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Jorge Drumond Silva

Publications and source records attributed to Jorge Drumond Silva.

At least 19 recordsLinked to original sources
arXiv2025

Elastic rigid rod in an expanding universe

We study the motion of a rigid elastic rod, initially set in its relaxed state along a spacelike geodesic, in an expanding Friedmann-Lemaître-Robertson-Walker universe. This leads to an initial boundary value problem (IBVP) for a nonlinear wave equation whose nonlinearity depends on a parameter $κ\geq 0$, related to the ratio between the rod's length and the cosmological scale. We show that if $κ$ is small enough then the solution to the IBVP is global in time and bounded, meaning that the rod's length oscillates around its initial value. For greater values of $κ$, however, the solution to the IBVP blows up in finite coordinate time, indicating that the rod is infinitely stretched by the cosmological expansion. This supports the widely held belief that sufficiently small bound systems do not follow the Hubble flow, whereas larger systems may do so. Similar conclusions apply to the tethered galaxy version of this problem, where the rod is used to connect two point masses (which results in nonlinear boundary conditions for the IBVP).

gr-qc↗
arXiv2025

Sharp local well-posedness for the Schrödinger-Korteweg-de Vries system

We prove a sharp local existence result for the Schrödinger-Korteweg-de Vries system with initial data in $H^k(\mathbb{R})\times H^s(\mathbb{R})$. The proof is based on the concept of \textit{integrated-by-parts strong solution}, which generalizes the classical notion of strong solution, and on frequency-restricted estimates. Moreover, we extend the known global well-posedness result to regularities $k,s>1/2$.

math.AP↗
arXiv2024

Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole

We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary PDE problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic to mechanical energy.

gr-qc↗
arXiv2023

Weighted Gagliardo-Nirenberg Interpolation Inequalities

In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupttype conditions. We also obtain improved results for some particular families of weights, including power-law weights $|x|^α$. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights $\langle x \rangle^α=(1+|x|^2)^{α/2}$.

math.CA↗
arXiv2023

Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov-Kuznetsov equation, the cubic and quintic nonlinear Schrödinger equation and the quartic Korteweg-de Vries equation.

math.AP↗
arXiv2019

Dispersive Blow-up for Solutions of the Zakharov-Kuznetsov equation

The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up inherited from the linear component part of the equation. Similar results are obtained for the generalized Zakharov-Kuznetsov equation.

math.AP↗
arXiv2019

Nonlinear smoothing for dispersive PDE: a unified approach

In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called nonlinear smoothing. We employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing for dispersive PDEs, contingent only on establishing two particular bounds. We then apply this theorem to show that the nonlinear smoothing property holds, depending on the regularity of the initial data, for five classical dispersive equations: in $\mathbb{R}$, the cubic nonlinear Schrödinger, the Korteweg-de Vries, the modified Korteweg-de Vries and the derivative Schrödinger equations; in $\mathbb{R}^2$, the modified Zakharov-Kuznetsov equation. For the aforementioned one-dimensional equations, this unifying methodology matches or improves the existing nonlinear smoothing results, while enlarging the ranges of Sobolev regularities where the property holds.

math.AP↗
arXiv2019

Solutions of the wave equation bounded at the Big Bang

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_gϕ=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(Σ)$ (where $Σ=\mathbb{R}^n, \mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $ϕ$ of the wave equation converging to $A$ in $H^1(Σ)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(Σ)$.

gr-qc↗
arXiv2018

On the occurrence of mass inflation for the Einstein-Maxwell-scalar field system with a cosmological constant and an exponential Price law

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-Nördstrom black hole at infinity. We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in $L^2_{\rm loc}$, thus violating the Christodoulou-Chruściel version of strong cosmic censorship.

gr-qc↗
arXiv2017

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 3: Mass inflation and extendibility of the solutions

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. In the first part of this series we established the well posedness of the characteristic problem, whereas in the second part we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when $Λ>0$.

gr-qc↗
arXiv2016

On the Unboundedness of Higher Regularity Sobolev Norms of Solutions for the Critical Schrödinger-Debye System with Vanishing Relaxation Delay

We consider the Schrödinger-Debye system in $\mathbb{R}^n$, for $n=3,4$. Developing on previously known local well-posedness results, we start by establishing global well-posedness in $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ for a broad class of initial data. We then concentrate on the initial value problem in $n=4$, which is the energy-critical dimension for the corresponding cubic nonlinear Schrödinger equation. We start by proving local well-posedness in $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$. Then, for the focusing case of the system, we derive a virial type identity and use it to prove that for radially symmetric smooth initial data with negative energy, there is a positive time $T_0$, depending only on the data, for which, either the $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ solutions blow-up in $[0,T_0]$, or the higher regularity Sobolev norms are unbounded on the intervals $[0, T]$, for $T>T_0$, as the delay parameter vanishes. We finish by presenting a global well-posedness result for regular initial data which is small in the $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ norm.

math.AP↗
arXiv2015

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 2: Structure of the solutions and stability of the Cauchy horizon

This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. In the first paper of this sequence, we established well posedness of the characteristic problem with general initial data. In this second paper, we generalize the results of Dafermos on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed by Dafermos, focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin. In the third and final paper, we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.

gr-qc↗
arXiv2014

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 1: Well posedness and breakdown criterion

This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $Λ$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a "suitably regular" Lorentzian manifold. In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals. In the second part of this series we study the stability of the radius function at the Cauchy horizon. In the third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

gr-qc↗
arXiv2014

Scattering for nonlinear Schrodinger equation under partial harmonic confinement

We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

math.AP↗
arXiv2013

Large time behavior in nonlinear Schrodinger equation with time dependent potential

We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev norms are bounded, and momenta grow at most polynomially in time, because the potential becomes negligible for large time: there is scattering, even though the potential is unbounded in space for fixed time.

math.AP↗
arXiv2012

Local and Global Well-Posedness for the Critical Schrodinger-Debye System

We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions $N=2, 3$ for data in $H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell \le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\times L^2$. Our results improve the previous ones obtained in \cite{Bidegaray1}, \cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case (N=2) and for initial data in $H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned in \cite{Fibich-Papanicolau} concerning the formation of singularities for these solutions.

math.AP↗
arXiv2012

On the Cauchy problem for a Boussinesq type system

We consider the initial value problem (IVP) associated to a Boussinesq type system. After rewriting the system in an equivalent form of coupled KdV-type equations, we prove that this is locally well-posed in $(H^s(\R^2))^4$, $s>3/2$, using sharp smoothing estimates. Consequently we obtain the local well-posedness result for the original system in $H^s\times \mathcal{V}^{s+1}$ for $s>3/2$ (see below for the definition of $\mathcal{V}^{s}$).

math.AP↗