{"id":12,"date":"2016-07-22T23:27:55","date_gmt":"2016-07-22T23:27:55","guid":{"rendered":"http:\/\/eventos.cmm.uchile.cl\/valdivia2016\/?page_id=12"},"modified":"2016-12-13T11:23:55","modified_gmt":"2016-12-13T14:23:55","slug":"abstracts","status":"publish","type":"page","link":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/abstracts\/","title":{"rendered":"Program and Abstracts (updated!)"},"content":{"rendered":"

\"valdivia_program_today-copy\"<\/a><\/h2>\n

Program Valdivia in pdf<\/a><\/p>\n

Abstracts i<\/a>n PDF<\/a><\/strong><\/p>\n

 <\/p>\n

Detailed Abstracts<\/strong><\/p>\n

Miguel A. Alejo<\/strong><\/p>\n

Title:\u00a0On the Variational Structure\u00a0of Breather solutions<\/strong><\/p>\n

Abstract:<\/strong> In this talk I will show some recent results about the variational structure and stability properties\u00a0for breathers solutions in different nonlinear models and I will support them with numerical results, computing the discrete spectra of the linearized operators around breather solutions of some nonlinear PDEs.<\/p>\n

<\/div>\n

 <\/p>\n

Hajer Bahouri<\/strong><\/p>\n

Title: Dispersive estimates for the Schr\u00f6dinger equation on 2-step stratified Lie groups<\/strong><\/p>\n

Abstract:<\/strong>\u00a0\u00a0The present work is dedicated to the proof of dispersive estimates on 2-step stratified Lie groups, for the linear \u00a0Schr\u00f6dinger equation involving \u00a0a sublaplacian. It turns out that the Schr\u00f6dinger\u00a0 propagator on 2-step stratified Lie groups behaves like a wave operator\u00a0\u00a0 on a space of the same dimension as the center of\u00a0 the group\u00a0 and like a Schr\u00f6dinger operator on a space of the same dimension as the radical of the canonical skew-symmetric form. This unusual behavior of the Schr\u00f6dinger\u00a0 propagator makes the analysis of\u00a0 the\u00a0 explicit representation of the solutions\u00a0 tricky\u00a0 and gives rise to uncommon dispersive estimates. It will also appear from our analysis that the optimal rate of decay is not always in accordance with the dimension of the center as it is the case for H-type groups: we will exhibit examples of\u00a0\u00a0 2-step\u00a0 stratified\u00a0 Lie groups with center of any dimension\u00a0\u00a0 and for which no dispersion phenomenon\u00a0 occurs for\u00a0 the\u00a0 Schr\u00f6dinger equation. We will identify a generic condition under which\u00a0 the optimal rate of decay is achieved.<\/p>\n

 <\/p>\n

Piotr Bizon<\/strong><\/p>\n

Title: Conformal flow on the 3-sphere<\/strong><\/p>\n

Abstract:<\/strong> For the conformally invariant cubic wave equation on the 3-sphere\u00a0 we construct an effective infinite-dimensional time-averaged dynamical system that approximates the dynamics of small solutions on long timescales. This effective system, which we call the conformal flow, was shown to display a rich phenomenology, including low-dimensional invariant subspaces, a wealth of stationary states, and periodic energy flows with alternating direct and inverse cascades. I will describe these results, as well as close parallels between the conformal flow and the cubic Szego equation.<\/p>\n

 <\/p>\n

Charles Collot<\/strong><\/p>\n

Title: On the stability of non-ODE blow-up for the energy supercritical semilinear heat equation<\/strong><\/p>\n

Abstract:\u00a0<\/strong>Two mechanisms are responsible for singularity formation at the origin for solutions to the focusing semilinear heat equation with power nonlinearity in the radial case. The first one is the concentration in finite time of stationary states by scale instability (type II blow-up), and the second one is the concentration in finite time of backward self-similar solutions. The latter involve a profile which shrinks according to the scaling law of the equation and at the diffusion speed. An example is given by solutions which are constant in space and which tend to infinity in finite as they solve the corresponding nonlinear ODE. In a range of parameters for which the equation is in the so called energy supercritical regime,\u00a0 Budd and Qi, Budd and Norbury, Troy, Lepin and Mizoguchi investigated the existence of backward self-similar solutions which are not constant in space. In some cases there exists a countable family of such radial solutions. In a joint work with Rapha\u00ebl and Szeftel we gave an alternative proof for the existence of these solutions which gave us tools to show the conditional non-radial and nonlinear stability of the underlying blow-up phenomenon.<\/p>\n

 <\/p>\n

Juan D\u00e1vila<\/strong><\/p>\n

Title: Finite time blowup for the harmonic map flow in 2 dimensions<\/strong><\/p>\n

