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2012 N 4 13 ijj 15:30  17:30@
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sw w@w 3 108
u
t l iBwj
u
ʉXyNg_Ƃ̖͊wnւ̉p
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2012 N 4 20 ijj 15:30  17:30
sw w@w 3 251
u
c isww@wȁj
u
Local well-posedness for the Navier-Stokes equations in the rotational framework
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In this talk, we consider the initial value problems for the Navier-Stokes equations with the Coriolis force. We prove the local in time existence and uniqueness of the mild solution in the framework of homogeneous Sobolev spaces. Furthermore, we give an exact characterization for the time interval of its local existence in terms of the Coriolis parameter. It follows from our characterization that the existence time of the solution can be taken arbitrarily large provided the speed of rotation is sufficiently fast.

2012 N 4 27 ijj 15:30  17:45
sw w@w 3 251
u
Frederic Rousset iIRMAR, Universite de Rennes 1j
u
Transverse stability of solitary waves in dispersive PDE
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<1>@15:3016:30
In the first talk, I will introduce the problem and present a general criterion for transverse linear instability. This criterion will be discussed on various examples like KP and Schrodinger type equations. I will also discuss the stability problem in the case of transverse periodic perturbations.
<2>@16:4517:45
In the second talk, I will focus on the study of nonlinear instability, the main example will be the water-waves system that describes the propagation of capillary-gravity waves.

2012 N 5 11 ijj 15:30  17:00
i֐m_Z~i[ƍJÁj
sw w@w 3 552
u
O ב i_ˑwj
u
AtpgŮ{ɂ
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2012 N 5 18 ijj 15:30  17:30
sw w@w 3 251
u
Ð Y iÉwj
u
Well-posedness of the KdV equation with almost periodic initial data
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First, we prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions satisfying f=f_1+f_2+...+f_N where f_j is in the Sobolev space of order s>?1/2N of a_j periodic functions. Note that f is not periodic when the ratio of periods a_i/a_j is irrational. Next, we prove an ill-posedness result in the sense that the flow map (if it exists) is not C2, which is related to the Diophantine problem. We also prove the global well-posedness in an almost periodic function space.

2012 N 5 22 iΗjj 16:45  18:15
sw w@w 3 552
u
Gustav Holzegel (Princeton University)
u
The Black Hole Stability Problem
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The first part of the talk will be an introduction to General Relativity with an emphasis on how the subject is linked to the subjects of partial differential equations and geometry. I will discuss some of the major results (Initial Value Formulation, Stability of Minkowski space) as well as future challenges and open problems. The second part will focus on recent progress in the context of the black hole stability problem. Here I will survey the recent results and techniques regarding the wave equation on Kerr black hole spacetimes, explain their relevance for the stability problem, and, finally, discuss some of my own work on so-called ultimately Schwarzschildean spacetimes".

2012 N 5 25 ijj 15:30  18:00
i15:30-16:30 1C17:00-18:00 2j
sw w@w 3 251
u
Franco Flandoli (University of Pisa)
u
The effect of noise on uniqueness and singularities of differential equations and the zero-noise limit problem
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The general theme of these two lectures is the regularization introduced by noise in ordinary and partial differential equations. The final examples we have in mind arise from fluid dynamics.

The first lecture will be devoted to a review of definitions and results of uniqueness for SDEs with additive noise and non smooth drift with emphasis on the fact that the deterministic equations with the same drift may have non unique solutions. We sketch the proof of the construction of a stochastic flow of diffeomorphisms in the case of Holder continuous drift. We also describe the zero-noise limit problem when the limit deterministic equation is not well posed and we recall a known result in dimension 1.

The second lecture is devoted to examples of SPDEs where similar regularization occurs. The role of bilinear multiplicative noise is discussed. At the PDE level the regularization due to noise may appear both for the problem of uniqueness and for the problem of singularities. We show in particular examples of linear transport equations and linear vector advection equations where noise prevents singularities which otherwise would emerge for the corresponding deterministic PDE. We give also an example where we may control the zero-noise limit.

Part of the literature related to these lecture is:

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour summer school lectures 2010, Lecture Notes in Mathematics n. 2015, Springer, Berlin 2011.

F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1--53.

