to study the asymptotic behavior of the linearized incompressible Euler and
Navier-Stokes at small viscosity equations about shear flows. We will focus
on the case of the mixing layer. Joint work with E Grenier, T. Nguyen and A. Soffer
Monday May 7, 2018
Abstract: Consider the optimal transport problem N-ples of points, in which the transport cost between N points equals an electrostatic-type energy such as $\sum_{i\neq j} |x_i-x_j|^{-s}$ with $0<s<d$. We are led to a minimization problem for probability measures on $(\mathbb R^d)^N$, which is an N-marginal Optimal Transport problem, and has a direct physical interpretation.
We prove the sharp large-N asymptotics for the above N-marginal transport problem at second order, namely beyond the mean-field continuum limit. To this aim we establish a general finite-range decomposition technique into positive-definite contributions, extending a paper by Fefferman from 1985. This kind of decomposition seems interesting to study in itself and may have further applications.
Our questions also appear in Density Functional Theory, an important model used in computational chemistry and material science (1998 Nobel prize for W. Kohn). Mathematically, our problem represents a link between the study of ground states for classical and quantum gases. The techniques of the proof give a new viewpoint on wavelet-type decompositions, here applied within optimal transport theory.
Our description of asymptotics (a) can also be seen as an Optimal Transport parallel to, and (b) gives a new angle on, the study of asymptotics for Coulomb and Riesz gases, as done by Serfaty and collaborators.
and recall formulations of conductivity and current-current correlation measure.
In this talk we will present a panoramic view and recent results on localized regime. We will focus in particular on the diagonal behaviour problem of the ccc-measure and explain how it is related to the localization length.
for the Yamabe problem
Title: On the 3D Ginzburg-Landau model of superconductivity.
Abstract: The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortices (similar to those in fluid mechanics, but quantized), which appear above a certain value of the applied magnetic field called the first critical field. We are interested in the regime of small ɛ, where ɛ>0 is the inverse of the Ginzburg-Landau parameter. In this regime, the vortices are at main order codimension 2 topological singularities.
In this talk I will present a quantitative 3D vortex approximation construction for the Ginzburg-Landau functional, which provides an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the ɛ-level. I will then apply these results to describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near the first critical field. I will also provide a quantitative product-type estimate for the study of Ginzburg-Landau dynamics.
Universität Stuttgart, Germany)
Title: The edge resonance in elastic media with zero Poisson coefficient
Abstract: A two-dimensional elastic semistrip with stress-free boundary conditions and
zero Poisson coefficients has an embedded eigenvalue on top of the continuous spectrum.
This effect is known as edge resonance. For an infinite plate of finite thickness with a drilling
hole $(R^2\setminus\Omega)\times I$ actually infinitely many edge resonances will occur.
This is related to the spectral problem of perturbations of symbols with strongly degenerated
minima, which also appear in BCS theory. I give an overview on some of our results in this
area, which still poses a number of mathematical challenges.
Monday March 12, 2018
Large time behavior of solutions of first-order Hamilton-Jacobi
equations in the unbounded case.
Abstract: I will discuss some results about the large time behavior for unbounded
solutions of Eikonal type Hamilton-Jacobi in the whole space R^n and
illustrate the additional difficulties with respect to the periodic
case on an explicit easy control problem. This is a joint work
with Guy Barles (Tours), Thi Tuyen Nguyen (Padova) and
Thanh Viet Phan (Ho Chi Minh City).
which move depending on time. Such singularities are removable if they are weak enough, but some equations do possess singular solutions with proper singular profile. In this talk, I will survey recent progress on the existence and removability of singularities in some parabolic equations. I also discuss asymptotic pro file of solutions near singular points.
Sala Multimedia 6to piso CMM
Low energy electronic excitations in graphene, a two-dimensional lattice of carbon atoms, are described effectively by a two–dimensional Dirac operator. For a bounded flake of graphene (a quantum dot), the choice of boundary conditions determines various properties of the spectrum. Several of these choices appear in the physics literature on graphene. For a simply connected flake and a family of boundary conditions, we obtain an explicit lower bound on the spectral gap around zero. We can also study the effect of the boundary conditions on eigenvalue sums in the semiclassical limit.
