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BEGIN:VEVENT
SUMMARY:Alessio Figalli (ETH Zurich)
DTSTART:20210906T120000Z
DTEND:20210906T122500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/1
DESCRIPTION:Title: The singular set in the Stefan problem\nby Alessio Figalli
(ETH Zurich) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge be
tween PDEs\, Analysis and Geometry\n\n\nAbstract\nThe Stefan problem descr
ibes phase transitions such as ice melting to water\, and it is among the
most classical free boundary problems. It is well known that the free boun
dary consists of a smooth part (the regular part) and singular points. In
this talk\, I will describe a recent result with Ros-Oton and Serra\, wher
e we analyze the singular set in the Stefan problem and prove a series of
fine results on its structure.\n
LOCATION:https://stable.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:20260404T041648Z
UID:CMO-21w5127/2
DESCRIPTION:Title: Stable cones in the thin one-phase free boundary problem\nb
y Xavier Ros-Oton (ICREA and Universitat de Barcelona (Spain)) as part of
CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis an
d Geometry\n\n\nAbstract\nWe study homogeneous stable solutions to the thi
n (or fractional) one-phase free boundary problem. The problem of classify
ing stable (or minimal) homogeneous solutions in dimensions $n\\geq3$ is c
ompletely open. In this context\, axially symmetric solutions are expected
to play the same role as Simons’ cone in the classical theory of minima
l surfaces\, but even in this simpler case the problem is open. The goal o
f 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 o
ne-phase problem. Quite surprisingly\, this requires the use of "large sol
utions" for the fractional Laplacian\, which blow up on the free boundary.
\nOn the other hand\, using our new stability condition\, we show that any
axially symmetric homogeneous stable solution in dimensions \\(n<6\\) is
one-dimensional\, independently of the parameter $s\\in(0\,1)$.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luciano Mari (Universtità di Torino (Italy))
DTSTART:20210906T130000Z
DTEND:20210906T132500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/3
DESCRIPTION:Title: Regularity for the prescribed Lorentzian mean curvature equatio
n with charges: the electrostatic Born-Infeld model\nby Luciano Mari (
Universtità di Torino (Italy)) as part of CMO-New Trends in Nonlinear Dif
fusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nIn el
ectrostatic Born-Infeld theory\, the electrostatic potential $u_\\rho$ gen
erated by a charge distribution $\\rho$ on $\\mathbb{R}^m$ (typically\, a
Radon measure) 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
\\rangle\n \\]\namong functions with a suitable decay at infinity and sati
sfying $|D\\psi| \\le 1$. Formally\, the Euler-Lagrange equation $(\\mathc
al{BI})$ prescribes $\\rho$ as being the Lorentzian mean curvature of the
graph of $u_\\rho$ in Minkowski spacetime $\\mathbb{L}^{m+1}$\; for instan
ce\, if $\\rho$ is a finite sum of Dirac deltas\, then the graph of $u_\
\rho$ is a maximal spacelike hypersurface with singularities in $\\mathbb{
L}^{m+1}$. While the existence/uniqueness of $u_\\rho$ follows from standa
rd variational arguments\, finding sharp conditions on $\\rho$ to guarante
e that $u_\\rho$ solves $(\\mathcal{BI})$ is an open problem that has been
addressed 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 stu
dy the solvability of $(\\mathcal{BI})$ and the regularity of $u_\\rho$ un
der mild conditions on $\\rho$. One of the main sources of difficulties is
the possible presence of light rays in the graph of $u_\\rho$\, which wil
l be discussed in detail.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilles Carron (Universite de Nantes)
DTSTART:20210906T133000Z
DTEND:20210906T135500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/4
DESCRIPTION:Title: Yamabe flow on singular spaces\nby Gilles Carron (Universit
e de Nantes) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge be
tween PDEs\, Analysis and Geometry\n\n\nAbstract\nIt is joint work with Bo
ris Vertman (Oldenburg) and Jørgen Olsen Lye (Oldenburg). We study the co
nvergence of the normalized Yamabe flow with positive Yamabe constant on a
class of pseudo-manifolds that includes stratified spaces with iterated c
one-edge metrics. We establish convergence under a low-energy condition. W
e also prove a concentration-compactness dichotomy\, and investigate what
the alternatives to convergence is.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (University of Chicago)
DTSTART:20210906T150000Z
DTEND:20210906T152500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/5
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without cutoff<
/a>\nby Luis Silvestre (University of Chicago) as part of CMO-New Trends i
n Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\n
Abstract\nWe study the regularization effect of the inhomogeneous Boltzman
n equation without cutoff. We obtain a priori estimates for all derivative
s of the solution depending only on bounds of its hydrodynamic quantities:
mass density\, energy density and entropy density. We use methods that or
