IMAS-UBA-CONICET\, Argentina)\, on the large-time behaviour of solutions t
o fully nonlocal heat equations involving a Caputo time derivative and a p
ower of the Laplacian. The Caputo time derivative introduces memory effect
s that yield new phenomena that are not present in classical diffusion equ
ations.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Nazaret (SAMM\, Université Paris 1)
DTSTART:20210908T150000Z
DTEND:20210908T152500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/21
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): A variat
ional point of view\nby Bruno Nazaret (SAMM\, Université Paris 1) as
part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Ana
lysis and Geometry\n\n\nAbstract\nIn this first lecture of a series of thr
ee\, we discuss stability results in Gagliardo-Nirenberg-Sobolev inequalit
ies\, from a joint project with M. Bonforte\, J. Dolbeault and N. Simonov.
The core of this approach is the use of a non scaling invariant form of t
he inequalities\, equivalent to entropy-entropy production inequalities ar
ising in the study of large time asymptotics for solutions to fast diffusi
on equations. We only use variational arguments\, leading to non construct
ive estimates\, but this paves the way for the constructive results given
in the next two lectures.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Simonov (Universitè Paris Duaphine (France))
DTSTART:20210908T153000Z
DTEND:20210908T155500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/22
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): Converge
nce in relative error for the fast diffusion equation\nby Nikita Simon
ov (Universitè Paris Duaphine (France)) as part of CMO-New Trends in Nonl
inear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstra
ct\nIn this talk\, I will discuss the asymptotic behavior of solutions to
the fast diffusion equation when the tails of the initial datum have a cer
tain decay. In this setting\, I will provide a fully constructive estimate
of the threshold time after which the solution enters a neighborhood of t
he Barenblatt profile in a uniform relative norm. This estimate plays a fu
ndamental role in obtaining a constructive stability result in Gagliardo-N
irenberg-Sobolev inequalities. The results are based on a joint work with
Matteo Bonforte\, Jean Dolbeault\, and Bruno Nazaret.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Dolbeault (Université Paris-Dauphine)
DTSTART:20210908T160000Z
DTEND:20210908T162500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/23
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): Entropy
methods and stability\nby Jean Dolbeault (Université Paris-Dauphine)
as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\,
Analysis and Geometry\n\n\nAbstract\nThis lecture is the third lecture on
stability issues in Gagliardo-Nirenberg-Sobolev inequalities\, a joint pro
ject with M. Bonforte\, N. Simonov and B. Nazaret. The results are based o
n entropy methods and the use of the fast diffusion equation (FDE) for stu
dying refined versions of the Gagliardo-Nirenberg-Sobolev inequalities. Us
ing the quantitative regularity estimates\, we go beyond the variational r
esults of the first lecture and provide fully constructive estimates\, to
the price of a small restriction of the functional space which is inherent
to the method.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shin-ichi Ohta (Osaka University)
DTSTART:20210909T120000Z
DTEND:20210909T122500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/24
DESCRIPTION:Title: Geometric analysis on Finsler manifolds\nby Shin-ichi Ohta (Osaka
University) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge be
tween PDEs\, Analysis and Geometry\n\n\nAbstract\nWe review developments i
n geometric analysis on Finsler manifolds of weighted Ricci curvature boun
ded below. We especially discuss a nonlinear analogue of the Gamma-calculu
s and its applications to isoperimetric and functional inequalities.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yao Yao (National University of Singapore)
DTSTART:20210909T123000Z
DTEND:20210909T125500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/25
DESCRIPTION:Title: Uniqueness and non-uniqueness of stationary solutions of aggregation-
diffusion equation\nby Yao Yao (National University of Singapore) as p
art of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Anal
ysis and Geometry\n\n\nAbstract\nIn this talk\, I will discuss a nonlocal
aggregation equation with degenerate diffusion\, which describes the mean-
