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Piero D'Ancona

Publications and source records attributed to Piero D'Ancona.

At least 37 records · Page 2Linked to original sources
arXiv2017

On large potential perturbations of the Schrödinger, wave and Klein--Gordon equations

We prove a sharp resolvent estimate in scale invariant norms of Amgon--Hörmander type for a magnetic Schrödinger operator on $\mathbb{R}^{n}$, $n\ge3$\begin{equation*} L=-(\partial+iA)^{2}+V \end{equation*}with large potentials $A,V$ of almost critical decay and regularity. The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein--Gordon flows associated to $L$.

math.AP↗
arXiv2017

On the cubic Dirac equation with potential and the Lochak--Majorana condition

We study a cubic Dirac equation on $\mathbb{R}\times\mathbb{R}^{3}$ \begin{equation*} i \partial _t u + \mathcal{D} u + V(x) u = \langle βu,u \rangle βu \end{equation*} perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in $H^{1}$ with additional angular regularity. The main tool is an endpoint Strichartz estimate for the perturbed Dirac flow. In particular, the result covers the case of spherically symmetric data with small $H^{1}$ norm. When the potential $V$ has a suitable structure, we prove global existence and scattering for \emph{large} initial data having a small chiral component, related to the Lochak--Majorana condition.

math.AP↗
arXiv2017

Stability properties of the regular set for the Navier--Stokes equation

We investigate the size of the regular set for small perturbations of some classes of strong large solutions to the Navier--Stokes equation. We consider perturbations of the data which are small in suitable weighted $L^{2}$ spaces but can be arbitrarily large in any translation invariant critical Banach space. We give similar results in the small data setting.

math.AP↗
arXiv2016

Weighted estimates for powers and Smoothing estimates of Schrödinger operators with inverse-square potentials

Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.

math.AP↗
arXiv2015

Global existence of small equivariant wave maps on rotationally symmetric manifolds

We introduce a class of rotationally invariant manifolds, which we call \emph{admissible}, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity $H^{\frac n2}$. The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces $\mathbb{H}^{n}$ for $n\ge3$.

math.AP↗
arXiv2015

Scattering in the energy space for the NLS with variable coefficients

We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\simeq|u|^{γ-1}u$. We assume that $L$ is a small, long range perturbation of $Δ$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow $e^{itL}$, we prove global well posedness in the energy space for subcritical powers $γ<1+\frac{4}{n-2}$, and scattering provided $γ>1+\frac4n$. When the domain is $\mathbb{R}^{n}$, by extending the Strichartz estimates due to Tataru [Tataru08], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.

math.AP↗
arXiv2015

On the regularity set and angular integrability for the Navier--Stokes equation

We investigate the size of the regular set for suitable weak solutions of the Navier--Stokes equation, in the sense of Caffarelli--Kohn--Nirenberg. We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space $\{t>0\}$ in an appropriate limit. In particular, we obtain that if the $L^{2}$ norm with weight $|x|^{-\frac12}$ of the data tends to 0, the regular set invades $\{t>0\}$.

math.AP↗
arXiv2014

Log-Sobolev inequalities for semi-direct product operators and applications

We prove logarithmic Sobolev inequalities for semi-direct product operators (see definition in Section 1). We apply our main results to examples of operators and provide some applications to ultracontractive bounds of semigroups. Hardy's inequalities are proved and use to study some operators of Grushin's type.

math.AP↗
arXiv2014

Sharp $L^p$ estimates for Schrödinger groups

Consider a non-negative self-adjoint operator $H$ in $L^2(\mathbb{R}^d)$. We suppose that its heat operator $e^{-tH}$ satisfies an off-diagonal algebraic decay estimate, for some exponents $p_0\in[0,2)$. Then we prove sharp $L^p\to L^p$ frequency truncated estimates for the Schrödinger group $e^{itH}$ for $p\in[p_0,p'_0]$. In particular, our results apply to every operator of the form $H=(i\nabla+A)^2+V$, with a magnetic potential $A\in L^2_{loc}(\mathbb{R}^d,\mathbb{R}^d)$ and an electric potential $V$ whose positive and negative parts are in the local Kato class and in the Kato class, respectively.

math.FA↗
arXiv2014

Kato smoothing and Strichartz estimates for wave equations with magnetic potentials

Let $H$ be a selfadjoint operator and $A$ a closed operator on a Hilbert space $\mathcal{H}$. If $A$ is $H$-(super)smooth in the sense of Kato-Yajima, we prove that $AH^{-\frac14}$ is $\sqrt{H}$-(super)smooth. This allows to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations. We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag \cite{ErdoganGoldbergSchlag09-a}, we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on $\mathbb{R}^{n}$, $n\ge3$.

math.AP↗
arXiv2014

Helmholtz and dispersive equations with variable coefficients on exterior domains

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equation $$ -Lu+zu=f, \qquad -Lu:=\nabla^{b}(a(x)\nabla^{b}u)-c(x)u, \qquad \nabla^{b}:=\nabla+ib(x) $$ with fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping. As an application, we prove smoothing estimates for the Schrödinger flow $e^{itL}$ and the wave flow $e^{it \sqrt{L}}$ with variable coefficients on exterior domains and Dirichlet boundary conditions.

math.AP↗
arXiv2013

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) ε_0, the same happens for the solution u(t,.) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows. We also get similar results for the Schroedinger equation with a real-analytic electromagnetic potential.

math.AP↗
arXiv2011

Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with angular integrability

We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of $L^{p}$ spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli-Kohn-Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions.

math.AP↗
arXiv2011

Weighted L^p estimates for powers of selfadjoint operators

We prove $L^{p}$ and weighted $L^{p}$ estimates for bounded functions of a selfadjoint operator satisfying both a pointwise gaussian estimate for its heat kernel and a finite speed of propagation property. As an application, we obtain weighted estimates for fractiional powers of an electromagnetic Schrödinger operator with singular coefficienta. The proofs are based on a modification of techniques due to Hebisch and Auscher and Martell. EDIT: second version, November 2010, a few inaccuracies have been corrected.

math.AP↗
arXiv2011

Endpoint estimates and global existence for the nonlinear Dirac equation with potential

We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial $H^{1}$ data with additional angular regularity. This implies in particular global existence in the critical energy space $H^{1}$ for small radial data.

math.AP↗
arXiv2010

Evolution equations on non flat waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-Δ_{x}-Δ_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $Ω$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $Ω$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-λu=f$. As consequences we prove smoothing estimates for the Schrödinger and wave equations associated to $H$, and Strichartz estimates for the Schrödinger equation. Additionally, we deduce that the operator $H$ does not admit eigenvalues.

math.AP↗
arXiv2010

Global well-posedness of the Maxwell-Dirac system in two space dimensions

In recent work, Grünrock and Pecher proved that the Dirac-Klein-Gordon system in 2d is globally well-posed in the charge class (data in $L^2$ for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell-Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part.

math.AP↗