- Tuesday October 10, 2017 (10:15am-11:45pm)
- Thirsday October 12, 2017 (10:15am-11:45pm)
- Tuesday October 17, 2017 (10:15am-11:45pm)
- Thirsday October 19, 2017 (10:15am-11:45pm)
Monday, September 11, 2017
- 4 pm. Fethi Mahmoudi (CMM, U. de Chile)
-
5:00 pm Matteo Rizzi (CMM, UChile)Cahn-Hilliard equation and Willmore surfacesAbstract:There are many results in the literature concerning the relation between the Allen-Cahn equation and minimal surfaces. In this talk I will present some analogue results for the Cahn-Hilliard equation, that is a fourth order PDE which is related to the Willmore energy through some Gamma-convergence results. I will present results in dimension 2 and 3, concerning the construction of solutions, and I will briefly discuss some qualitative properties.
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Monday, September 4, 2017
Sala 2, Facultad de Matemáticas PUC.
- 4 pm. Rémy Rodiac (Université Catholique de Louvain, Belgium)
Axially symmetric minimizers of the neo-Hookean energy in 3D
Abstract:
The neo-Hookean energy is an energy broadly used by physicists and engineers to describe the behavior of elastic materials undergoing large deformations. However to prove the existence of minimizers of this energy is still an open problem. We consider this problem in an axisymmetric setting and show that if the domain do not contain the axis of symmetry then minimizers do exist. Ours axisymmetric minimizers are also solutions of the weak form of the Euler-Lagrange equations of elasticity. This is a joint work with Duvan Henao.
- 5 pm. Mauricio Romo (Institute for Advanced Study, Princeton, USA)
Title: Derived Categories, gauge theories and analytic continuation
Monday, July 17, 2017
Sala de seminarios del 5to. piso del DIM U.Chile.
- 4 p.m. Jorge Faya (CMM U. Chile)
Towers of nodal bubbles for the Bahri-Coron problem in punctured domains
Abstract (pdf).
-
5 pm. Claudio Muñoz (DIM-CMM U. Chile)Decay in 1+1 dimensional scalar field equations arising in PhysicsAbstract: In this talk I will review some results in collaboration with Kowalczyk and Martel, and a couple of very recent results worked together with Poblete and Pozo, and with Alejo, where we prove decay for small perturbations of scalar equations arising either in shallow water waves, nonlinear electrodynamics or quantum field theory. All these models share similar difficulties (and similar proofs): either there is no general decay, or decay is very weak because of long range nonlinearities, weak linear dispersion, or because of internal modes appearing from the linear dynamics. We will exemplify these results in the case of the nonlinear Klein-Gordon, phi^4 model, Boussinesq, and Born-Infeld models.
Friday, June 30, 2017
Sala de seminarios del 5to. piso del DIM U.Chile.
- 3:00pm
Abstract In this talk, we report complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. Such solutions, that differ to the usual multi-soliton ones, originate from degenerate energy solutions of the Schrödinger equation . We will describe how degenerate multi-soliton are obtained by the application of Darboux-Crum transformations involving Jordan states, or alternatively from a limiting process of Hirota’s direct method or Bäcklund transformations. We will also discuss the lateral displacements and time-delays for a scattering processes of complex multi-soliton solutions. The structure of the asymptotic behaviour resulting from the integrability of the model together with its PT-symmetry ensure the reality of all of these charges, including in particular the mass, the momentum and the energy.
This talk is based in the references
F. Correa, A. Fring, JHEP 09 008 (2016), arXiv:1605.06371.
J. Cen, F. Correa, A. Fring, JMP 58, 032901 (2017), arXiv:1608.01691.
-
4pm. Marcel Clerc (DFI, U. de Chile)Title: Liquid-solid and liquid-liquid like transition in driven granular media
Abstract : The theory of non-ideal gases at thermodynamic equilibrium, for instance the van der Waals gas model, has played a central role in our understanding of coexisting phases, as well as the transitions between them. In contrast, the theory fails with granular matter because collisions between the grains dissipate energy, and their macroscopic size renders thermal fluctuations negligible. When a mass of grains is subjected to mechanical vibration, it can make a transition to a fluid state. In this state, granular matter exhibits patterns and instabilities that resemble those of molecular fluids. Here, we report a granular solid–liquid phase transition in a vibrating granular monolayer. Unexpectedly, the transition is mediated by waves and is triggered by a negative compressibility, as for van der Waals phase coexistence, although the system does not satisfy the hypotheses used to understand atomic systems. The dynamic behaviour that we observe—coalescence, coagulation and wave propagation—is common to a wide class of phase transitions. We have combined experimental, numerical and theoretical studies to build a theoretical framework for this transition.