field limit of interacting particles driven by nonlocal interactions and l
ocalized repulsion. When the interaction potential is attractive\, it is p
reviously known that all stationary solutions must be radially decreasing
up to a translation\, but uniqueness (for a given mass) within this class
was open\, except for some special interaction potentials. For general att
ractive potentials\, we show that the uniqueness/non-uniqueness criteria a
re determined by the power of the degenerate diffusion\, with the critical
power being $m=2$. Namely\, for $m \\geq 2$\, we show the stationary solu
tion for any given mass is unique for any attractive potential\, by tracki
ng the associated energy functional along a novel interpolation curve. And
for $1< m < 2 $\, we construct examples of smooth attractive potentials\,
such that there are infinitely many radially decreasing stationary soluti
ons of the same mass. This is a joint work with Matias Delgadino and Xukai
Yan.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria del Mar Gonzalez (Universidad Autonoma de Madrid)
DTSTART:20210909T130000Z
DTEND:20210909T132500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/26
DESCRIPTION:Title: Non-local ODEs in conformal geometry\nby Maria del Mar Gonzalez (
Universidad Autonoma de Madrid) as part of CMO-New Trends in Nonlinear Dif
fusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nWhen
one looks for radial solutions of an equation with fractional Laplacian\,
it is not generally possible to use standard ODE methods. If such equation
has some conformal invariances\, then one may rewrite it in Emden-Fowler
(cylindrical) coordinates and use the properties of the conformal fraction
al Laplacian on the cylinder\, which involves some complex analysis techni
ques. After giving the necessary background\, we will briefly consider two
particular applications of this technique: 1. Symmetry breaking\, non-deg
eneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequa
lity (joint work with W. Ao and A. DelaTorre). 2. Existence and regularity
for fractional Laplacian equations with drift and a critical Hardy potent
ial (joint with H. Chan\, M. Fontelos and J. Wei).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monica Musso (University of Bath)
DTSTART:20210909T133000Z
DTEND:20210909T135500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/27
DESCRIPTION:Title: Infinite-time blowing-up solutions to small perturbations of the Yama
be flow\nby Monica Musso (University of Bath) as part of CMO-New Trend
s in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\
n\nAbstract\nUnder the validity of the positive mass theorem\, the Yamabe
flow on a smooth compact Riemannian manifold of dimension greater or equal
to $3$ is known to exist for all time and converges to a solution to the
Yamabe problem at infinity. In this talk I will present a result\, obtain
ed in collaboration with Seunghyeok Kim\, in which we prove that if a suit
able perturbation\, which may be smooth and arbitrarily small\, is imposed
on the Yamabe flow on any given Riemannian manifold M of dimension bigger
or equal to $5$\, the resulting flow may blow up at multiple points on M
in the infinite time. We construct such a flow by using solutions of the Y
amabe problem on the unit sphere as blow-up profiles. We also examine the
stability of the blow-up phenomena under a negativity condition on the Ric
ci curvature at blow-up points.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramon Plaza (Universidad Nacional Autónoma de México)
DTSTART:20210909T150000Z
DTEND:20210909T152500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/28
DESCRIPTION:Title: Spectral stability of monotone traveling fronts for reaction diffusio
n-degenerate Nagumo equations\nby Ramon Plaza (Universidad Nacional Au
tónoma de México) as part of CMO-New Trends in Nonlinear Diffusion: a Br
idge between PDEs\, Analysis and Geometry\n\n\nAbstract\nThis talk address
es the spectral stability of monotone traveling front solutions for reacti
on-diffusion equations where the reaction function is of Nagumo (or bistab
le) type and with diffusivities which are density dependent and degenerate
at zero (one of the equilibrium points of the reaction). Spectral stabili
ty is understood as the property that the spectrum of the linearized opera
tor around the wave\, acting on an exponentially weighted space\, is conta
ined in the complex half plane with non-positive real part. The degenerate
