C1 – Speaker: Prof. Dr. Roberto Triggiani (University of Virginia – EUA)
Title: Interior and boundary stabilization of the Navier-Stokes equations.
Abstract: This work considers the Navier Stokes equations defined on a bounded domain in the physical dimensions d=2 and d=3, subject to two types of closed-loop feedback:
(i) interior feedback acting on an arbitrary interior subdomain, while holding no-slip (Dirichlet) boundary conditions;
(ii) boundary feedback acting on (possibly only part of) the boundary, in the Dirichlet B.C.
In all cases, a Riccati-based feedback control is established that exponentially stabilizes a steady-state (stationary) solution in a neighbourhood of it. The topologies are optimal.
In the case of the interior control, the feedback is in fact finite-dimensional, of optimal dimension. This is also the case of the boundary control case in dimension d=2 under a finite dimensional spectral assumption. Instead, in contrast, in the case d=3, the feedback control must be infinite dimensional.
For d=3, the solution of this problem requires a delicate extension of the optimal control theory and related Riccati equation, above the critical threshold of the literature.
C2 - Speaker: Prof. Dr. Djairo Guedes de Figueiredo (IMECC–UNICAMP – Brasil)
Title: Sistemas elípticos não variacionais abstratos.
Abstract: Uso do método de blow-up para obter limitações a priori para as soluções positivas e utilização de métodos topológicos para obtenção das soluções.
C3-Speaker: Prof. Dra. Valéria Neves Domingos Cavalcanti (UEM – Brasil)
Title: Global existence and asymptotic stability for the wave equation with nonlinear source and boundary damping terms.
Abstract: We study the global existence and uniform decay rates of solutions of the following problem:
where is a bounded star-shaped domain of Rn , n 1, with a smooth boundary
= 01.
We also consider that and, for each M>0, is such that the initial energy E (0) satisfies E(0) < M and is a positive constante which verifies , where B1 > 0 is the optimal constant of Sobolev immersion .
C4- Speaker: Prof. Dra. Irena Lasiecka (University of Virginia – EUA)
Title: Finite dimensional attractors for second order evolutions.
Abstract: This talk discusses developments and results in the area of long time behaviour of second order evolutions. Particular examples include semilinear wave equations and Von Karman evolutions with boundary nonlinear dissipation. Structure and properties of global attractors will be discussed. Special attention will be given to questions such as dimensionality and structure of trajectories on attractors. It has been known that the issue of finite-dimensionality of attractors for hyperbolic-like flows with a nonlinear dissipation has been a long standing problem. We shall show that, in fact, by using a concept of Kolmogorov entropy and recent developments in observability theory of hyperbolic boundary problems this question can be now settled – even for the case of strongly nonlinear boundary dissipation. This Talk is based on a joint work with Igor Chueskov and Matthias Eller.
C5-Speaker: Prof. Dr. Raul Manásevich (Universidad del Chile - Chile)
Titled: Non-trivial solutions for a Dirichlet B. V. P. with a p-Laplace –like operator.
Abstract: We consider the problem
where is a bounded smooth domain in RN, (s) = sa(s) is an increasing homeomorphism from R onto R, and f(0) = 0. We are interested in non-trivial solutions for this problem.
These solutions are obtained by means of some asymptotic interaction, at zero and at infinity, of the nonlinearity with the eigenvalues of the p Laplace operator. Sufficient conditions for this are impose on the functions and f. The setting is that of Orlicz-Sobolev spaces.
C6- Speaker: Prof. Dr. Gleb Germanovitch Doronin (UEM – Brasil )
Title: Mathematics for a dusty gas: models and results.
Abstract: There are a number of mathematical models describing two-phase flows of gas-particles mixtures, known as ``dusty gases''. Within various physical and numerical studies, a qualitative mathematical analysis (i.e., well-posedness, uniqueness, global solvability, stability, etc.) of such models is not satisfactory. We briefly review in this communication a failure of hyperbolicity and non-existence theorems for nonviscous two-phase models of a dusty gas. Then, we concentrate our attention for possible regularizations of this model: use of viscosity terms leads to local solvability and well-posedness of initial-boundary value problems. However, global-in-time solvability and asymptotic behavior of solutions for this kind of models were lacking. Some steps towards a global solvability were made for incompressible models including Kuramoto-Sivashinsky approach. The present work deals with a dusty gas model in which the carrier phase is assumed to be a compressible viscous fluid, and the dust is described by hyperbolic conservation laws for the velocity and local concentration
of the dust particles. We prove global-in-time unique solvability of the Cauchy problem for sufficiently small initial data, and exponential decay of the associated energy provided small total concentration of the dust. These results can be interpreted as an asymptotic stability of a steady state of a dusty gas flow.
C7-Speaker: Prof. Dr. Yoshikazu Giga (Hokkaido University–Japão)
Title: Local solvability of a constrained gradient system of total variation
Abstract: A 1-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in $\bf{R}^N$ is
formulated by use of subdifferentials of a singular energy - the total variation.
