Luis Vega

University of the Basque Country | UPV/EHU · Departamento de Matemáticas

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann’s non-differentiable functions Rx0(t)=∑n≠0e2πi(n2t+nx0)n2,x0∈[0,1].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{ams...

We recall some recent results concerning the Initial Value Problem of 1d-cubic non-linear Schrödinger equation (NLS) and other related systems as the Schrödinger Map. For the latter we prove the existence of a cascade of energy. Finally, some new examples of the Talbot effect at the critical level of regularity are given.

We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$ for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite tim...

We study spectral properties of Dirac operators on bounded domains \(\Omega \subset {\mathbb {R}}^3\) with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter \(\tau \in \mathbb {R}\); the case \(\tau = 0\) corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing function...

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence, this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution...

We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically, it is a flow of curves in three dimensions, explicitly connected to the one-dimensional Schrödinger map with values on the two-dimensional sphere, and to the one-dimensional cubic Schrödinger equation. Although these equa...

We study spectral properties of Dirac operators on bounded domains $\Omega \subset \mathbb{R}^3$ with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter $\tau\in\mathbb{R}$; the case $\tau = 0$ corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of $\tau$,...

We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schr\"odinger map with values on the 2-D sphere, and to the 1-D cubic Schr\"odinger equation. Although these equations are completely integrable w...

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution o...

The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schrödinger equation on \(\mathbb {R}\) in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally we prove the existence of a unique solution of the binormal flow with datum a polygonal line. This equation is...

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensiona...

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schrödinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time....

We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocusing case, the self-similar profiles are solutions to the Painlevé II equation; although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. These Four...

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,\,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$ such that for some non-empty open set $\Omega\subset \mathbb R^n\times[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) \in \O...

We prove a local well-posedness result for the modified Korteweg–de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an asymptotic description of small solutions as $t \to +\infty$.

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schr\"odinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite tim...

We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and cons...

We prove that if $u_1,\,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)\in\R \times [0,T]$ which agree in an open set $\Omega\subset \R \times [0,T]$, then $u_1\equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary valu...

The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schr{\"o}dinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally we prove the existence of a unique solution of the binormal flow with datum a polygonal line. This equation is used as a...

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensiona...

We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W^{1,\infty} around a carefully chosen, two term ansatz. Such knowledge is crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flo...

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered...

In this note we consider the 1-D cubic Schr\"odinger equation with data given as small perturbations of a Dirac-$\delta$ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schr\"odinger map to define the solution beyond the singularity ti...

We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we intr...

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat...

This work is concerned with special regularity properties of solutions to the k k -generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial datum is u 0 ∈ H l ( ( b , ∞ ) ) u_0\in H^l((b,\infty )) for some l ∈ Z + l\in \mathbb {Z}^+ and b ∈ R b\in \mathbb {R} , the...

In this article we investigate spectral properties of the coupling $H+V_\lambda$, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_\lambda$ is an electrostatic shell potential (which depends on a parameter $\lambda\in\mathbb R$) located on the boundary of a smooth domain in $\mathbb R^3$. Our main resu...

Using Carleman estimates, we give a lower bound for solutions to the discrete Schr\"odinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of the solutions.

This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the so-called binormal flow. The case of a regular polygon studied in collaboration with F. de la Hoz is also considered.

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for Schroedinger operators in all dimensions under v...

In this paper, we study the evolution of the localized induction approximation (LIA), also known as vortex filament equation, $$ \mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss}, $$ for $\mathbf X(s, 0)$ a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is al...

In this paper, we consider the so-called local induction approximation (LIA): $$ \Xt = \Xs\wedge\Xss, $$ X t = X s ∧ X ss , where ∧ is the usual cross product, and s denotes the arc-length parametrization. We study its evolution, taking planar regular polygons of M sides as initial data. Assuming uniqueness and bearing in mind the invarian...

Spectral properties and the confinement phenomenon for the coupling $H+V$ are studied, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\mathcal{R}^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given in terms of surface measures on the boundary of bounded regular domains in $\mathcal{R}^3$....

In this paper, we consider the evolution of the so-called vortex filament equation (VFE), \begin{equation*} \mathbf X_t = \mathbf X_s\wedge\mathbf X_{ss}, \end{equation*} taking a planar regular polygon of $M$ sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator...

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...

In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies...

The self-adjointness of $H+V$ is studied, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\R^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence o...

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not...

In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the ma...

We study the forward problem of the magnetic Schr\"odinger operator with potentials that have a strong singularity at the origin. We obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem.