- 5 pm. Norelys Aguila Camacho (Department of Electrical Engineering, University of Chile)
Title: Stability analysis of fractional adaptive systems: Motivation, advances and main challenges.
Monday, May 29, 2017
- 4pm.: Felipe Poblete (Universidad Austral de Chile, Valdivia, Chile)
L^p-Maximal Regularity of Second Order Evolution Equations on the Line. Abstract (pdf)
- 5 pm.: Juan Carlos Pozo (Universidad de La Frontera, Temuco, Chile)
Fundamental Solutions of Fully Nonlocal Diffusion Problems. Abstract (pdf)
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Monday, May 8, 2017
Sala 2, Facultad de Matemáticas PUC.
- 4.pm: Maria Medina (Pontificia Universidad Católica de Chile)
- 5.10 pm: Erwin Topp (USACH)
Title: Parabolic equations with Caputo time derivative. Abstract (Pdf)
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- 4.pm: Alexander Quaas (Universidad Técnica Federico Santa María)
- 5.10 pm: Cristobal Quiñinao (Instituto de Ciencias de la Ingeniera – Universidad de O’Higgins)
Some results on large-scale dynamics for FitzHugh-Nagumo neurons: the effects of coupling on the fully excitatory case. Abstract (Pdf)
Monday, April 10, 2017
Auditorio Ninoslav Bralic de la Facultad de Matemáticas de la PUC.
- 4 pm. : Roberto Cortez (U. Valparaíso y CMM)
Abstract
- 5pm,: Fernando Cortez (Escuela Politécnica de Quito, Ecuador)
We consider the Cauchy problem for a particular equation posed on the circle $S^1$. In the literature, this equation represents the compressible hyper-elastic rod equation. We prove that a breakdown will occur as soon as a quantity is strictly negative in at least one point of the circle. Our analysis relies on the application of new local-in-space blow-up criteria for periodic solutions, which involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities. We will also discuss about the impact of this result in other Cauchy problems.
Monday, March 27, 2017
Sala de seminarios del 5to. piso del DIM, U. de Chile.
- 4:00pm.: Tetsu Mizumachi (Hiroshima University, Japan)
In the case where the surface tension is weak or negligible, we fi nd a curve of
resonant continuous eigenvalues near 0. Time evolution of these resonant modes
is described by a 1D damped wave equation in the transverse variable and it gives a
linear approximation of the local phase shifts of modulating line solitary waves.
- 4:00pm: Serena Dipierro(University of Melbourne)
Title: Nonlinear free boundary problems
Abstract:
This type of problems presents some severe instability: namely, minimizers may not exist; when they exist,they may show a lack of regularity; and minimizers vary in general from scale to scale. We discuss these instability properties and present some regularity results.
- 5:10pm: Enrico Valdinoci (U. Melbourne)
We present some recent results on a nonlocal version of the Allen-Cahn equation, which is also related, at a large scale, to surfaces which minimize a nonlocal perimeter functional. We discuss some features, such as symmetry property, regularity and rigidity results.
- 4pm.:Dominique Spehner, Institut Fourier and LPMMC, Universite Grenoble Alpes, France
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one expects an interaction-driven transition between a delocalized state (the particles are independent and live in both wells) and a localized state (half of the particles live in each well). We start from the full many-body Schroedinger Hamiltonian in a large-filling situation where the on-site interactions and kinetic energies are comparable. When tunneling is negligible against the interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly reduced. The modes in which the particles condense are minimizers of nonlinear Schroedinger-type functionals.
- 5:10pm: Nicolás Torres (DIM U. Chile)
Chulkwang Kwak
Title: Fifth-order modified KdV equation
4:55
Coffee break
5:10pm
Nikola Kamburov
Title: The space of one-phase free boundary solutions in the plane
Abstract: In joint work with David Jerison we study the compactness of the space of solutions to the one-phase free boundary problem in the disk, whose positive phase is of a fixed genus. We describe the local structure of the free boundary and obtain rigidity estimates on its shape. Via a correspondence due to Traizet, our results are direct counterparts to theorems by Colding and Minicozzi for minimal surfaces.
Abstract:
Monday October 17, 2016
Sala del 7mo. piso del Centro Modelamiento Matemático, U. de Chile.
