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.
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
C3-
solutions of the following problem:
u tt  u   u  u in  x  0,

u  0 on 0 x  0,

 u
  g (u t )  0 on 1 x 0,
 v
u ( x,0)  u 0 ( x), u t ( x, 0)  u 1 ( x);
where  is a bounded star-shaped domain of Rn , n  1, with a smooth boundary
= 01.
2
, n  3;   0, n  1, 2 and, for each M>0,
We also consider that 0   
n2
u 0 , u 1 , is such that the initial energy E (0) satisfies E(0) < M and  is a positive





constante which verifies   B1(   2 ) 2 M (   2)  1 2 , where B1 > 0 is the optimal
constant of Sobolev immersion v   2  B1 v 2 , v  u  H 1 (); u  0 on 1 .
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.
C4-
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
 div a u u   f (u ),

u  0 x 
x
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.
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 twophase flows of gas-particles mixtures, known as ``dusty gases''. Within various
physical and numerical studies, a qualitative mathematical analysis (i.e., wellposedness, 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.
C6-
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
w
2w
i
 r 2  w  w * v  0
t
x
v
 2v
w2
i
 s 2  v 
0
t
2
x
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
wxe
it

, vx e 2it .
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
P( z, x ), z  , x   , its Poisson Kernel. We will discuss estimates
K   Dz Dx Pz, x   Cn1    , z, x    x ,
xz
where   1 , ,  n  and   1 , ,  n1  are multi-indexes. These inequalities
are important in the study of Hardy spaces H p , 0  p   , 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 GrassinSerre 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

 1   . 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:= \ 1 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
1u ï"  Cijkl u k ,l , j  0
 2 u ï"  Gijkl vk ,l , j  g (vi' )  0
satisfying the boundary conditions
v0
in 1 xR 
(0.1)
in  2 xR 
on
(0.2)
xR  ,
u  v , Cijkl uk ,l v j  Gijkl vk ,l v j
on
1 xR ,
and initial data
u ( x,0)  u 0 ( x),
u t ( x,0)  u 1 ( x) in 1 ,
u ( x,0)  v0 0 ( x), vt ( x,0)  v1 ( x) in  2 .
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-2 u) = B(|x|)|u|q-20u, x  RN,
where q > p > 1 and A, B are positive functions in C 1(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 Q  0,1 x 0, T  the following problem
(1)
ut  uu x  u xxx  0, in Q,
(2)
u(0, t )  u(1, t )  u x (1, t )  0, t  0
(3)
u) x, 0  u0 ( x ), x  (0, 1).
To solve (1)-(3) we approximate it by the mixed problems for the KuramotoSivashinsky equation
ut  uux  uxxx   uxxxx  0,
u (0, t )  u (1, t )  0,
 uxx (0, t )   u xxx (1, t )  ux (1, t )  0,
u ( x, 0)  u0 ( x ),
(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).
.
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
u tt  u  0 in x 0,
u  0 on  x 0,
0

 u
 v  g (u t )  f (u ) on 1 x 0,

0
1
u ( x,0)  u ( x), u t ( x, 0)  u ( x);
where  is a bounded star-shaped domain of Rn , n  1, with a smooth boundary
= 01.
2

0  
, n  3;   0, n  1, 2, f ( s )  s s, s  1, we prove
Assuming that
n2
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 C1 s  g (s)  C2 s for s  1.
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.