We prove that if a solution of an equation of KdV type is bounded above by a traveling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the traveling wave.

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics i...

We prove unique continuation principles for solutions of evolution Schr\"odinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a large class of semi-linear Schr\"odinger equation.

We prove that solutions to non-linear Schr\"odinger equations in two dimensions and in the exterior of a bounded and smooth star-shaped obstacle scatter in the energy space. The non-linear potential is defocusing and grows at least as the quintic power.

In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the exponential decay in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $...

In this paper we will study the stability properties of self-similar solutions of 1-d cubic NLS equations with time-dependent coefficients of the form iu_t+u_{xx}+\frac{u}{2} (|u|^2-\frac{A}{t})=0, A\in \R (cubic). The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some si...

We review some recent results concerning the evolution of a vortex filament and its relation to the cubic non-linear Schr\"odinger equation. Selfsimilar solutions and questions related to their stability are studied.

We prove the existence of maximizers of Sobolev-Strichartz estimates for a general class of propagators, involving relevant examples, as for instance the wave, Dirac and the hyperbolic Schrodinger flows.

Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally L2-stable, and asymptotically stable in the sense of Martel-Merle. The proof is surprisingly simple and combines the Gardner transform, which links the Gardner and KdV equations, together with the Martel-Merle-Tsai and Martel-Merle recent results on stability a...

We prove unique continuation properties for solutions of the evolution Schrödinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schrödinger equations.

We give a new proof of Hardy uncertainty principle, up to the endpoint case, which is only based on calculus. The method allows us to extend Hardy uncertainty principle to Schrödinger equations with nonconstant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.

We prove the existence of maximizers for a general family of restriction operators, up to the end-point. We also provide some counterxamples in the end-point case.

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n⩾3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the p...

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.

In this note we review some recent work with F. Planchon on bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities firstly obtained by Colliander, Keel, Staffilani, Takaoka and Tao. We recover and extend known bilinear Strichartz inequalities and some classical estimates on O...

The motion of incompressible and ideal fluids is studied in the plane. The stability in L 1 of circular vortex patches is established among the class of all bounded vortex patches of equal strength. For planar incompressible and ideal fluid flow, the theory of Yudovich [9] establishes global well-posedness of the initial value problem with initial...

We prove unique continuation properties for solutions of evolutionary Schrödinger's equation with time-dependent potentials. In the case of the free solution, these correspond to uncertainty principles referred to as being of Morgan type. As an application of our method, we also obtain results concerning the possible concentration profiles of solut...

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schr\"odinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schr\"odinger equations.

In this paper we study the stability of the self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions $\chi_a(t,x)$ form a family of evolving regular curves of $\mathbb R^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We cons...

We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L 2( d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)−γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequenc...

In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski e...

We present a numerical study of the self-similar solutions of the Localized Induction Approximation of a vortex filament. These self-similar solutions, which constitute a one-parameter family, develop a singularity at finite time. We study a number of boundary conditions that allow us reproduce the mechanism of singularity formation. Some related q...

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresp...

We extend in a nonlinear context previous results obtained in [L. Vega and N. Visciglia, Proc. Am. Math. Soc. 135, No. 1, 119–128 (2007; Zbl 1173.35107); Indiana Univ. Math. J. 56, No. 5, 2265–2304 (2007; Zbl 1171.35117); Commun. Math. Phys. 279, No. 2, 429–453 (2008; Zbl 1155.35098)]. In particular, we present a precised version of Morawetz type e...

We investigate the regularization of Moore’s singularities by surface tension in the evolution of vortex sheets and its dependence on the Weber number (which is inversely proportional to surface tension coefficient). The curvature of the vortex sheet, instead of blowing up at finite time t 0, grows exponentially fast up to a O(We) limiting value cl...

We show a family of virial-type identities for the Schr\"odinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension $n\geq3$, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trappin...

In this article we will study the initial value problem for some Schrödinger equations with Dirac-like initial data and therefore with infinite L^{2} mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This a...

We study some qualitative properties of global solutions to the following focusing and defocusing critical $NLW$: \begin{equation*} \Box u+ \lambda u|u|^{2^*-2}=0, \hbox{} \lambda\in {\mathbf R} \end{equation*} $$\hspace{2cm} u(0)=f\in \dot H^1({\mathbf R}^n), \partial_t u(0)=g\in L^2({\mathbf R}^n)$$ on ${\mathbf R}\times {\mathbf R}^n$ for $n\geq...