iginated in the study of nonlocal elliptic and parabolic equations: a weak
Harnack inequality in the style of De Giorgi\, and a Schauder-type estima
te.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yannick Sire (Johns Hopkins University)
DTSTART:20210906T153000Z
DTEND:20210906T155500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/6
DESCRIPTION:Title: KAM theory for ill-posed PDEs\nby Yannick Sire (Johns Hopki
ns University) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge
between PDEs\, Analysis and Geometry\n\n\nAbstract\nI will review some res
ults for the construction of invariant tori in infinite dimensional system
s modeled on lattices and (some) PDEs\, with an emphasis on ill-posed PDEs
arising in fluids. I will in particular work out the details for the Bous
sinesq equation and some other long-wave approximations of the water wave
system.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Arnold (Technische Universitaet Wien)
DTSTART:20210906T160000Z
DTEND:20210906T162500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/7
DESCRIPTION:Title: Optimal non-symmetric Fokker-Planck equation for the convergenc
e to a given equilibrium\nby Anton Arnold (Technische Universitaet Wie
n) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs
\, Analysis and Geometry\n\n\nAbstract\nWe are concerned with finding Fokk
er-Planck equations in whole space with the fastest exponential decay towa
rds a given equilibrium. For a prescribed\, anisotropic Gaussian we determ
ine a non-symmetric Fokker-Planck equation with linear drift that shows th
e highest exponential decay rate for the convergence of its solutions towa
rds equilibrium. At the same time it has to allow for a decay estimate wit
h a multiplicative constant arbitrarily close to its infimum. This infimum
is $1$\, corresponding to the high-rotational limit in the Fokker-Planck
drift.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elvise Berchio (Politecnico di Torino)
DTSTART:20210906T163000Z
DTEND:20210906T165500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/8
DESCRIPTION:Title: Optimization of eigenvalues of partially hinged composite plate
s and related theoretical issues\nby Elvise Berchio (Politecnico di To
rino) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between P
DEs\, Analysis and Geometry\n\n\nAbstract\nWe consider the spectrum of non
-homogeneous \npartially hinged plates having structural engineering \napp
lications. A possible way to prevent instability \nphenomena is to optimiz
e the frequencies of certain \noscillating modes with respect to the densi
ty function of \nthe plate\; we prove existence of optimal densities and w
e \ninvestigate their qualitative properties. The analysis is \ncarried ou
t by showing fine properties of the involved \nfourth order operator\, suc
h as the validity of the \npositivity preserving property.\n\nBased on a j
oint work with Alessio Falocchi.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel del Pino (University of Bath)
DTSTART:20210907T120000Z
DTEND:20210907T122500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/9
DESCRIPTION:Title: Dynamics of concentrated vorticities in 2d and 3d\nby Manue
l del Pino (University of Bath) as part of CMO-New Trends in Nonlinear Dif
fusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nA cla
ssical problem that traces back to Helmholtz and Kirchhoff is the understa
nding of the dynamics of solutions to the Euler equations of an inviscid i
ncompressible fluid\, when the vorticity of the solution is initially conc
entrated near isolated points in 2d or vortex lines in 3d. We discuss some
recent results on the existence and asymptotic behaviour of these solutio
ns. We describe\, with precise asymptotics\, interacting vortices\, and tr
avelling helices. We rigorously establish the law of motion of "leapfroggi
ng vortex rings"\, originally conjectured by Helmholtz in 1858. This is jo
int work with Juan Davila\, Monica Musso\, and Juncheng Wei.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatsuki Kawakami (Ryukoku University (Japan))
DTSTART:20210907T123000Z
DTEND:20210907T125500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/10
DESCRIPTION:Title: The large diffusion limit for the heat equation with a dynamic
al boundary condition\nby Tatsuki Kawakami (Ryukoku University (Japan)
) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\
, Analysis and Geometry\n\n\nAbstract\nWe study the heat equation in a hal
f-space or the exterior of the unit ball with a dynamical boundary conditi
on. In this talk\, we construct a global-in-time solution of this problem
and show that\, if the diffusion coefficient tends to infinity\, then the
solutions converge (in a suitable sense) to solutions of the Laplace equat
ion with the same dynamical boundary condition. Furthermore\, we give the
optimal rate of convergence.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikaela Iacobelli (ETH-Zurich)
DTSTART:20210907T130000Z
DTEND:20210907T132500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/11
DESCRIPTION:Title: Singular limits for Vlasov equations via kinetic-type Wasserst
ein distances\nby Mikaela Iacobelli (ETH-Zurich) as part of CMO-New Tr
