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Nicola Visciglia

Publications and source records attributed to Nicola Visciglia.

At least 37 records · Page 2Linked to original sources
arXiv2014

Modified scattering for the cubic Schrödinger equation on product spaces and applications

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq d\leq 4)$. The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\mathbb{R}\times \mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobolev norms $H^s$.

math.AP↗
arXiv2014

Ground states for semi-relativistic Schrödinger-Poisson-Slater energies

We prove the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy $$I^{α,β}(ρ)=\inf_{\substack{u\in H^\frac 12(\R^3) \int_{\R^3}|u|^2 dx=ρ}} \frac{1}{2}\|u\|^2_{H^\frac 12(\R^3)} +α\int\int_{\R^{3}\times\R^{3}} \frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-β\int_{\R^{3}}|u|^{\frac{8}{3}}dx$$ $α,β>0$ and $ρ>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $α, β>0$ such that $I^{α, β}(ρ)>-\infty$ for every $ρ>0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$\frac{1}{S}\frac{\|φ\|_{L^\frac 83(\R^3)}}{\|φ\|_{\dot H^\frac 12(\R^3)}^\frac 12}\leq \left (\int\int_{\R^3\times \R^3} \frac{|φ(x)|^2|φ(y)|^2}{|x-y|}dxdy\right)^\frac 18 $$ for all $φ\in C^\infty_0(\R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $\|u\|^2_{H^\frac 12(\R^3)}$ by the homogeneous one $\|u\|_{\dot H^\frac 12(\R^3)}$.

math-ph↗
arXiv2014

An Improvement on the Brézis-Gallouët technique for 2D NLS and 1D half-wave equation

We revise the classical approach by Brézis-Gallouët to prove global well posedness for nonlinear evolution equations. In particular we prove global well--posedness for the quartic NLS posed on general domains $Ω$ in $\R^2$ with initial data in $H^2(Ω)\cap H^1_0(Ω)$, and for the quartic nonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$.

math.AP↗
arXiv2013

Relativistic Hardy inequalities in magnetic fields

We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best constant as in the free case. This leaves naturally open an interesting question whether there exist magnetic fields for which a Hardy inequality with a better constant than the usual one, in connection with the well known diamagnetic phenomenon arising in non-relativistic models.

math-ph↗
arXiv2013

Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb's Translation Lemma and a Riesz energy version of the Brézis--Lieb lemma.

math.FA↗
arXiv2013

Invariant measures and long time behaviour for the Benjamin-Ono equation II

As a continuation of our previous work on the subject, we prove new measure invariance results for the Benjamin-Ono equation. The measures are associated with conservation laws whose leading term is a fractional Sobolev norm of order larger or equal than 5/2. The new ingredient, compared with the case of conservation laws whose leading term is an integer Sobolev norm of order larger or equal than 3 (that has been studied in our previous work), is the use of suitable orthogonality relations satisfied by multilinear products of centered complex independent gaussian variables. We also give some partial results for the measures associated with the two remaining conservation laws at lower regularity. We plan to complete the proof of their invariance in a separated article which will be the final in the series. Finally in an appendix, we give a brief comparing of the recurrence properties of the flows of Benjamin-Ono and KdV equations.

math.AP↗
arXiv2013

Invariant measures and long time behaviour for the Benjamin-Ono equation

We study the Benjamin-Ono equation, posed on the torus. We prove that an infinite sequence of weighted gaussian measures, constructed in our previous work, are invariant by the flow of the equation. These measures are supported by Sobolev spaces of increasing regularities. As a by product we deduce informations on the long time behaviour of regular solutions. To our knowledge this is the first result which gives an evidence about recurrence properties of the Benjamin-Ono equation flow.

math.AP↗
arXiv2012

Decay and scattering of small solutions of pure power NLS in $\R$ with $p>3$ and with a potential

We prove decay and scattering of solutions of the Nonlinear Schröding-er equation (NLS) in ${\mathbf R}$ with pure power nonlinearity with exponent $3<p<5$ when the initial datum is small in $Σ$ (bounded energy and variance), in the presence of a linear inhomogeneity represented by a linear potential which is a real valued Schwarz function. We assume absence of discrete modes. The proof is analogous to the one for the translation invariant equation. In particular we find appropriate operators commuting with the linearization.

math.AP↗
arXiv2012

The NLS ground states on product spaces

We study the nature of the Nonlinear Schrödinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.

math.AP↗
arXiv2012

Constrained energy minimization and ground states for NLS with point defects

We investigate the ground states of the one-dimensional nonlinear Schrödinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability. This problem had been already considered for the delta and for the delta prime defect with a different constrained minimization problem, i.e. defining ground states as the minimizers of the action on the Nehari manifold. The case of dipole defect is entirely new.

math.AP↗
arXiv2011

The lack of compactness in the Sobolev-Strichartz inequalities

We provide a general method to decompose any bounded sequence in $\dot H^s$ into linear dispersive profiles generated by an abstract propagator, with a rest which is small in the associated Strichartz norms. The argument is quite different from the one proposed by Bahouri-Gérard and Keraani in the cases of the wave and Schrödinger equations, and is adaptable to a large class of propagators, including those which are matrix-valued.

math.AP↗