We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrodinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresp...

We give bounds on sup t |u(x,t)| for solutions u of dispersive equations of the type i 2π∂ t u+1 2πi ∂ x α u=0 with α=2,3,⋯ on the one-dimensional torus. They are obtained from some improvements on bilinear types of estimate.

We prove bilinear virial identities for the nonlinear Schrodinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.

We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - \Delta u + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of id...

We study convexity properties of solutions to the free Schrodinger equation with Gaussian decay.

We prove some Hardy type inequalities related to the Dirac operator by elementary methods, for a large class of potentials, which even includes measure valued potentials. Optimality is achieved by the Coulomb potential. When potentials are smooth enough, our estimates provide some spectral information on the operator. oui

We obtain conditions on the measure µ so that the L 2 (µ)-norm of a function is controlled by the L 2 -norms of the function and its gradient. Applications to eigenvalues of the Schrodinger operator and to other inequalites are also given.

In this article we will study the initial value problem for some Schr\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This all...

We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<\xi>^s u^_0||_{L^r'_\xi}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached....

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg–de Vries equation of a given (small) mean.

In this paper we establish the equivalence of solutions between Schr\"odinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schr\"odinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric m...

opening < 180,, then u1 = u2.In [8] one of the key steps in the proof was a uniform exponential decay estimate in the time interval [0, 1] obtained under the assumption that the corresponding solution has the same decay at times t0 = 0, t1 = 1 (see Lemma 2.1 below). The proof of this estimate follows by combining energy estimates for the Fourier tr...

We consider the general quasilinear Schrödinger equation whose second order coefficients are given by a real symmetric non-degenerate matrix. We deduce conditions which guarantee that the associated initial value problem is locally well posed.

We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.

We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like \(n(x)\rightarrow n_{\infty} (x/|x|)\) as \(|x| \rightarrow \infty\). Under some appropriate assumptions on this convergence and on n ∞ we prove that the Sommerfeld condition at infinity...

We prove the local smoothing effect for Schrödinger equations with repulsive potentials for n⩾3. The estimates are global in time and are proved using a variation of Morawetz multipliers. As a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay, which turns out to be different whether dimension...

In each dimension n ≥ 2, we construct a class of nonnegative potentials that are homogeneous of order −σ, chosen from the range 0 ≤ σ < 2, and for which the perturbed Schrödinger equation does not satisfy global-in-time Strichartz estimates.

In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference u1−u2 of two solutions u1,u2 of (1.1) at two different times t0=0 and t1=1 which guarantee that u1≡u2.

It is conjectured that the solution to the Schrödinger equation in ℝn+1 converges almost everywhere to its initial datum f, for all f ∈ Hs (ℝn), if and only if s ≥ 1/4. It is known that there is an s < 1/2 for which the solution converges for all f ∈ Hs(ℝ2). We show that the solution to the nonelliptic Schrödinger equation, i∂tu + (∂x2 - ∂y2)u = 0,...

I will review some recent work done in collaboration with C. E. Kenig, G. Ponce and C. Rolvung on a general method to solve locally in time the initial value problem for non-linear Schrödinger equations under some natural hypotheses of decay and regularity of the coefficients. Also some non-trapping conditions of the solutions of the hamiltonian fl...

It is shown that a function $u$ satisfying, $|\Delta u+\partial_tu|\le M(|u|+|\nabla u|)$, $|u(x,t)|\le Me^{M|x|^2}$ in $\R^n\times [0,T]$ and $|u(x,0)|\le C_ke^{-k|x|^2}$ in $\R^n$ and for all $k\ge 1$, must vanish identically in $\R^n\times [0,T]$.

We study uniqueness properties of solutions of Schr\"odinger equations. The aim is to obtain sufficient conditions on the decay behavior of the difference of two solution $u_1-u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee the uniqueness of the solution, i.e. that $u_1\equiv u_2$.

We consider Schrödinger flows which are given by a real symmetric non-degenerate matrix of variable coefficient second order differential operators. After establishing the local smoothing effect we treat non-linear perturbations for first and zero order terms. A fundamental step is the construction of an integrating factor using some non-standard s...

In this work we establish a local existence theory for the initial value problem associated to the general quasi-linear ultrahyperbolic Schr\"odinger equation.

We introduce a new method to prove averaging lemmas, i.e., prove a regularizing effect on the average in velocity of a solution to a kinetic equation. The method does not require the use of Fourier transform and the whole procedure is performed in the ‘real space’. We are consequently able to improve the known result when the integrability of the s...