3pm
Serena Dipierro (The University of Melbourne)
Title: Chaotic orbits for systems of nonlocal equations
Abstract:
We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is a joint work with S. Patrizi and E. Valdinoci.
3:50pm
Pausa Café
4:05pm
Matthew Gursky (University of Notre Dame)
Title: Some geometric variational problems and associated Riemannian structures
Abstract:
In this talk I want to describe two problems from geometric analysis that can be re-interpreted as variational problems in an infinite-dimensional Riemannian space. The first problem is the well known uniformization theorem for surfaces, and the associated Riemannian structure turns out to be equivalent to the Mabuchi-Semmes-Donaldson metric on the natural Kahler class. In four dimensions the problem I will describe is a special case of the k–Yamabe
problem, a fully nonlinear version of the Yamabe problem. I will explain how the Riemannian interpretation gives unexpected information about the variational properties of this equation.
5pm
Enrico Valdinoci (The University of Melbourne)
Title: Quantitative regularity results for nonlocal minimal surfaces
Abstract:
We present some quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the nonlocal perimeter as a particular case. These results include BV-estimates in every dimension for stable sets
and flatness estimates for local minimizers in dimension 2 and 3. The detailed results have been given in a recent paper joint with
Eleonora Cinti and Joaquim Serra https://arxiv.org/abs/1602.00540
Thursday October 13, 2016
Sala del 5to. piso del Departamento de Ingeniería Matemática DIM, U. de Chile
5pm
Martín de Borbón (Universidad de San Luis)
Title: Blow-up limits of Kahler-Einstein mettrics with cone singularities
Abstract:
In Geometric Analysis it is often the case where one is interested in solutions to an elliptic equation which have a notion of scale invariant energy. Under suitable assuptions the only way in which solutions can degenerate is by accumulating energy at a finite number of points. Scaling the solutions at these points gives rise to the so-called blow-up limits or `bubbles’. These blow-up limits are then useful to model the degenration of solutions and to understand the cooresponding compactification of the moduli space of solutions. This general framework arises in the study of Harmonic maps, J-holomorphic curves, Yang-Mills theory and Einstein metrics on four-manifolds, among others. I will talk about this blow-up analysis in the context of Kahler-Einstein metrics on a complex surface with cone singularities along a smooth complex curve.
The study of Kahler metrics with cone singularities has been a very active area of research after a fundamental work of Donaldson in 2011 which laid the foundations for the relevant linear theory. These metrics play a key role in the recent proof of the Calabi conjecture for Fano manifolds by Chen-Donaldson-Sun and Tian. Besides that, Kahler-Einstein metrics with cone singularities have intrinsic interest; as they interact with the study of pairs of complex manifolds together with a divisor in Algebraic Geometry.
Monday September 4, 2016
Sala de seminarios del 5to. piso del Departamento de Ingeniería Matemática DIM, U. de Chile
4:00pm
Karina Vilches (U. Católica del Maule)
Title: Simultaneous Blow-up for two species Patlack-Keller-Segel System in $\mathbb R^2$.
Abstract: See the PDF.
4:55 Pausa Café
5:10pm
Gianmarco Sperone (DIM U. Chile)
Title: Further remarks on the Luo-Hou’s ansatz for a self-similar solution to the 3D Euler equations
Abstract:
It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a blow-up solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profi les, to an over-determined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consisent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refi nement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors find the trivial class of solutions under a rather unjusti fied decay condition of the blow-up profiles.
Monday August 22, 2016
Sala de Seminarios del 5to piso DIM U. de Chile
4:00pm.
Boumediene Abdellaoui (Tlemcen University)
Title: Towards a deterministic KPZ equation with fractional diffusion: the stationary case
Abstract: see the PDF.
Monday August 8, 2016
Sala 5, Facultad de Matemáticas PUC
4pm
Mariel Sáez (Pontificia Universidad Católica de Chile)
Title: Fractional Laplacians and extension problems: the higher rank case (joint with M.M. Gonzalez)
Abstract:
In previous work by A.Chang and M.M. Gonzalez the authors studied the connection between the fractional laplacian defined via the extension problem proposed in work of Caffarelli and Silvestre and a class of conformally covariant operators in conformal geometry.