2012 N 6 1 ijj 15:30  17:30
sw w@w 3 251
u
⟺ i iwj
u
Ill-posedness for the nonlinear Schrodinger equations in one space dimension
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2012 N 6 8 ijj 15:30  17:30
sw w@w 3 251
u
OY pV iwj
u
Asymptotics of small exterior Navier-Stokes flows with nonhomogeneous boundary data
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We consider the 3D incompressible Navier-Stokes flows in an exterior domain with small boundary data which do not necessarily decay in time. We prove that the spatial asymptotics of a time periodic flow is given by a Landau solution. We next show that if the boundary datum is time-periodic and the initial datum is asymptotically homogeneous with order -1, the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity. This is joint work with Kyungkuen Kang and Tai-Peng Tsai.

2012 N 6 15 ijj 15:30  16:30
sw w@w 3 251
u
Luc Molinet @iUniversité François-Rabelais de Toursj
u
Dispersive limit from the Kawahara to the KdV equation
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We will discuss the limit behavior of the solutions to the Kawahara equation
$u_t + u_{3x} + \varepsilon u_{5x} + u u_x = 0, \varepsilon>0$
as $\varepsilon \to 0$. In this equation, the terms $u_{3x}$ and $\varepsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\varepsilon}$. This prohibits the use of a standard dispersive approach for this problem. Nevertheless, by combining different dispersive approaches according to the range of spaces frequencies, we will see that the solutions to this equation converge in $C([0,T];H^1(R))$ towards the solutions of the KdV equation for any fixed $T>0$.

2012 N 6 19 iΗjj 15:00  17:00
sw w@w 3 552
iʏƗjEEꂪقȂ܂Dꂪ킩ɂ̂łӂDj
u
iwK@wj
u
On minimal non-scattering solution for focusing mass-subcritical NLS equation
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We consider time global behavior of solutions to focusing mass-subcritical NLS equation in framework of weighted $L^2$ space. We prove that there exists an initial data such that (i) corresponding solution does not scatter (non-scattering data); (ii) with respect to a certain scale-invariant quantity, this attains minimum value in all non-scattering data. Here, we call a solution with the above data as a minimal non-scattering solution. In mass-critical and -supercritical cases, it is known that the ground states are this kind of minimal non-scattering solutions. However, in this case, we can show that the non-scattering solution is NOT a standing wave solution such as ground state or excited state.

2012 N 6 22 ijj 15:30  17:30
sw w@w 3 251
u
c D i_ˑwj
u
Decay structure for symmetric hyperbolic systems with non-symmetric relaxation
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In this talk, we consider the initial value problem for symmetric hyperbolic systems. When the systems satisfy the Shizuta-Kawashima condition, we can obtain the asymptotic stability result and the explicit rate of convergence. There are, however, some physical models which do not satisfy the Shizuta-Kawashima condition (cf. Timoshenko system, Euler-Maxwell system). Moreover, it had already known that the dissipative structure of these systems is weaker than the standard type. Our purpose of this talk is to construct a new condition which include the Shizuta-Kawashima condition, and to analyze the weak dissipative structure.

2012 N 6 29 ijj 15:30  17:30
sw w@wE 3 251
u
Sanghyuk Lee iSeoul National Universityj
u
On space time estimates for the Schrödinger operator
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We discuss about the mixed-norm space-time estimates for the Schrödinger operator including their relation to Fourier restriction estimate and recent developments related to multilinear restriction estimates.

2012 N 7 13 ijj 15:30  17:30
sw w@w 3 251
u
aco G i򕌑wj
u
Remarks on logarithmic Hardy inequality in critical Sobolev-Lorentz spaces
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The classical Hardy inequality is a weighted inequality for the lower order Sobolev space. On the other hand, the logarithmic Hardy type inequality is known for the critical Sobolev space. In this talk, we give a variant of the logarithmic Hardy inequality in terms of the critical Sobolev-Lorentz space. In particular, we can establish the logarithmic Hardy inequality for the weak critical Sobolev space. At the same time, we investigate the optimality for the exponents appearing in the inequality. This is a joint work with Professor Tohru Ozawa in Waseda University and Professor Shuji Machihara in Saitama University.

2012 N 7 27 ijj 15:30  17:30
sw w@w 3 251
u
pY icwj
u
Stationary Navier-Stokes equations in multi-connected domains
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In multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stokes equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leray's inequality and the topological type of the domain is also clarified. This talk is based on the joint work with Prof.Taku Yanagisawa at Nara Women University.