ends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry
\n\n\nAbstract\nThe Vlasov-Poisson system with massless electrons (VPME) i
s widely used in plasma physics to model the evolution of ions in a plasma
. It differs from the classical Vlasov-Poisson system in that the Poisson
coupling has an exponential nonlinearity that creates several mathematical
difficulties. In this talk\, we will discuss the well-posedness of VPME\,
the stability of solutions\, and its behaviour under singular limits. The
n\, we will introduce a new class of Wasserstein-type distances specifical
ly designed to tackle stability questions for kinetic equations. As we sha
ll see\, these distances allow us to improve classical stability estimates
by Loeper and Dobrushin and to obtain\, as a consequence\, improved rates
in quasi-neutral limits.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Edoardo Mainini (Università di Genova (Italy))
DTSTART:20210907T133000Z
DTEND:20210907T135500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/12
DESCRIPTION:Title: Fractional PDEs and steady states for aggregation-diffusion mo
dels\nby Edoardo Mainini (Università di Genova (Italy)) as part of CM
O-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and
Geometry\n\n\nAbstract\nAggregation-diffusion models describe the motion o
f interacting agents towards states of overall balance between diffusion e
ffects and mutual attraction. The Newtonian and the Riesz interaction pote
ntials provide relevant examples of aggregation modeling with long range e
ffects. They give rise to local and nonlocal PDEs for the characterization
of stationary states: we will focus on existence\, uniqueness and regular
ity properties of radial entire solutions to the equilibrium equations. Th
is is a joint work with H. Chan\, M.D.M. González\, Y. Huang and B. Volzo
ne.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Loss (Georgia Institute of Technology)
DTSTART:20210907T150000Z
DTEND:20210907T152500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/13
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Michael Los
s (Georgia Institute of Technology) as part of CMO-New Trends in Nonlinear
Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nI
present some results concerning the size of magnetic fields that support
zero modes for the three dimensional Dirac equation and related problems f
or spinor equations. The critical quantity\, is the $3/2$ norm of the magn
etic field $B$. The point is that the spinor structure enters the analysis
in a crucial way. This is joint work with Rupert Frank at Caltech.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Gualdani (University of Texas Austin)
DTSTART:20210907T153000Z
DTEND:20210907T155500Z
DTSTAMP:20260404T041648Z
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://stable.researchseminars.org/talk/CMO-21w5127/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Schmeiser (University of Vienna)
DTSTART:20210907T160000Z
DTEND:20210907T162500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/15
DESCRIPTION:Title: A kinetic model for myxobacteria with binary reversal and alig
nment interaction and with Brownian forcing\nby Christian Schmeiser (U
niversity of Vienna) as part of CMO-New Trends in Nonlinear Diffusion: a B
ridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nThe competition
between directional dispersal caused\nby Brownian forcing and tendency tow
ards concentration caused\nby alignment is studied. Main results are the s
tability of uniform\nstates for dominating Brownian forcing (proven by hyp
ocoercivity\nmethods) as well as the existence of nontrivial steady states
(shown\nby a bifurcation approach).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Craig (University of California Santa Barbara)
DTSTART:20210907T163000Z
DTEND:20210907T165500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/16
DESCRIPTION:Title: A blob method for nonlinear diffusion and applications to samp
ling and two layer neural networks\nby Katy Craig (University of Calif
ornia Santa Barbara) as part of CMO-New Trends in Nonlinear Diffusion: a B
ridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nGiven a desired
target distribution and an initial guess of that distribution\, composed o
f finitely many samples\, what is the best way to evolve the locations of
the samples so that they accurately represent the desired distribution? A
classical solution to this problem is to allow the samples to evolve accor
ding to Langevin dynamics\, the stochastic particle method corresponding t
o the Fokker-Planck equation. In today’s talk\, I will contrast this cla
ssical approach with a deterministic particle method corresponding to the
porous medium equation. This method corresponds exactly to the mean-field
dynamics of training a two layer neural network for a radial basis functio
n activation 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 equati
on. As a consequence\, we obtain both a novel method for sampling probabil
ity distributions as well as insight into the training dynamics of two lay
er neural networks in the mean field regime.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuhiro Ishige (The University of Tokyo)