In this talk I will describe a new family of fractional operators that arise from extending the work of A.Chang and M.M. Gonzalez to products of manifolds. These fractional operators can be understood from the view point of L.Caffarelli and L.Silvestre as considering extensions into two different directions and studying a Dirichlet to Neumann map type of problem associated to a degenerate elliptic pde system.
5pm
Duvan Henao (Pontificia Universidad Católica de Chile)
Title: Existence theorems for geometrically nonlinear models of nematic elastomers
Abstract:
We will discuss recent models for nematic elastomers by Barchiesi & DeSimone and by Calderer, Garavito, & Yan, which involve simultaneously an elastic energy defined in the reference configuration and an energy that penalizes spatial variations of the nematic orientation in the deformed configuration. A local invertibility property for orientation-preserving Sobolev maps is used in order to establish the existence of minima. We present a refinement of the theories in which the hypotheses on the coercivity of the energy functions are slightly weakened. This is made possible by the results on the regularity of the inverses of Sobolev maps obtained by Henao & Mora-Corral in the analysis of cavitation and fracture in nonlinear elasticity.
Monday June 18, 2016
Sala de Seminarios DIM U. de Chile 5º piso, Beauchef 851, Edificio Norte
5pm
Andrés Zúñiga (Indiana University, http://pages.iu.edu/~ajzuniga/)
Title: On the heteroclinic connection problem for multi-well gradient systems
Abstract:
In this talk we revisit the existence problem of vector-valued heteroclinic connections associated with Hamiltonian systems involving potentials W having several global minima in R^N. First we will review the standard results on this problem. Secondly, we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor W^(1/2). Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of P. Sternberg and represents a more geometric alternative to the approaches of e.g. N.D. Alikakos-G. Fusco and J. Byeon- P. Montecchiari- P. Rabinowitz, for finding such connections.
Monday June 6, 2016
Sala 5 Facultad de Matemáticas, Pontificia Universidad Católica de Chile
5:00pm
Gyula Csató (Universidad de Concepción)
Title: About Hardy-Sobolev, Moser-Trudinger and isoperimetric inequalities with densities
Abstract:
The standard isoperimetric inequality states that among all sets with a given fixed volume (or area in dimension 2) the ball has the smallest perimeter. [See the attached pdf version of the abstract for a more precise statement.] The isoperimetric problem with density is a generalization of this question: given two positive functions f and g from R^2 to R, one studies the existence of minimizers of $\int_{\partial \Omega} g(x)$ among all domains $\Omega$ such that $\int_\Omega f(x)$ equals a fixed given constant. I will talk about some results when f(x)=|x|^q and g(x)=|x|^p, where p,q are real numbers. This is a rich problem with strong variations in difficulties depending on the values of p and q. I will first give an overview on Sobolev, Hardy-Sobolev and Moser-Trudinger inequalities and establish a different kind of connections to isoperimetric inequalities with densities. Finally I will present some of the results appearing in the following references:
- Csató G., An isoperimetric problem with density and the Hardy-Sobolev inequality in R^2, Differential Integral Equations, 28, Number 9/10 (2015), 971-988.
- Csató G. and Roy P., Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54, Issue 2 (2015), 2341-2366.
- Csató G. and Roy P., The singular Moser-Trudinger inequality on simply connected domains, Comm. Partial Differential Equations, to appear.
Monday May 2, 2016
Sala 2 Facultad de Matemáticas, P.U.C.
5:00 pm
María Medina (Pontificia Universidad Católica de Chile)
Title: The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian
Abstract:
In this talk we will consider a non-autonomous problem (see attached pdf) involving the fractional Laplacian, a source term f(x), and a homogeneous Dirichlet condition outside a bounded domain. We will study its solvability, and the summability of the corresponding solution, according to the integrability of f(x), in the spirit of the classical results of Stampacchia. In particular, we will try to find the optimal conditions on f(x) and a parameter lambda (of eigenvalue type) in order to obtain a solution, seeing that the techniques applied in Boccardo-Orsina-Peral’ 06, where the authors study the problem for the classical Laplacian, do not provide optimality in this setting.
This work can be found in [Abdellaoui-Medina-Primo-Peral, JDE 260 (2016) 8160-8206].