An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result a local-in-time solution of 1-harmonic map flow equation
is constructed as a limit of the solutions of $p$-harmonic ($p$ > 1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.
C8-Speaker: Prof. Dr. Orlando Lopes (IMECC–UNICAMP)
Title: Stability of standing waves for some coupled Schrodinger systems.
Abstract: In this talk we consider the so-called x2 SHG equations
where r, s, are positive real parameters and w(x) and v(x) are complex functions.
A solitary wave is a solution of the form
In this talk we presents results of solvability and of instability of those standing waves. The main tool is a theorem about the spectrum of the linearized operator.
C9- Speaker: Prof. Dr. Felipe Linares (IMPA-Brazil)
Title: On well-posedness issues for the Schrödinger-Korteweg-de Vries equation.
Abstract: In this talk we will discuss a local well-posedness result for the Schrödinger-Korteweg-de Vries equation for data in low regularity spaces. We will explain why we believe this result is sharp. The method of proof allows us to employ the conserved quantities associated to this system to extend the results globally in time. In particular, in the resonant case we obtain better global results. This is a joint work with A. Corcho Fernandez.
C10-Speaker:: Prof. Dr. Michael Renardy (Virginia Tech – USA)
Title: On damping in two-layer elastic viscoelastic media.
Abstract: Two different kinds of damping for the wave equation are given by linear damping, utt+ ut= uxx, and Kelvin-Voigt damping: utt= uxx+ uxxt. There is a fundamental difference between the two examples: For linear damping, the damping rate approaches a fixed limit as the frequency of a disturbance tends to infinity. For Kelvin-Voigt damping, on the other hand, high frequency modes are strongly damped, i.e. the damping rate tends to infinity with frequency.
About a decade ago, the question was raised what happens to damping if the damping mechanism is only active in a part of the physical domain, for instance if we consider the equation utt+a(x)ut= uxx, where a(x) is positive only on a part of the interval. It is known that solutions still converge to zero exponentially. The same question can be raised for two-layer systems composed of two di_erent materials, where one is elastic with no damping and the other is viscoelastic. In this lecture, we shall study the question whether it is possible in this situation to have a damping rate which tends to infinity with frequency. The answer is not obvious, since what might be the first guess, a two-layer system of an elastic medium and a Kelvin-Voigt medium, actually has a damping rate which tends to zero with increasing frequency! We shall show that a damping rate which tends to infinity with frequency occurs in the following two situation:
1. Kelvin-Voigt damping with a smooth transition rather than a sharp interface.
2. A viscoelastic medium of Boltzmann type, if the wave speed is matched to that of the elastic medium and the derivative of relaxation modulus is infinite at zero.
C11-Speaker: Prof. Dr. Jorge Guilhermo Hounie (UFSCAR–Brasil)
Title: Boundary estimates for the Poisson kernel of smooth domains.
Abstract: Let Rn be a domain with smooth boundary and denote by , its Poisson Kernel. We will discuss estimates
where and are multi-indexes. These inequalities are important in the study of Hardy spaces , defined on the boundary of a smooth open subset or Rn and allow to prove that Hardy spaces can be defined either through the intrinsic maximal function or through Poisson integrals, yielding identical spaces. This extends to any smooth open subset of Rn results that were known for the unit ball. As an application, a characterization of the weak boundary values of functions that belong to holomorphic. Hardy spaces is given, which implies an F. and M. Riesz type theorem.
C12-Speaker: Prof. Dr. Paul Godin (Université Libre Bruxelles–Bélgica)
Title: Global centered waves and contact discontinuities for the axisymmetric isentropic Euler equations of perfect gases in 2 space dimensions.
Abstract: Global existence results have been obtained by Serre and Grassin-Serre for smooth solutions to the Euler equations of a perfect gas, provided the initial data belong to suitable spaces, the initial sound speed is small, and the initial velocity forces particles to spread out. We work in 2 space dimensions and consider suitable perturbations of initial data of the type considered by Serre ans Grassin-Serre which are rotation invariant around O and jump on a given circle centered at O, in such a way that the solution presents two centered waves and one contact discontinuity for small
positive times. We show that this solution is global in positive time and keeps the same structure.
C13-Speaker: Prof. Dr. Higidio Portillo Oquendo (UFPR–Brasil)
Titled: Decay for a transmission problem with nonlinear internal damping.