Abstract:<\/strong>\u00a0We study singularity formation in the harmonic map flow
\nfrom a two dimensional domain into the sphere.
\nWe show that for suitable initial conditions the flow develops a type
\nII singularity at some point in finite time, and obtain the rate and profile.
\nWe show also that this is stable
\nunder small perturbations of the initial condition. The the rate and
\nprofile of blow up was derived formally\u00a0by van den Berg, Hulshof and King (2003) and proved by Raphael and\u00a0Schweyer (2013) in the class\u00a0of 1-corrotationally symmetric maps.
\nThis is joint work with Manuel del Pino (Universidad de Chile) and
\nJuncheng Wei (University of British Columbia).<\/p>\n

 <\/p>\n

Anne Sophie de Suzzoni<\/strong><\/p>\n

Title: \u00a0The relativistic dynamics of an electron coupled with a classical nucleus.<\/strong>
\nJoint work with F. Cacciafesta, D. Noja and E. S\u00e9r\u00e9<\/strong><\/p>\n

Abstract:<\/strong> This talk is about the Dirac equation. We consider an electron modeled by a wave function and evolving in the Coulomb field generated by a nucleus. In a very rough way, this should be an equation of the form
\n$$
\ni\\partial_t u = -\\Delta u + V( \\cdot – q(t)) u
\n$$
\nwhere $u$ represents the electron while $q(t)$ is the position of the nucleus. When one considers relativitic corrections on the dynamics of an electron, one should replace the Laplacian in the equation by the Dirac operator. Because of limiting processes in the chemistry model from which this is derived, there is also a cubic term in $u$ as a correction in the equation. What is more, the position of the nucleus is also influenced by the dynamics of the electron. Therefore, this equation should be coupled with an equation on $q$ depending on $u$.<\/p>\n

I will present this model and give the first properties of the equation. Then, I will explain why it is well-posed on $H^2$ with a time of existence depending only on the $H^1$ norm of the initial datum for $u$ and on the initial datum for $q$. The linear analysis, namely the properties of the propagator of the equation $i\\partial_t u = D u + V( \\cdot – q(t))$ where $D$ is the Dirac operator is based on works by Kato, while the non linear analysis is based on a work by Canc\u00e8s and Lebris.<\/p>\n

It is possible to have more than one nucleus. I will explain why.<\/p>\n

 <\/p>\n

Zaher Hani<\/strong><\/p>\n

TBA <\/strong><\/del> \u00a0<\/strong>(<\/strong>Cancelled)<\/strong><\/p>\n

 <\/p>\n

Oana Ivanovici<\/strong><\/p>\n

TBA<\/strong><\/p>\n

 <\/p>\n

Jacek Jendrej<\/strong><\/p>\n

\n

Title:\u00a0On two-bubble solutions for energy-critical dispersive equations<\/strong><\/p>\n<\/div>\n

Abstract:\u00a0<\/strong>The Soliton Resolution Conjecture predicts that, generically, solutions of nonlinear dispersive equations decompose asymptotically into a superposition of a finite number n of solitons and a linear radiation term. In the case of absence of the radiation term, such a solution is called a pure multi-soliton or a pure n-soliton. Motivated by the recent progress on this conjecture for energy-critical equations, I consider the problem of existence of pure radial two-solitons for the energy critical wave equation and the energy-critical Schr\u00f6dinger equation with a focusing power nonlinearity.<\/div>\n

 <\/p>\n

Herbert Koch<\/strong><\/p>\n

Title: Conserved energies for NLS, mKdV and KdV.<\/strong><\/p>\n

Abstract:\u00a0<\/strong>In the talk I will explain the construction of a family of conserved
\n<\/strong>energies for all three equations and consequences.<\/p>\n

 <\/p>\n

Yang Lan<\/strong><\/p>\n

Title: Stable self-similar blow up dynamics for slightly $L^2$<\/strong>
\nsupercritical gKdV equations<\/strong><\/p>\n

Abstract:<\/strong> We consider the focusing generalized KdV equations with slightly
\n$L^2$\u00a0supercritical nonlinearity. We will use the self-similar profile
\nconstructed by H. Koch to prove the existence and stability of a blow up
\ndynamics with self-similar blow up rate in the energy space $H^1$. We
\nwill also give a specific description of the formation of singularity
\nnear the blow up time.<\/p>\n

 <\/p>\n

Andrew Lawrie<\/strong><\/p>\n

Title:\u00a0<\/strong>Energy subcritical nonlinear wave equations.\u00a0<\/strong><\/p>\n

Abstract:<\/strong> \u00a0In this talk we’ll describe recent joint work with B. Dodson on the energy subcritical radial cubic wave equation and forthcoming work with Dodson, Mendleson, and Murphy on the same equation in the non-radial setting. We prove that all solutions scatter as long as the critical norm of the evolution stays bounded using technique inspired by the work of Kenig and Merle and Duyckaerts, Kenig, and Merle. We\u2019ll focus on the new methods we introduced to treat the energy subcritical case and on how our results complement the classic work of Merle and Zaag in this setting.<\/div>\n