Monday April 25, 2016
Sala de seminarios D.I.M. (5to piso), U. de Chile
5:00 pm
Carlos Román (Université Pierre et Marie Curie – Paris VI,
https://www.ljll.math.upmc.fr/~roman/)
Title: On the first critical magnetic field in the three-dimensional Ginzburg-Landau model of superconductivity
Abstract:
In this talk I will describe the behavior of global minimizers, close to the first critical field, of the Ginzburg-Landau energy with external magnetic field, for extreme type-II superconductors. I will present results by Sylvia Serfaty and Etienne Sandier in a two-dimensional reduced problem and new results concerning the full three-dimensional GL-functional.
Tuesday April 19, 2016
Sala de seminarios del Departamento de Ingeniería Matemática de la Universidad de Chile
5º piso, Beauchef 851, Edificio Norte
3:00 pm
Michel Chipot (Universität Zürich, http://user.math.uzh.ch/chipot/)
Title: Nonhomogeneous boundary value problems for the stationary Navier-Stokes equations in two-dimensional domains with semi-infinite outlets
Abstract:
We would like to present existence results for stationary nonhomogeneous Navier-Stokes systems (see attached pdf). The boundary value for the fluid velocity is assumed to have compact support and the domain occupied by the fluid is supposed to be unbounded and having outlets to infinity. The core of the technique is the construction of solenoidal extensions of the boundary value satisfying the so-called Leray-Hopf condition.
Tuesday April 12, 2016
Sala de seminarios del Departamento de Ingeniería Matemática de la Universidad de Chile 5º piso, Beauchef 851, Edificio Norte
3:00 pm
Miguel Ángel Alejo (Universidade Federal de Santa Catarina, https://sites.google.com/site/miguelalejo/)
Title: Stability of mKdV breathers and numerical results
Abstract:
In this talk I will show some recent results about the stability of breather solutions of mKdV. I will also present some numerical results about the study of the spectra of linearized operators around breather solutions of other nonlinear PDEs bearing this kind of solution.
Monday March 14, 2016
Sala de seminarios del Departamento de Ingeniería Matemática de la Universidad de Chile
5º piso, Beauchef 851, Edificio Norte
5:00pm
Marek Fila (Comenius University, http://www.iam.fmph.uniba.sk/institute/fila/)
Title: Extinction of solutions of the fast diffusion equation with critical and subcritical exponents
Abstract:
We discuss the asymptotic behaviour near extinction of positive solutions of the fast diffusion equation with critical and subcritical exponents. By a suitable rescaling, the equation is transformed to a nonlinear Fokker-Planck equation. Among other things, we show that the rate of convergence to regular and singular steady states of the transformed equation can be arbitrarily slow for suitable initial data. Our results reveal the difference in the rates occurring in the critical and the subcritical case. They also yield new classes of extinction rates for the fast diffusion equation.
Wednesday January 13, 2016
Sala de seminarios del Departamento de Ingeniería Matemática de la Universidad de Chile
4:00pm
Sylvain Ervedoza (U. Toulouse)
Title: Local exact controllability for compressible Navier-Stokes equations
around constant trajectories
Abstract:
In this talk, I will present a recent result obtained on the local
exact controllability of the 3d compressible Navier-Stokes equation
around a constant trajectory with non-zero velocity, when the control
is exerted on the whole boundary of the domain. The proof of this
result is based on an observability inequality for the adjoint of the
linearized system. The main ingredient to obtain this inequality is
the tricky combination of the equations to reduce the problem to a
closed subsystem easier to deal with, and obtained by the introduction
of a new quantity corresponding to the effective viscous flux for the
adjoint equations. This subsystem will allow to understand
independently the hyperbolic and parabolic effects of the system. We
will then introduce Carleman estimates with weight functions following
the characteristics allowing to handle the non-linearity of the model
using a fixed point argument. This result has been obtained in
collaboration with Olivier Glass (Univ. Paris Dauphine) and Sergio
Guerrero (Univ. Pierre et Marie Curie).
4:50pm
Pausa Café
5:05pm
André de Laire (U. Lille)
Title: Global well-posedness for a nonlocal Gross-Pitaevskii equation with
nonzero condition at infinity
Abstract:
In this talk we consider the Gross-Pitaevskii equation involving a nonlocal
interaction potential. Our aim is to give sufficient conditions that cover
a variety of nonlocal interactions such that the associated Cauchy problem
is globally well-posed with non-zero boundary condition at infinity, in
any dimension. We focus on even potentials that are positive definite or
positive tempered distributions.
We also discuss some nonexistence results for traveling waves solutions
to this equation.