Abstract: We consider an anisotropic body constituted by two different types of materials: a part is simple elastic while the other has a nonlinear internal damping. We show that dissipation caused by the damped part is strong enough to produce uniform decay of the energy. That is, if we denote by and 1, two bounded open sets in Rn with smooth boundaries and 1 respectively such that . We shall assume that an anisotropic body, in equilibrium, occupy the region and it is constituted by two types of materials: in 1 the body is simple elastic while ists complementary part 2:= \ has a nonlinear damping. If we denote by u(x, t) and v(x, t) the displacement vectors in 1 and 2 at the time t, the equations that model this problem is given by
(0.1)
(0.2)
satisfying the boundary conditions
and initial data
C14-Speaker: Prof. Dr. Fanghua Lin (Courant Institute – EUA)
Title: Multiple time scale dynamics in Ginzburg-Landau-Schrodinger Equations.
Abstract: In this talk,we should give a brief describtion of some recent works concerning multiple time scales involved in some typical coupled Ginzburg-Landau and nonlinear Schrodinger equations.Solutions to these equations general possess multiparticle like(solitons) concentrations as well as other sharp concentrations(say domain walls etc).The Hamiltonian energy of such solutions would also contain various levels describing points and wall defects as well as configurations connecting them.Other challenging mathematical issues involving radiations and sound waves will be also addressed.
C15-Speaker:Prof. Dra. Marta Garcia-Huidobro (Universidad del Chile - Chile)
Title: On positive solutions to a quasilinear elliptic equation with weights
Abstract. We study the behavior of the radial non negative solutions of the equation
-div(A(|x|)|u|p-2u) = B(|x|)|u|q-20u, x RN,
where q > p > 1 and A, B are positive functions in C1(0, ) satisfying some growth assumptions.
We study the associated radial initial value problem and classify the solutions as either crossing, slowly decaying or rapidly decaying. We use Pohozaev-Pucci-Serrin type identities.
C16- Speaker: Prof. Dr. Nickolai Andreevich Larkine (UEM–Brasil)
Title: Korteweg-de Vries Equation in Bounded Domains.
Abstract: We consider in the following problem
(1)
(2)
(3)
To solve (1)-(3) we approximate it by the mixed problems for the Kuramoto-Sivashinsky equation
(4)
(5)
(6)
(7)
where > 0.
We prove the existence of strong solutions to (4)-(7) for any > 0. Then, passing to the limit as tends to 0, we prove the existence of a weak solution to (1)-(3). Finally, we prove that a weak solution is a strong one and that a solution of (1)-(3) decays exponentially as t .
C17-Speaker: Prof. Dr. Gustavo Perla Menzala (LNCC/UFRJ - Brazil)
Title: Some results on resonances in wave propagation phenomenon
Abstract: We describe some results we obtained concerning the location of resonances (scattering frequencies) for a class of wave propagation phenomenon perturbed either by a compact obstacle or under the presence of an impurity represented by a real-valued function q(x). We find large regions of the complex plane which are pole-free of resonances. The main tool we use is the analytic version of Fredholm theory. This work is part of recent research done in colaboration with C.Fernandez (Catholic University, Chile), M.A.Astaburuaga (Catholic University ,Chile), R.Coimbra (Federal University of Santa Catarina, Brasil)and L.Cortes-Vega (Concepcion, Chile).
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C18- Speaker: Prof. Dr Jaime Munoz Rivera (LNCC/UFRJ - Brazil)
Title: Stability of system with weak dissipation
Abstract:In this talk we consider the estability of systems with weak dissipation, that is systems that are not exponentially stable. We will introduce some examples of such system with weak dissipation, then we show that there exists norms, for which the system has a polynomial decay of type 1/t. Our examples include models of plates with frictional damping models in magnetoelasticity with memory, thermoelasticity among others.
C19-Speaker: Prof. Dr. Juan Amadeo Soriano (UEM – Brasil)
Title: Global existence and uniform decay rates for the wave equation with nonlinear source and boundary damping terms.
Abstract: We study the global existence and uniform decay rates of solutions of the following problem
where is a bounded star-shaped domain of Rn , n 1, with a smooth boundary
= 01.
Assuming that we prove existence of weak solutions in both cases:
(i)If, without imposing any restrictions on the initial data.
(ii)If , considering the initial data taken inside the ‘Potential Well’.
For this end, we make use of a different approach than the one used by Vitillaro [J. Diff. Equat., 2003], by considering arguments due Lasiecka and Tataru [Diff. Integral Equations, 1993]. This allows us to obtain uniform decay rates of the energy, without imposing a polynomial growth on the feedback near the origin and assuming that
This is a joint work with Marcelo M. Cavalcanti and Valéria N. Domingos Cavalcanti.
C20-Speaker: Prof. : Prof. Dra. Vanilde Bisognin (UNIFRA – Brasil)
Title: Exponential stabilization of a coupled system of Korteweg– de Vries equations with localized damping
Abstract: A locally uniform stabilization result of the solutions of a coupled system of Korteweg- de Vries equations in a bounded domain is established. The main novelty is that internally only a localized damping mechanism is considered. We use the multiplier method combined with compactness arguments together with recent results on gain of regularity and the (horizontal) unique continuation property valid for the above system.
This is joint work with E. Bisognin and Perla Menzala.