DTSTART:20210908T120000Z
DTEND:20210908T122500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/17
DESCRIPTION:Title: Power concavity and Dirichlet heat flow\nby Kazuhiro Ishig
e (The University of Tokyo) as part of CMO-New Trends in Nonlinear Diffusi
on: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nWe show t
hat log-concavity is the weakest power concavity preserved by the Dirichle
t heat flow in $N$-dimensional convex domains\, where $N\\ge 2$. Jointly w
ith what we already know\, i.e. that log-concavity is the strongest power
concavity preserved by the Dirichlet heat flow\, we see that log-concavity
is indeed the only power concavity preserved by the Dirichlet heat flow.\
n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Goro Akagi (Tohoku University)
DTSTART:20210908T123000Z
DTEND:20210908T125500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/18
DESCRIPTION:Title: Rates of convergence to non-degenerate asymptotic profiles for
fast diffusion equations via an energy metho\nby Goro Akagi (Tohoku U
niversity) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge betw
een PDEs\, Analysis and Geometry\n\n\nAbstract\nThis talk is concerned wit
h a quantitative analysis of\nasymptotic behavior of solutions to the Cauc
hy-Dirichlet problem\nfor the fast diffusion equation posed on bounded dom
ains with\nSobolev subcritical exponents. More precisely\, rates of conver
gence\nto non-degenerate asymptotic profiles will be discussed via an ener
gy method.\nThe rate of convergence for positive profiles was recently dis
cussed\nbased on an entropy method by Bonforte and Figalli (2021\, CPAM).\
nAn alternative proof will also be provided to their result.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabio Punzo (Politecnico di Milano (Italy))
DTSTART:20210908T130000Z
DTEND:20210908T132500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/19
DESCRIPTION:Title: Global existence for a class of nonlinear reaction-diffusion e
quations on Riemannian manifolds: an approach via Sobolev and Poincaré in
equalities\nby Fabio Punzo (Politecnico di Milano (Italy)) as part of
CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis an
d Geometry\n\n\nAbstract\nWe discuss existence of global-in-time solutions
to the porous medium equation with a reaction term on Riemannian manifold
s\, where Sobolev and Poincaré inequalities are assumed to hold. Smoothin
g estimates are also established. The results have been recently obtained
jointly with Gabriele Grillo and Giulia Meglioli (Politecnico di Milano).\
n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Quirós (Universidad Autónoma de Madrid (Spain))
DTSTART:20210908T133000Z
DTEND:20210908T135500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/20
DESCRIPTION:Title: Large-time behaviour in nonlocal heat equations with memory\nby Fernando Quirós (Universidad Autónoma de Madrid (Spain)) as part o
f CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis
and Geometry\n\n\nAbstract\nIn this talk we will review several recent res
ults\, in collaboration with Carmen Cortázar (PUC\, Chile) and Noemí Wol
anski (IMAS-UBA-CONICET\, Argentina)\, on the large-time behaviour of solu
tions to fully nonlocal heat equations involving a Caputo time derivative
and a power of the Laplacian. The Caputo time derivative introduces memory
effects that yield new phenomena that are not present in classical diffus
ion equations.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Nazaret (SAMM\, Université Paris 1)
DTSTART:20210908T150000Z
DTEND:20210908T152500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/21
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): A
variational point of view\nby Bruno Nazaret (SAMM\, Université Paris
1) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDE
s\, Analysis and Geometry\n\n\nAbstract\nIn this first lecture of a series
of three\, we discuss stability results in Gagliardo-Nirenberg-Sobolev in
equalities\, from a joint project with M. Bonforte\, J. Dolbeault and N. S
imonov. The core of this approach is the use of a non scaling invariant fo
rm of the inequalities\, equivalent to entropy-entropy production inequali
ties arising in the study of large time asymptotics for solutions to fast
diffusion equations. We only use variational arguments\, leading to non co
nstructive estimates\, but this paves the way for the constructive results
given in the next two lectures.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Simonov (Universitè Paris Duaphine (France))
DTSTART:20210908T153000Z
DTEND:20210908T155500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/22
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): C
onvergence in relative error for the fast diffusion equation\nby Nikit
a Simonov (Universitè Paris Duaphine (France)) as part of CMO-New Trends
in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\
nAbstract\nIn this talk\, I will discuss the asymptotic behavior of soluti
ons to the fast diffusion equation when the tails of the initial datum hav
e a certain decay. In this setting\, I will provide a fully constructive e
stimate of the threshold time after which the solution enters a neighborho
od of the Barenblatt profile in a uniform relative norm. This estimate pla
