Page 468 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 468
ANALYSIS OF PDES ON NETWORKS (MS-26)
Discontinuous ground states for the NLSE on R with a Fülöp-Tsutsui δ
interaction
Alice Ruighi, alice.ruighi@polito.it
Politecnico di Torino, Italy
We analyse the existence and the stability of the ground states of the one-dimensional nonlinear
Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. A
ground state is intended as a global minimizer of the action functional on the Nehari’s manifold
and the defect considered is a Fülöp-Tsutsui δ type, namely a δ condition that allows disconti-
nuities. The existence of ground states is proved by variational techniques, while the stability
results follow from the Grillakis-Shatah-Strauss’ theory.
This is a joint work with Riccardo Adami.
Local minimizers in absence of ground states for the critical NLS energy
on metric graphs
Nicola Soave, nicola.soave@gmail.com
Politecnico di Milano, Italy
Coauthors: Dario Pierotti, Gianmaria Verzini
We consider the mass-critical nonlinear Schrödinger equation on non-compact metric graphs.
A quite complete description of the structure of the ground states, which correspond to global
minimizers of the energy functional under a mass constraint, has been recently provided by R.
Adami, E. Serra and P. Tilli (Comm. Math. Phys. 352, no.1, 387-406, 2017). They proved
that existence and properties of ground states depend in a crucial way on both the value of the
mass, and the topological properties of the underlying graph. In this talk I present some results
regarding cases when ground states do not exist and show that, under suitable assumptions,
constrained local minimizers of the energy do exist. This result paves the way to the existence
of stable solutions in the time-dependent equation in cases where the ground state energy level
is not achieved.
Initial-boundary value problems for transport equations in one space
dimension with very rough coefficients
Laura V. Spinolo, spinolo@imati.cnr.it
IMATI-CNR, Italy
Coauthors: Elio Marconi, Simone Dovetta
I will discuss new existence, uniqueness and regularity propagation results for solutions of
transport equations defined in one-dimensional domains with boundaries. The only assump-
tions imposed on the coefficient are boundedness and near incompressibility, which means that
the coefficient supports a nonnegative and bounded density. This analyis is motivated by appli-
cations to a source-destination model for traffic flows on road networks. The talk will be based
on joint work with Simone Dovetta and Elio Marconi.
466
Discontinuous ground states for the NLSE on R with a Fülöp-Tsutsui δ
interaction
Alice Ruighi, alice.ruighi@polito.it
Politecnico di Torino, Italy
We analyse the existence and the stability of the ground states of the one-dimensional nonlinear
Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. A
ground state is intended as a global minimizer of the action functional on the Nehari’s manifold
and the defect considered is a Fülöp-Tsutsui δ type, namely a δ condition that allows disconti-
nuities. The existence of ground states is proved by variational techniques, while the stability
results follow from the Grillakis-Shatah-Strauss’ theory.
This is a joint work with Riccardo Adami.
Local minimizers in absence of ground states for the critical NLS energy
on metric graphs
Nicola Soave, nicola.soave@gmail.com
Politecnico di Milano, Italy
Coauthors: Dario Pierotti, Gianmaria Verzini
We consider the mass-critical nonlinear Schrödinger equation on non-compact metric graphs.
A quite complete description of the structure of the ground states, which correspond to global
minimizers of the energy functional under a mass constraint, has been recently provided by R.
Adami, E. Serra and P. Tilli (Comm. Math. Phys. 352, no.1, 387-406, 2017). They proved
that existence and properties of ground states depend in a crucial way on both the value of the
mass, and the topological properties of the underlying graph. In this talk I present some results
regarding cases when ground states do not exist and show that, under suitable assumptions,
constrained local minimizers of the energy do exist. This result paves the way to the existence
of stable solutions in the time-dependent equation in cases where the ground state energy level
is not achieved.
Initial-boundary value problems for transport equations in one space
dimension with very rough coefficients
Laura V. Spinolo, spinolo@imati.cnr.it
IMATI-CNR, Italy
Coauthors: Elio Marconi, Simone Dovetta
I will discuss new existence, uniqueness and regularity propagation results for solutions of
transport equations defined in one-dimensional domains with boundaries. The only assump-
tions imposed on the coefficient are boundedness and near incompressibility, which means that
the coefficient supports a nonnegative and bounded density. This analyis is motivated by appli-
cations to a source-destination model for traffic flows on road networks. The talk will be based
on joint work with Simone Dovetta and Elio Marconi.
466
