Jorge Drumond Silva

Instituto Superior Técnico · Centro de Análise Matemática Geometria e Sistemas Dinâmicos (CAMGSD) - Departamento de Matemática

We consider the large time behavior of solutions to defocusing nonlinear Schrödinger 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...

We consider the nonlinear Schrödinger 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...

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 $\Lambda$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner Nordstr\"{o}m black hole event horizon, stu...

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 initi...

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 g...

We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a g...

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 Muckenhou...

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 solut...

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 g...

We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a g...

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 solut...

We consider the quadratic Schr\"odinger system $$iu_t+\Delta_{\gamma_1}u+\overline{u}v=0$$ $$2iv_t+\Delta_{\gamma_2}v-\beta v+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $\gamma_1,\gamma_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blo...

By solving a singular initial value problem, we prove the existence of solutions of the wave equation □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 ∈ H ³ (ς) (where ς = ℝn, n or ℍ ⁿ models the spatial hypersurfaces) there exists a unique so...

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n, \mathbb{S}^n$ or $\mathbb{H}^n$ models...

We consider the Schrödinger-Debye system in R^n, for n=3,4. Developing on previously known local well-posedness results, we start by establishing global well-posedness in H^1(R^3)xL^2(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 no...

Resumo: Pretende-se, com este texto, fazer uma introdução resumida ao prob-lema de valor inicial para as equações de Einstein, a um nível bastante elementar, sem recorrer a demonstrações ou definições matematicamente técnicas, assumindo apenas conhecimentos básicos de equações diferenciais parciais e geometria Rie-manniana. Devido ao carácter hiper...

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 $\Lambda$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstr\"{o}m black hole event horizon, st...

Motivated by the strong cosmic censorship conjecture, we consider the Einstein- Maxwell-scalar field system with a cosmological constant Λ (of any sign), under spherical symmetry, for characteristic initial conditions, with outgoing data prescribed by a (complete) subextremalReissner-Nordstrom black hole event horizon. We study the structure of the...

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–Nordström black hole event horizon, study the futur...

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 × H^l, with s and l satisfying max{0, s − 1} ≤ l ≤ min{2s, s + 1}. In particular, these include the energy space H^1×L^2. Our results improve the previous ones obtained by B. Bidégaray, and by A. J...

In this work we study the linear instability of periodic traveling waves associated with a generalization of the Regularized Boussinesq equation. By using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of per...

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 $...

We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order operators, with L_1 non-degenerate and L_2 elliptic.

In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for...

We use the Wigner transform to study properties of solutions to the Schrödinger equation. In particular, we present an elementary proof of the time decay inequalities for the free particle and harmonic oscillator. Furthermore, we estimate certain localized integral quantities in phase space, involving the Winner transform of solutions to the Schröd...

The I-method in its first version as developed by Colliander et al. is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H^s, provided s>-1/42.

We consider a system of Korteweg–de Vries (KdV) equations coupled through nonlinear terms, called the Hirota–Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restr...

We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov’s theorem remains true if, instead of just considering the principal symbols in S^m/S^m−1 for the pseudodifferential operators, on...

In this dissertation we show how the celebrated theorem of Egorov, on conjugation of pseudodifferential operators by Fourier integral operators, can be sharpened if ideas from the Weyl calculus are brought in. We build up the motivation for the result, focusing on the symplectic invariance properties and self-adjointness of the Weyl operator with r...