ys a fundamental role in obtaining a constructive stability result in Gagl
iardo-Nirenberg-Sobolev inequalities. The results are based on a joint wor
k with Matteo Bonforte\, Jean Dolbeault\, and Bruno Nazaret.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Dolbeault (Université Paris-Dauphine)
DTSTART:20210908T160000Z
DTEND:20210908T162500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/23
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg-Sobolev inequalities (GNS): E
ntropy methods and stability\nby Jean Dolbeault (Université Paris-Dau
phine) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between
PDEs\, Analysis and Geometry\n\n\nAbstract\nThis lecture is the third lect
ure on stability issues in Gagliardo-Nirenberg-Sobolev inequalities\, a jo
int project with M. Bonforte\, N. Simonov and B. Nazaret. The results are
based on entropy methods and the use of the fast diffusion equation (FDE)
for studying refined versions of the Gagliardo-Nirenberg-Sobolev inequalit
ies. Using the quantitative regularity estimates\, we go beyond the variat
ional results of the first lecture and provide fully constructive estimate
s\, to the price of a small restriction of the functional space which is i
nherent to the method.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shin-ichi Ohta (Osaka University)
DTSTART:20210909T120000Z
DTEND:20210909T122500Z
DTSTAMP:20260404T041648Z
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 Br
idge between PDEs\, Analysis and Geometry\n\n\nAbstract\nWe review develop
ments in geometric analysis on Finsler manifolds of weighted Ricci curvatu
re bounded below. We especially discuss a nonlinear analogue of the Gamma-
calculus and its applications to isoperimetric and functional inequalities
.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yao Yao (National University of Singapore)
DTSTART:20210909T123000Z
DTEND:20210909T125500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/25
DESCRIPTION:Title: Uniqueness and non-uniqueness of stationary solutions of aggre
gation-diffusion equation\nby Yao Yao (National University of Singapor
e) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs
\, Analysis and Geometry\n\n\nAbstract\nIn this talk\, I will discuss a no
nlocal aggregation equation with degenerate diffusion\, which describes th
e mean-field limit of interacting particles driven by nonlocal interaction
s and localized repulsion. When the interaction potential is attractive\,
it is previously known that all stationary solutions must be radially decr
easing up to a translation\, but uniqueness (for a given mass) within this
class was open\, except for some special interaction potentials. For gene
ral attractive potentials\, we show that the uniqueness/non-uniqueness cri
teria are determined by the power of the degenerate diffusion\, with the c
ritical power being $m=2$. Namely\, for $m \\geq 2$\, we show the stationa
ry solution for any given mass is unique for any attractive potential\, by
tracking the associated energy functional along a novel interpolation cur
ve. And for $1< m < 2 $\, we construct examples of smooth attractive poten
tials\, such that there are infinitely many radially decreasing stationary
solutions of the same mass. This is a joint work with Matias Delgadino an
d Xukai Yan.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria del Mar Gonzalez (Universidad Autonoma de Madrid)
DTSTART:20210909T130000Z
DTEND:20210909T132500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/26
DESCRIPTION:Title: Non-local ODEs in conformal geometry\nby Maria del Mar Gon
zalez (Universidad Autonoma de Madrid) as part of CMO-New Trends in Nonlin
ear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract
\nWhen one looks for radial solutions of an equation with fractional Lapla
cian\, it is not generally possible to use standard ODE methods. If such e
quation has some conformal invariances\, then one may rewrite it in Emden-
Fowler (cylindrical) coordinates and use the properties of the conformal f
ractional Laplacian on the cylinder\, which involves some complex analysis
techniques. After giving the necessary background\, we will briefly consi
der two particular applications of this technique: 1. Symmetry breaking\,
non-degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg
inequality (joint work with W. Ao and A. DelaTorre). 2. Existence and reg
ularity for fractional Laplacian equations with drift and a critical Hardy
potential (joint with H. Chan\, M. Fontelos and J. Wei).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monica Musso (University of Bath)
DTSTART:20210909T133000Z
DTEND:20210909T135500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/27
DESCRIPTION:Title: Infinite-time blowing-up solutions to small perturbations of t
he Yamabe flow\nby Monica Musso (University of Bath) as part of CMO-Ne
w Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and Geom
etry\n\n\nAbstract\nUnder the validity of the positive mass theorem\, the
Yamabe flow on a smooth compact Riemannian manifold of dimension greater o
r 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\,
obtained in collaboration with Seunghyeok Kim\, in which we prove that if