 <\/p>\n

Mathieu Lewin<\/strong><\/p>\n

Title: Long time dynamics for the Hartree equation<\/strong><\/p>\n

Abstract:<\/strong> This talk will be a review of known results and open problems concerning the nonlinear Hartree equation used to describe the electrons in an atom or a molecule. It is a dispersive equation of Schr\u00f6dinger type with (long range) Coulomb forces in 3D. We will discuss the existence of stationary states, formulate a “soliton resolution” type conjecture and present a theorem proved with Enno Lenzmann on the long time dynamics, based on novel Virial type arguments. If time permits, we will also mention open problems for the infinite Coulomb plasma.<\/p>\n

 <\/p>\n

Felipe Linares<\/strong><\/p>\n

\n
\n

Title: On the fractional KP equation.<\/strong><\/p>\n<\/div>\n

Abstract:<\/strong> We will discuss a recent result\u00a0 regarding local well-posedness for the fractional KP equation. This is a joint work with D. Pilod (UFRJ, Brazil) and J-C. Saut (Orsay).<\/p>\n<\/div>\n

 <\/p>\n

Fethi Mahmoudi<\/strong><\/p>\n

Title:<\/strong>\u00a0Concentration on submanifolds for an Ambrosetti-Prodi type problem<\/strong><\/p>\n

Abstract:<\/strong> see pdf<\/a>.<\/p>\n

 <\/p>\n

Frank Merle<\/strong><\/p>\n

Title: An example of insolatedness of caracteristic points for the nonlinear wave equation in dimension two<\/strong><\/p>\n

Abstract:\u00a0<\/strong>We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in R^2. The blow-up surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one-dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two-dimensional stationary solution, whose existence is a by-product of the proof.
\nAt the origin, it behaves like the sum of 4 solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blow-up solution with a characteristic point in higher dimensions, showing a really two-dimensional behavior.\u00a0Moreover, the points of the bisectrices outside the origin give us the first example of non-characteristic points where the blow-up surface is non-differentiable.<\/p>\n

 <\/p>\n

Kenji Nakanishi<\/strong><\/p>\n

Title: Scattering for the Gross-Pitaevskii equation in the 3D radial\u00a0<\/strong>energy space<\/strong>
\nAbstract:<\/strong> This is joint work with Zihua Guo and Zaher Hani. We\u00a0consider long-time behavior of solutions for the Gross-Pitaevskii\u00a0equation (GP), or the nonlinear Schrodinger equation (NLS) with\u00a0non-zero constant amplitude at spatial infinity, in three space\u00a0dimensions. The main result is the scattering for small initial data\u00a0in the energy space with radial symmetry or angular regularity.\u00a0For NLS, it means asymptotic stability for small energy perturbation\u00a0(with the symmetry) of the plane wave solutions.\u00a0The interaction with the plane wave is very long range, which makes\u00a0the scattering for GP much harder than NLS.\u00a0We introduce a quadratic transform to remove its effect around zero\u00a0frequency, which is slightly different from those in the previous work\u00a0of Gustafson, Tsai and myself.\u00a0After the transform, we can make a global iteration by the Strichartz
\nestimate for the linearized equation which is improved under the\u00a0symmetry.\u00a0The scattering can not extend to the entire energy space, since GP\u00a0admits traveling wave solutions. Under the radial symmetry, however,\u00a0one might expect large-data scattering, as the traveling waves are not\u00a0radial.\u00a0Concerning this question, we have an interesting observation that the\u00a0focusing energy-critical wave equation appears in the zero-frequency\u00a0limit, which suggests that its ground state might be the lowest energy\u00a0obstruction for the scattering of GP.<\/p>\n

 <\/p>\n

Nejla Nouali<\/strong><\/p>\n

Title:\u00a0Construction of blow-up solution for\u00a0 complex Ginzburg-Landau equation in the critical case<\/strong><\/p>\n

Abstract: We construct a solution for the complex Ginzburg-Landau equation in the critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude.<\/p>\n

 <\/p>\n

Didier Pilod<\/strong><\/p>\n

Title: Construction of a minimal mass blow up solution of the modified Benjamin-Ono equation<\/strong><\/p>\n

Abstract: <\/strong>see pdf<\/a><\/p>\n

 <\/p>\n

Fabrice Planchon<\/strong><\/p>\n

Title: Positive and negative results for Strichartz estimates in a 2D convex model domain<\/strong><\/div>\n
\n
<\/div>\n
Abstract:<\/strong> We will report on how the sharp parametrix construction for the wave equation inside a convex 2D domain may be used to tighten the gap between known Strichartz estimates and counterexamples. In particular, a new set of (better) counterexamples will be discussed. This is joint work with Oana Ivanovici and Gilles Lebeau.<\/div>\n<\/div>\n