fronts under consideration travel with positive speed above a threshold v
alue and connect the (diffusion-degenerate) zero state with the unstable e
quilibrium point of the reaction function. In this case\, the degeneracy o
f the diffusion coefficient is responsible of the loss of hyperbolicity of
the asymptotic coefficient matrices of the spectral problem at one of the
end points\, precluding the application of standard techniques to locate
the essential spectrum (cf. Kapitula\, Promislow\, 2013). This difficulty
is overcome with a suitable partition of the spectrum\, a generalized conv
ergence of operators technique\, the analysis of singular (or Weyl) sequen
ces and the use of energy estimates. The monotonicity of the fronts\, as w
ell as detailed descriptions of the decay structure of eigenfunctions on a
case by case basis\, are key ingredients to show that all traveling front
s under consideration are spectrally stable in a suitably chosen exponenti
ally weighted $L^2$ energy space. This is joint work with J. F. Leyva (Ben
emérita Universidad Autónoma de Puebla) y L. F. López Ríos (IIMAS-UNAM
).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihaela Ignatova (Temple University)
DTSTART:20210909T153000Z
DTEND:20210909T155500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/29
DESCRIPTION:Title: Nernst-Planck-Navier-Stokes equations\nby Mihaela Ignatova (Templ
e University) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge b
etween PDEs\, Analysis and Geometry\n\n\nAbstract\nI will describe results
on global existence\, stability and interior electroneutrality for Nernst
-Planck equations coupled with Navier-Stokes and related equations.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michele Coti Zelati (Imperial College London)
DTSTART:20210909T160000Z
DTEND:20210909T162500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/30
DESCRIPTION:Title: Stationary Euler flows near the Kolmogorov flow\nby Michele Coti
Zelati (Imperial College London) as part of CMO-New Trends in Nonlinear Di
ffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nWe e
xhibit a large family of new\, non-trivial stationary states of\nanalytic
regularity\, that are arbitrarily close to the Kolmogorov flow on the\nsqu
are torus. Our construction of these stationary states builds on a\ndegene
racy in the global structure of the Kolmogorov flow.\nThis has surprising
consequences in the context of inviscid\ndamping in 2D Euler and enhanced
dissipation in Navier-Stokes.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José A. Carrillo (University of Oxford)
DTSTART:20210909T163000Z
DTEND:20210909T165500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/31
DESCRIPTION:Title: Nonlocal Aggregation-Diffusion Equations: entropies\, gradient flows\
, phase transitions and applications\nby José A. Carrillo (University
of Oxford) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge bet
ween PDEs\, Analysis and Geometry\n\n\nAbstract\nThis talk will be devoted
to an overview of recent results understanding the bifurcation analysis o
f nonlinear Fokker-Planck equations arising in a myriad of applications su
ch as consensus formation\, optimization\, granular media\, swarming behav
ior\, opinion dynamics and financial mathematics to name a few. We will pr
esent several results related to localized Cucker-Smale orientation dynami
cs\, McKean-Vlasov equations\, and nonlinear diffusion Keller-Segel type m
odels in several settings. We will show the existence of continuous or dis
continuous phase transitions on the torus under suitable assumptions on th
e Fourier modes of the interaction potential. The analysis is based on lin
ear stability in the right functional space associated to the regularity o
f the problem at hand. While in the case of linear diffusion\, one can wor
k in the $L^2$ framework\, nonlinear diffusion needs the stronger $L^\\inf
ty$ topology to proceed with the analysis based on Crandall-Rabinowitz bif
urcation analysis applied to the variation of the entropy functional. Expl
icit examples show that the global bifurcation branches can be very compli
cated. Stability of the solutions will be discussed based on numerical sim
ulations with fully explicit energy decaying finite volume schemes specifi
cally tailored to the gradient flow structure of these problems. The theor
etical analysis of the asymptotic stability of the different branches of s
olutions is a challenging open problem. This overview talk is based on sev
eral works in collaboration with R. Bailo\, A. Barbaro\, J. A. Canizo\, X.