a suitable 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 point
s on M in the infinite time. We construct such a flow by using solutions o
f the Yamabe problem on the unit sphere as blow-up profiles. We also exami
ne the stability of the blow-up phenomena under a negativity condition on
the Ricci curvature at blow-up points.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramon Plaza (Universidad Nacional Autónoma de México)
DTSTART:20210909T150000Z
DTEND:20210909T152500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/28
DESCRIPTION:Title: Spectral stability of monotone traveling fronts for reaction d
iffusion-degenerate Nagumo equations\nby Ramon Plaza (Universidad Naci
onal Autónoma de México) as part of CMO-New Trends in Nonlinear Diffusio
n: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstract\nThis talk
addresses the spectral stability of monotone traveling front solutions for
reaction-diffusion equations where the reaction function is of Nagumo (or
bistable) type and with diffusivities which are density dependent and deg
enerate at zero (one of the equilibrium points of the reaction). Spectral
stability is understood as the property that the spectrum of the linearize
d operator around the wave\, acting on an exponentially weighted space\, i
s contained in the complex half plane with non-positive real part. The deg
enerate fronts under consideration travel with positive speed above a thre
shold value and connect the (diffusion-degenerate) zero state with the uns
table equilibrium point of the reaction function. In this case\, the degen
eracy of the diffusion coefficient is responsible of the loss of hyperboli
city 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 diff
iculty is overcome with a suitable partition of the spectrum\, a generaliz
ed convergence of operators technique\, the analysis of singular (or Weyl)
sequences and the use of energy estimates. The monotonicity of the fronts
\, as well as detailed descriptions of the decay structure of eigenfunctio
ns on a case by case basis\, are key ingredients to show that all travelin
g fronts under consideration are spectrally stable in a suitably chosen ex
ponentially weighted $L^2$ energy space. This is joint work with J. F. Ley
va (Benemérita Universidad Autónoma de Puebla) y L. F. López Ríos (IIM
AS-UNAM).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihaela Ignatova (Temple University)
DTSTART:20210909T153000Z
DTEND:20210909T155500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/29
DESCRIPTION:Title: Nernst-Planck-Navier-Stokes equations\nby Mihaela Ignatova
(Temple University) as part of CMO-New Trends in Nonlinear Diffusion: a B
ridge between 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://stable.researchseminars.org/talk/CMO-21w5127/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michele Coti Zelati (Imperial College London)
DTSTART:20210909T160000Z
DTEND:20210909T162500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/30
DESCRIPTION:Title: Stationary Euler flows near the Kolmogorov flow\nby Michel
e Coti Zelati (Imperial College London) as part of CMO-New Trends in Nonli
near Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstrac
t\nWe exhibit a large family of new\, non-trivial stationary states of\nan
alytic regularity\, that are arbitrarily close to the Kolmogorov flow on t
he\nsquare torus. Our construction of these stationary states builds on a\
ndegeneracy in the global structure of the Kolmogorov flow.\nThis has surp
rising consequences in the context of inviscid\ndamping in 2D Euler and en
hanced dissipation in Navier-Stokes.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José A. Carrillo (University of Oxford)
DTSTART:20210909T163000Z
DTEND:20210909T165500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/31
DESCRIPTION:Title: Nonlocal Aggregation-Diffusion Equations: entropies\, gradient
flows\, phase transitions and applications\nby José A. Carrillo (Uni
versity of Oxford) as part of CMO-New Trends in Nonlinear Diffusion: a Bri
dge between PDEs\, Analysis and Geometry\n\n\nAbstract\nThis talk will be
devoted to an overview of recent results understanding the bifurcation ana
lysis of nonlinear Fokker-Planck equations arising in a myriad of applicat
ions such as consensus formation\, optimization\, granular media\, swarmin
g behavior\, opinion dynamics and financial mathematics to name a few. We
will present several results related to localized Cucker-Smale orientation
dynamics\, McKean-Vlasov equations\, and nonlinear diffusion Keller-Segel
type models in several settings. We will show the existence of continuous
or discontinuous phase transitions on the torus under suitable assumption
s on the Fourier modes of the interaction potential. The analysis is based
on linear stability in the right functional space associated to the regul
arity of the problem at hand. While in the case of linear diffusion\, one
can work in the $L^2$ framework\, nonlinear diffusion needs the stronger $
L^\\infty$ topology to proceed with the analysis based on Crandall-Rabinow
itz bifurcation analysis applied to the variation of the entropy functiona
l. Explicit examples show that the global bifurcation branches can be very