2012 N 9 14 ijj 15:30  17:45
sw w@w 3 251
u
Søren Fournais iAarhus Universityj
u
Part I i15:30  16:30j : On the third critical field in Ginzburg-Landau theory
Part II i16:45  17:45j : Semiclassics in self-generated magnetic fields
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[Part I] This talk concerns the theory of superconductivity in the Ginzburg-Landau model. It is a classical result that when a superconducting material is submitted to a sufficiently strong magnetic field, the field will completely penetrate the sample and the material will loose its superconducting properties. The value of the external magnetic field strength for which such a transition takes place is called the third critical field. We will discuss problems concerning the precise definition of this critical field and both positive and negative results on its calculation. This is joint work with B. Helffer and M. Persson.
[Part II] Consider a gas of non-interacting electrons in an external electric potential and a magnetic field. The novelty is that we consider the magnetic field as self-generated, i.e. we include the classical field energy of the magnetic field and minimize over the total system (electrons)+(field). Under appropriate assumptions on the electric potential, this combined system will be stable, i.e. its energy will be bounded from below. We further study the system in a semi-classical limit. The results depend on the magnitude of the constant in front of the magnetic field energy. For certain regimes, one can obtain leading order (Weyl) asymptotic formula and even higher order semiclassics. This is joint work with L. Erdös and J.P. Solovej.

2012 N 10 12 ijj 15:30  17:00 @y֐m_Z~i[ƋÁz
sw w@w 3 552
u
] K icwj
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@LgUʂ̊m_ItƂ̉p
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2012 N 10 19 ijj 15:30  17:00
sw w@w 3 251
uE
Dongho Chae iChung-Ang Universityj
u
On the blow-up problem for the Euler equations and the Liouville type results for the fluid equations
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In the first part of the talk we discuss some new observations on the blow-up problem in the 3D Euler equations. We consider the scenarios of the self-similar blow-ups and the axisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for the self-similar Euler equations. For the axisymmetric case we show that some uniformity condition for the pressure is not consistent with the global regularity. In the second part we present Liouville type theorems for the steady Navier-Stokes equations for both of the incompressible and the compressible cases. In the time dependent case we prove that some pressure integrals have definite sign unless the solution is trivial.

2012 N 10 26 ijj 15:30  17:30
sw w@w 3 251
u
ac u iF{wj
u
Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger Equations
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We consider Schrödinger equations in $\R^{1+2}$ with electro-magnetic potentials. The potentials belong to $H^1$, and typically they are time-independent or determined as solutions to inhomogeneous wave equations. We prove Kato type smoothing estimates for solutions. We also apply this result to the Maxwell-Schrödinger equations in the Lorenz gauge and prove unique solvability of this system in the energy space.

2012 N 11 6 iΗjj 15:00  17:15
sw w@w 3 552
u
Jean-Claude Saut iUniversité Paris-Sudj
u
Part I i15:00  16:00j : Well-posedness of the Euler-Poisson system and rigorous justification of the Zakharov-Kuznetsov equation
Part II i16:15  17:15j : Initial boundary value problem and control of the Zakharov-Kuznetsov equation
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yPart Iz
We consider the rigorous derivation of the Zakharov-Kuznetsov (ZK) equation from the Euler-Poisson (EP) system for uniformly magnetized plasmas. We first provide a proof of well-posedness of the Cauchy problem for the EP system in long time, in dimensions two and three. Then we prove that the long wave small amplitude limit is described by the ZK equation. This is done first in the case of a cold plasma; then we show how to extend the result in presence of the isothermal pressure term with uniform estimates when this latter tends to zero.
yPart IIz
Imposing suitable boundary conditions yields strong dissipative effects on dispersive equations. We will develop this scenario for the Zakharov-Kuznetsov equation posed on some semi-bounded or bounded domains. The results will be applied to various control problems for the Zakharov-Kuznetsov equation.