Chen\, P. Degond\, R. Gvalani\, J. Hu\, G. Pavliotis\, A. Schlichting\, Q
. Wang\, Z. Wang\, and L. Zhang. This research has been funded by EPSRC EP
/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriele Grillo (Politecnico di Milano)
DTSTART:20210910T120000Z
DTEND:20210910T122500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/32
DESCRIPTION:Title: Nonlinear characterizations of stochastic completeness\nby Gabrie
le Grillo (Politecnico di Milano) as part of CMO-New Trends in Nonlinear D
iffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nA m
anifold is said to be stochastically complete if the free heat semigroup p
reserves probability. It is well known that this property is equivalent to
nonexistence of nonnegative\, bounded solutions to certain (linear) ellip
tic problems\, and to uniqueness of solutions to the heat equation corresp
onding to bounded initial data. We prove that stochastic completeness is a
lso equivalent to similar properties for certain nonlinear elliptic and pa
rabolic problems. This fact\, and the known analytic-geometric characteriz
ations of stochastic completeness\, allow to give new explicit criteria fo
r existence/nonexistence of solutions to certain nonlinear elliptic equati
ons on manifolds\, and for uniqueness/nonuniqueness of solutions to certai
n nonlinear diffusions on manifolds.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asuka Takatsu (Tokyo Metropolitan University (Japan))
DTSTART:20210910T123000Z
DTEND:20210910T125500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/33
DESCRIPTION:Title: Spectral convergence of high-dimensional spheres to Gaussian spaces\nby Asuka Takatsu (Tokyo Metropolitan University (Japan)) as part of CM
O-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and
Geometry\n\n\nAbstract\nIt is known that the projection of a uniform proba
bility measure on the $N$-dimensional sphere to the first $n$ coordinates
approximates the $n$-dimensional Gaussian measure.\nIn this talk\, I will
present that the spectral structure on the $N$-dimensional sphere compatib
le with the projection to the first $n$ coordinates approximates the spect
ral structure on the $n$-dimensional Gaussian space.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diana Stan (Universidad de Cantabria (Spain))
DTSTART:20210910T130000Z
DTEND:20210910T132500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/34
DESCRIPTION:Title: The fast p-Laplacian evolution equation. Global Harnack principle and
fine asymptotic behavior\nby Diana Stan (Universidad de Cantabria (Sp
ain)) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between P
DEs\, Analysis and Geometry\n\n\nAbstract\nWe study fine global properties
of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian
evolution equation on the whole Euclidean space\, in the so-called "good
fast diffusion range". It is well known that non-negative solutions behave
for large times as B\, the Barenblatt (or fundamental) solution\, which h
as an explicit expression. We prove the so-called Global Harnack Principle
(GHP)\, that is\, precise global pointwise upper and lower estimates of n
onnegative solutions in terms of B. This can be considered the nonlinear c
ounterpart of the celebrated Gaussian estimates for the linear heat equati
on. To the best of our knowledge\, analogous issues for the linear heat eq
uation\, do not possess such clear answers\, only partial results are know
n. Also\, we characterize the maximal (hence optimal) class of initial dat
a such that the GHP holds\, by means of an integral tail condition\, easy
to check. Finally\, we derive sharp global quantitative upper bounds of th
e modulus of the gradient of the solution\, and\, when data are radially d
ecreasing\, we show uniform convergence in relative error for the gradient
s. This is joint work with Matteo Bonforte (UAM-ICMAT\, Madrid\, Spain) an
d Nikita Simonov (Ceremade-Univ. Paris-Dauphine\, Paris\, France).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincenzo Ferone (Università di Napoli Federico II)
DTSTART:20210910T133000Z
DTEND:20210910T135500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/35
DESCRIPTION:Title: Symmetrization for fractional elliptic problems: a direct approach\nby Vincenzo Ferone (Università di Napoli Federico II) as part of CMO-N
ew Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geo
metry\n\n\nAbstract\nWe provide new direct methods to establish symmetriza
tion results in the form of mass concentration (i.e. integral) comparison
for fractional elliptic equations of the type $ (-\\Delta)^s u =f \\ $ ($
0 < s < 1 $) in a bounded domain $ \\Omega $\, equipped with homogeneous
Dirichlet boundary conditions. The classical pointwise Talenti rearrangem
ent inequality is recovered in the limit $ s\\rightarrow1 $. Finally\, exp
licit counterexamples constructed for all $ s\\in(0\,1) $ highlight that t
he same pointwise estimate cannot hold in a nonlocal setting\, thus showin
g the optimality of our results. The results are contained in a joint pape