complicated. Stability of the solutions will be discussed based on numeri
cal simulations with fully explicit energy decaying finite volume schemes
specifically tailored to the gradient flow structure of these problems. Th
e theoretical analysis of the asymptotic stability of the different branch
es of solutions is a challenging open problem. This overview talk is based
on several works in collaboration with R. Bailo\, A. Barbaro\, J. A. Cani
zo\, X. Chen\, P. Degond\, R. Gvalani\, J. Hu\, G. Pavliotis\, A. Schlicht
ing\, Q. Wang\, Z. Wang\, and L. Zhang. This research has been funded by E
PSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriele Grillo (Politecnico di Milano)
DTSTART:20210910T120000Z
DTEND:20210910T122500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/32
DESCRIPTION:Title: Nonlinear characterizations of stochastic completeness\nby
Gabriele Grillo (Politecnico di Milano) as part of CMO-New Trends in Nonl
inear Diffusion: a Bridge between PDEs\, Analysis and Geometry\n\n\nAbstra
ct\nA manifold is said to be stochastically complete if the free heat semi
group preserves probability. It is well known that this property is equiva
lent to nonexistence of nonnegative\, bounded solutions to certain (linear
) elliptic problems\, and to uniqueness of solutions to the heat equation
corresponding to bounded initial data. We prove that stochastic completene
ss is also equivalent to similar properties for certain nonlinear elliptic
and parabolic problems. This fact\, and the known analytic-geometric char
acterizations of stochastic completeness\, allow to give new explicit crit
eria for existence/nonexistence of solutions to certain nonlinear elliptic
equations on manifolds\, and for uniqueness/nonuniqueness of solutions to
certain nonlinear diffusions on manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asuka Takatsu (Tokyo Metropolitan University (Japan))
DTSTART:20210910T123000Z
DTEND:20210910T125500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/33
DESCRIPTION:Title: Spectral convergence of high-dimensional spheres to Gaussian s
paces\nby Asuka Takatsu (Tokyo Metropolitan University (Japan)) as par
t of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analys
is and Geometry\n\n\nAbstract\nIt is known that the projection of a unifor
m probability measure on the $N$-dimensional sphere to the first $n$ coord
inates approximates the $n$-dimensional Gaussian measure.\nIn this talk\,
I will present that the spectral structure on the $N$-dimensional sphere c
ompatible with the projection to the first $n$ coordinates approximates th
e spectral structure on the $n$-dimensional Gaussian space.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diana Stan (Universidad de Cantabria (Spain))
DTSTART:20210910T130000Z
DTEND:20210910T132500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/34
DESCRIPTION:Title: The fast p-Laplacian evolution equation. Global Harnack princi
ple and fine asymptotic behavior\nby Diana Stan (Universidad de Cantab
ria (Spain)) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge be
tween PDEs\, Analysis and Geometry\n\n\nAbstract\nWe study fine global pro
perties of nonnegative solutions to the Cauchy Problem for the fast $p$-La
placian 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 has an explicit expression. We prove the so-called Global Harnack Pr
inciple (GHP)\, that is\, precise global pointwise upper and lower estimat
es of nonnegative solutions in terms of B. This can be considered the nonl
inear counterpart of the celebrated Gaussian estimates for the linear heat
equation. To the best of our knowledge\, analogous issues for the linear
heat equation\, do not possess such clear answers\, only partial results a
re known. Also\, we characterize the maximal (hence optimal) class of init
ial data such that the GHP holds\, by means of an integral tail condition\
, easy to check. Finally\, we derive sharp global quantitative upper bound
s of the modulus of the gradient of the solution\, and\, when data are rad
ially decreasing\, we show uniform convergence in relative error for the g
radients. This is joint work with Matteo Bonforte (UAM-ICMAT\, Madrid\, Sp
ain) and Nikita Simonov (Ceremade-Univ. Paris-Dauphine\, Paris\, France).\
n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincenzo Ferone (Università di Napoli Federico II)
DTSTART:20210910T133000Z
DTEND:20210910T135500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/35
DESCRIPTION:Title: Symmetrization for fractional elliptic problems: a direct appr
oach\nby Vincenzo Ferone (Università di Napoli Federico II) as part o
f CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis
and Geometry\n\n\nAbstract\nWe provide new direct methods to establish sym
metrization results in the form of mass concentration (i.e. integral) comp
arison for fractional elliptic equations of the type $ (-\\Delta)^s u =f \
\ $ ($ 0 < s < 1 $) in a bounded domain $ \\Omega $\, equipped with homo
geneous Dirichlet boundary conditions. The classical pointwise Talenti rea
rrangement inequality is recovered in the limit $ s\\rightarrow1 $. Finall
y\, explicit counterexamples constructed for all $ s\\in(0\,1) $ highlight