2012 N 11 13 iΗjj 15:30  17:00
sw w@w 3 552
u
Viktor I. Burenkov iCardiff Universityj
u
Sharp spectral stability estimates for uniformly elliptic differential operators
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2012 N 11 16 ijj 15:30  17:30
sw w@w 3 251
u
j iEcwj
u
Asymptotic stability of boundary layers in plasma physics with fluid-boundary interaction
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We study the asymptotic stability of a boundary layer called sheath, which appears over a material in contact with plasma. Two types of boundary conditions on the electrostatic potential are considered depending on the physical situation. In our previous works with Prof. Shinya Nishibata and Prof. Masahiro Suzuki at Tokyo Institute of Technology, we formulated the sheath by a monotone stationary solution to the Euler-Poisson system over a half space under Bohm's criterion and proved its asymptotic stability with a boundary condition which fixes potential value on the wall. In this talk, we take into account the accumulation of charged particles on the boundary due to the flux from the inner fluid, which changes potential gradient on the boundary and further influences the entire fluid. Our main results claim the asymptotic stability of the stationary solution also with this fluid-boundary interaction.

2012 N 12 14 ijj 15:30  17:30
sw w@w 3 251
u
ikwj
u
Energy solutions for dissipative wave equations with weighted nonlinear terms
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The Cauchy problem for dissipative wave equations with weighted nonlinear terms is considered. The nonlinear terms are power type with a singularity at the origin of Coulomb type. The local and global solutions are shown in the energy class by the use of the Caffarelli-Kohn-Nirenberg inequality. The exponential type nonlinear terms are also considered in the critical two-spatial dimensions.

2012 N 12 21 ijj 15:00  17:00
sw w@w 3 251
u
gA ikCwj
u
A local regularity theorem on varifold mean curvature flow
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A family of k-dimensional surfaces in the n-dimensional Euclidean space (or more generally in a Riemannian manifold) is called the mean curvature flow (MCF) if the velocity of motion is equal to its mean curvature. One can define a weak notion of MCF using a language of geometric measure theory called varifold. Recently we proved a general local epsilon-regularity theorem in this setting, which shows among other things almost everywhere smoothness for the so called unit density MCF in Riemannian manifold for any codimensions. The Allard regularity theorem on generalized minimal submanifolds turns out to be a special case of our theorem. I will spend most time explaining the background materials and main results, and sketch the outline of proof.

2013 N 1 11 ijj 15:30  17:30
sw w@w 3 251
u
ēc OY iLwj
u
Inverse and direct bifurcation problems for nonlinear elliptic equations
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We consider the semilinear elliptic equations which are motivated by logistic equation of population dynamics. The purpose of this talk is to consider the inverse and direct bifurcation problem. For the direct problems, we establish precise asymptotic formulas for the bifurcation curves in $L^q$-framework. As for inverse problems, taking some asymptotic properties of bifurcation curves into account, we propose new concept of inverse bifurcation problem. This problem, in some sense, corresponds to the linear inverse eigenvalue problems, which determine unknown potential from the information about eigenvalues.

2013 N 1 18 ijj 15:00  17:00@@ys͊wnZ~i[Ƃ̋Áz
sw w@w 6 609
u
Yancong Xu iHangzhou Normal Universityj
u
Snakes and isolas in non-reversible conservative systems
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Reversible variational partial differential equations such as the Swift?Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas.

2013 N 1 25 ijj 15:30  17:30
sw w@w 3 251
u
ēc ǍO icwj
u
On the $R$-sectoriality of the Stokes operators in a general domain
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I will talk about the $R$-sectoriality of the Stokes operators, which implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity at the same time. The notion of $R$-sectoriality is related to the $R$-bound of the operator families, which plays an important role to use the Weis operator valued Fourier multiplier theorem. The main assumption on the domain is the unique solvability of weak Dirichlet-Neumann problem. According to our theory, a local in time existence of the Navier-Stokes equations follows from the unique solvability of weak Dirichlet-Neumann problem.

2013 N 2 1 ijj 15:30 ` 17:30
sw w@w 3 251
u
iwj
u
The $L^{\infty}$-Stokes semigroup in exterior domains
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Analyticity of the Stokes semigroup is well understood on $L^{p}$ space for various kinds of domains including bounded and exterior domains with smooth boundaries. The situation is different for the case $p=\infty$ since the Helmholtz projection is not bounded on $L^{\infty}$ anymore. In this talk, we give an a priori $L^{\infty}$-estimate of the Stokes flow by a blow-up argument for the analyticity of the semigroup on $L^{\infty}$. This talk is based on a joint work with Professor Yoshikazu Giga.