r with Bruno Volzone [Ferone\, V.\; Volzone\, B.\, Symmetrization for frac
tional elliptic problems: a direct approach. Arch. Ration. Mech. Anal. 239
(2021)\, 1733-1770].\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexis Vasseur (University of Texas at Austin)
DTSTART:20210910T150000Z
DTEND:20210910T152500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/36
DESCRIPTION:Title: Uniform stability of viscous shocks for the compressible Navier-Stoke
s equation\nby Alexis Vasseur (University of Texas at Austin) as part
of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis
and Geometry\n\n\nAbstract\nWe show the stability of viscous shocks of th
e 1D compressible Navier-Stokes equation. This stability holds uniformly w
ith respect to the viscosity\, up to the inviscid limit. Stability results
for shocks of the Euler equation are then inherited at the inviscid limit
. These stability results hold in the class of wild perturbations of invis
cid limits\, without any regularity restriction. This shows that the class
of inviscid limits of Navier-Stokes equations is better behaved than the
larger class of weak entropic solutions to the Euler equation. The result
is based on the theory of a-contraction with shifts. This is a joint work
with Moon-Jin Kang.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Azahara DelaTorre (University of Granada)
DTSTART:20210910T153000Z
DTEND:20210910T155500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/37
DESCRIPTION:Title: : The fractional Lane–Emden equation with Serrin’s critical expon
ent\nby Azahara DelaTorre (University of Granada) as part of CMO-New T
rends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometr
y\n\n\nAbstract\nIn this talk we will focus on the the existence\, multipl
icity and local behavior of singular solutions of the fractional Lane–Em
den equation with Serrin’s critical exponent and homogeneous Dirichlet e
xterior condition. These will provide the profile to construct singular me
trics with constant (non-local) curvature. We will show radial symmetry cl
ose to the origin\, a Liouville-type result without any assumption on its
asymptotic behavior (showing the necessity of imposing the Dirichlet condi
tion) and the existence of multiple solutions in a bounded domain with any
prescribed closed singular set. Moreover\, we will show that the singular
behavior of the profile is unique\, presenting new methods based on the c
onnection between the non-local equation and its associated first order OD
E in one dimension. \nThis is a joint work with H. Chan.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix del Teso (Universidad Complutense de Madrid (Spain))
DTSTART:20210910T160000Z
DTEND:20210910T162500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/38
DESCRIPTION:Title: The Liouville Theorem and linear operators satisfying the maximum pri
nciple\nby Felix del Teso (Universidad Complutense de Madrid (Spain))
as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\,
Analysis and Geometry\n\n\nAbstract\nThe classical Liouville Theorem state
s that bounded harmonic functions\nare constant. The talk will revisit thi
s result for the most general class of\nlinear operators with constant coe
fficients satisfying the maximum principle\n(characterized by Courrège in
[P. Courrège\, Générateur infinitésimal d’un semi-groupe de convolu
tion sur $R^n$ \, et formule de Lévy-Khinchine. Bull. Sci. Math. (2)\, 88
:3–30\, 1964]). The class includes local and nonlocal and\nnot necessari
ly symmetric operators among which you can find the fractional\nLaplacian\
, Relativistic Schrödinger operators\, convolution operators\, CGMY\,\nas
well as discretizations of them.\nWe give a full characterization of the
operators in this class satisfying the\nLiouville property. When the Liouv
ille property does not hold\, we also establish\nprecise periodicity sets
of the solutions.\nThe techniques and proofs of [N. Alibaud\, F. del Teso\
, J. Endal\, and E. R. Jakobsen\, The Liouville\ntheorem and linear operat
ors satisfying the maximum principle. Journal de\nMathématiques Pures et
Appliquées\, 142:229–242\, 2020] combine arguments from PDEs\, group th
e-\nory\, number theory and numerical analysis (and still\, they are simpl
e\, short\,\nand very intuitive).\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Luis Vazquez (Universidad Autonoma de Madrid (Spain))
DTSTART:20210910T163000Z
DTEND:20210910T165500Z
DTSTAMP:20260422T185332Z
UID:CMO-21w5127/39
DESCRIPTION:Title: Nonlinear fractional Laplacian operators and equations\nby Juan L
uis Vazquez (Universidad Autonoma de Madrid (Spain)) as part of CMO-New Tr
ends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry
\n\n\nAbstract\nWe consider a number of equations involving nonlinear frac
tional \nLaplacian operators where progress has been obtained in recent ye
ars. \nExamples include fractional $p$-Laplacian operators appearing in el
liptic \nand parabolic equations and a number of variants. Numerical analy
sis is \nperformed.\n
LOCATION:https://researchseminars.org/talk/CMO-21w5127/39/
END:VEVENT
END:VCALENDAR