that the same pointwise estimate cannot hold in a nonlocal setting\, thus
showing the optimality of our results. The results are contained in a joi
nt paper with Bruno Volzone [Ferone\, V.\; Volzone\, B.\, Symmetrization f
or fractional elliptic problems: a direct approach. Arch. Ration. Mech. An
al. 239 (2021)\, 1733-1770].\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexis Vasseur (University of Texas at Austin)
DTSTART:20210910T150000Z
DTEND:20210910T152500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/36
DESCRIPTION:Title: Uniform stability of viscous shocks for the compressible Navie
r-Stokes equation\nby Alexis Vasseur (University of Texas at Austin) a
s part of CMO-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, A
nalysis and Geometry\n\n\nAbstract\nWe show the stability of viscous shock
s of the 1D compressible Navier-Stokes equation. This stability holds unif
ormly with respect to the viscosity\, up to the inviscid limit. Stability
results for shocks of the Euler equation are then inherited at the invisci
d limit. These stability results hold in the class of wild perturbations o
f inviscid limits\, without any regularity restriction. This shows that th
e class of inviscid limits of Navier-Stokes equations is better behaved th
an 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 join
t work with Moon-Jin Kang.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Azahara DelaTorre (University of Granada)
DTSTART:20210910T153000Z
DTEND:20210910T155500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/37
DESCRIPTION:Title: : The fractional Lane–Emden equation with Serrin’s critica
l exponent\nby Azahara DelaTorre (University of Granada) as part of CM
O-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and
Geometry\n\n\nAbstract\nIn this talk we will focus on the the existence\,
multiplicity and local behavior of singular solutions of the fractional La
ne–Emden equation with Serrin’s critical exponent and homogeneous Diri
chlet exterior condition. These will provide the profile to construct sing
ular metrics with constant (non-local) curvature. We will show radial symm
etry close to the origin\, a Liouville-type result without any assumption
on its asymptotic behavior (showing the necessity of imposing the Dirichle
t condition) and the existence of multiple solutions in a bounded domain w
ith any prescribed closed singular set. Moreover\, we will show that the s
ingular behavior of the profile is unique\, presenting new methods based o
n the connection between the non-local equation and its associated first o
rder ODE in one dimension. \nThis is a joint work with H. Chan.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix del Teso (Universidad Complutense de Madrid (Spain))
DTSTART:20210910T160000Z
DTEND:20210910T162500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/38
DESCRIPTION:Title: The Liouville Theorem and linear operators satisfying the maxi
mum principle\nby Felix del Teso (Universidad Complutense de Madrid (S
pain)) as part of CMO-New Trends in Nonlinear Diffusion: a Bridge between
PDEs\, Analysis and Geometry\n\n\nAbstract\nThe classical Liouville Theore
m states that bounded harmonic functions\nare constant. The talk will revi
sit this result for the most general class of\nlinear operators with const
ant coefficients satisfying the maximum principle\n(characterized by Courr
ège in [P. Courrège\, Générateur infinitésimal d’un semi-groupe de
convolution sur $R^n$ \, et formule de Lévy-Khinchine. Bull. Sci. Math. (
2)\, 88:3–30\, 1964]). The class includes local and nonlocal and\nnot ne
cessarily symmetric operators among which you can find the fractional\nLap
lacian\, Relativistic Schrödinger operators\, convolution operators\, CGM
Y\,\nas well as discretizations of them.\nWe give a full characterization
of the operators in this class satisfying the\nLiouville property. When th
e Liouville property does not hold\, we also establish\nprecise periodicit
y sets of the solutions.\nThe techniques and proofs of [N. Alibaud\, F. de
l Teso\, J. Endal\, and E. R. Jakobsen\, The Liouville\ntheorem and linear
operators satisfying the maximum principle. Journal de\nMathématiques Pu
res et Appliquées\, 142:229–242\, 2020] combine arguments from PDEs\, g
roup the-\nory\, number theory and numerical analysis (and still\, they ar
e simple\, short\,\nand very intuitive).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Luis Vazquez (Universidad Autonoma de Madrid (Spain))
DTSTART:20210910T163000Z
DTEND:20210910T165500Z
DTSTAMP:20260404T041648Z
UID:CMO-21w5127/39
DESCRIPTION:Title: Nonlinear fractional Laplacian operators and equations\nby
Juan Luis Vazquez (Universidad Autonoma de Madrid (Spain)) as part of CMO
-New Trends in Nonlinear Diffusion: a Bridge between PDEs\, Analysis and G
eometry\n\n\nAbstract\nWe consider a number of equations involving nonline
ar fractional \nLaplacian operators where progress has been obtained in re
cent years. \nExamples include fractional $p$-Laplacian operators appearin
g in elliptic \nand parabolic equations and a number of variants. Numerica
l analysis is \nperformed.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5127/39/
END:VEVENT
END:VCALENDAR