\u00a0<\/strong><\/p>\n

 <\/p>\n

Marjolaine Puel<\/strong><\/p>\n

Title:\u00a0Asymptotic limits for collisional kinetic equations<\/strong><\/p>\n

Abstract:<\/strong> In several domain of applied math as nuclear industry, aerodynamic, biology, \u00a0gas dynamics may be modeled by some kinetic equations. Their structure is complex and a real challenge consists in providing simpler models that are more performant for numerics.
\nWe \u00a0first \u00a0try to explain how kinetic equations may be linked to particle trajectories and introduce two particular cases, the Boltzmann equation and the Fokker Planck equation. Then we will give the context in which kinetic equations may be approximated by more macroscopic equations. At the end, we will focus on the diffusion approximation and in particular on the anomalous diffusion approximation for \u00a0both Boltzmann and Fokker Planck.<\/p>\n

 <\/p>\n

Pierre Rapha\u00ebl<\/strong><\/p>\n

TBA<\/strong><\/p>\n

 <\/p>\n

Fr\u00e9d\u00e9ric Rousset<\/strong><\/p>\n

Title: Quasineutral limit for the Vlasov-Poisson system<\/strong><\/p>\n

Abstract:<\/strong>\u00a0We will study the Vlasov Poisson system for electrons or ions in the quasineutral regime.\u00a0In this regime, there is a \u00a0small parameter in front of the Laplacian in the Poisson equation and the aim is to describe the limit when this parameter tends to zero.\u00a0This is a singular limit where many instabilities occur, in particular the limit system is not always well-posed. \u00a0We will describe recent results obtained with D. Han-Kwan about
\nthe justification of this limit under some stability conditions.<\/p>\n

 <\/p>\n

Julien Sabin<\/strong><\/p>\n

Title: Maximizers for the Stein-Tomas inequality<\/strong><\/p>\n

Abstract:<\/strong> We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein\u2013Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the\u00a0 existence of an optimizer in the Stein\u2013Tomas inequality. Our result is valid in any dimension. This is a joint work with Rupert Frank (Caltech) and Elliott Lieb (Princeton).<\/p>\n

 <\/p>\n

Jean-Claude Saut<\/strong><\/p>\n

Title: Long time existence for some water wave models<\/strong><\/p>\n

Abstract:<\/strong> Most of dispersive equations or systems are not derived\u00a0from first principles but as asymptotic models derived to\u00a0zoom at some specific regimes of amplitudes, wavelengths…in order to explain the dynamics of more complex systems.\u00a0They are not supposed to be “good” models for all time but\u00a0only on “long” time scales, in term of inverse powers\u00a0 of\u00a0a small parameter.\u00a0Those long time issues cannot be solved in general by using\u00a0the elaborate “dispersive” techniques that have been developed to study the
\nlocal Cauchy problem in large spaces and one should use instead\u00a0kind of “hyperbolic” techniques.\u00a0This\u00a0 will be illustrated on various dispersive systems arising in\u00a0the theory of water waves.<\/p>\n

 <\/p>\n

Luis Vega<\/strong><\/p>\n

Title: “Singular solutions of the Binormal Flow: transfer of energy and momentum\u201d.<\/strong><\/div>\n
<\/div>\n
Abstract:<\/strong> I shall present some recent work done with V. Banica on singular solutions of the binormal curvature flow of curves in 3d. On one hand although these solutions develop a singularity in finite time they can be uniquely continued in an appropriate sense. On the other hand some lack of continuity of a norm related to the energy of the solution occurs at the singularity time. Similarly, there is no preservation of the linear momentum. Some recent numerical simulations done with F. de la Hoz will be also given.<\/div>\n

\u00a0<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Program Valdivia in pdf Abstracts in PDF   Detailed Abstracts Miguel A. Alejo Title:\u00a0On the Variational Structure\u00a0of Breather solutions Abstract: In this talk I will show some recent results about the variational structure and stability properties\u00a0for breathers solutions in different nonlinear models and I will support them with numerical results, computing the discrete spectra of … Continue reading Program and Abstracts (updated!)<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":42,"featured_media":0,"parent":0,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-12","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/pages\/12","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/users\/42"}],"replies":[{"embeddable":true,"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/comments?post=12"}],"version-history":[{"count":43,"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/pages\/12\/revisions"}],"predecessor-version":[{"id":205,"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/pages\/12\/revisions\/205"}],"wp:attachment":[{"href":"https:\/\/eventos.cmm.uchile.cl\/valdivia2016\/wp-json\/wp\/v2\/media?parent=12"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}