# sw NLPDE Z~i[

## 2011Nx̃Z~i[̋L^

EFuy[WiSFVÍGj]

2011 N 4 15 ijj 15:30  17:30
sw w@w 3 552
u
RRb iswj
u
Measure valued solutions of the 2D Keller-Segel system
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We deal with the two-dimensional Keller-Segel system describing chemotaxis in a bounded domain with smooth boundary under the nonnegative initial data. As for the Keller-Segel system, the $L^1$-norm is the scaling invariant one for the initial data, and so if the initial data is sufficiently small in $L^1$, then the solution exists globally in time. On the other hand, if its $L^1$-norm is large, then the solution blows up in a finite time. The first purpose of my talk is to construct a time global solution as a measure valued function beyond the blow-up time even though the initial data is large in $L^1$. The second purpose is to show the existence of two measure valued solutions of the different type depending on the approximation, while the classical solution is unique before the blow-up time.

2011 N 4 22 ijj 15:30  17:30
sw w@w 3 251
u
J isw͌j
u
Parametrices and Strichartz estimates for Schrodinger equations on scattering manifolds
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Ul̂ƌĂ΂QߓIɐ^\l̏̃VfBK[̏lA ɁAWpfɑ΂pgbNX̍\уXgbJ[c]ɂčl@B ܂ŃXgbJ[c]̏ؖɑ΂Ă͎gǏĉ͂s@pĂA ͋ԕϐǏ萸ȉ͂KvƂȂB ̋c_𒆐SɐsiQߓIɕRȌvʂ𔺂[NbhԁAQߓIoȌ^ĺj ̏ЉȂ炨bB

2011 N 5 6 ijj 15:00  16:00
sw w@w 3 251
u
ѓc Bm iRwj
u
A characterization of a multiple weights class
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This talk is oriented to an introduction of the newest results of the speaker. Before the speaker formulates his results, an introductory discussion of fractional integral operators, multilinear fractional integral operators and their relation with other mathematics is included.

2011 N 5 6 ijj 16:30  17:30
sw w@w 3 251
u
Vc L i򕌑wj
u
iBGEXg[NXւ̒a͊wIAv[
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S񈳏k̂̉^LqiBGEXg[NX SԂōlD llXȊ֐Ԃɗ^̊炩ȎԋǏ݂̑ƈӐɂčl@D ܂C֘AbɂĂD

2011 N 5 13 ijj 15:30  17:30
sw w@w 3 251
u
gc ĊC iwj
u
Asymptotic behavior of solutions to Cauchy problem for scalar viscous conservation law with partially linearly degenerate flux
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ꎟPƔSۑ̉̋ɊւĂIflin-Oleinik̊󔖔gՌgւ̑Q߂ɑ΂ ʂɎn܂AEt̏ꍇłLiu-Matsumura-Nishihara Hashimoto-MatsumuraɂĊ󔖔gƒł鋫Ew𓙂̏dˍɊւQߌʂ ܂ŒmĂBRɋEtȂꍇɑ΂āA ȏ́A󔖔gɑ\gmݍpNl Qߋɑ΂Ă͍܂ŊFłB ]Ă̂ƂɈ΂𓊂ׂ PƂł̔Sۑ̕Ŕ̈ꕔމꍇ̍l@sB ͔M̎ȑiSڐGgjA^ 󔖔gւ̑Q߂҂A]đŜƂĂ͗҂̏dˍւ̑Q߂\zA ۂɗ҂̍gɊւ錋ʂ𓾂Bۑ̏lœ̔gɊւ gւ̑Q߈萫𓾂邽߂ɂ͔gm̑ݍp𐸖ɕ]Ȃ΂Ȃ炸A ꂪłȓ_łB {uł͂̂Ƃɏd_uؖ̊TďqׂB

2011 N 5 20 ijj 15:30  17:30
sw w@w 3 251
u
c EEl iʑwj
u
Bifurcation from semi-trivial standing waves and ground states for a system of nonlinear Schrodinger equations
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We consider a system of nonlinear Schrodinger equations related to the Raman amplification in a plasma. We study the orbital stability and instability of standing waves bifurcating from the semi-trivial standing wave of the system. The stability and instability of the semi-trivial standing wave at the bifurcation point are also studied. Moreover, we determine the set of the ground states completely. (This is a joint work with Mathieu Colin.)

2011 N 6 3 ijj 15:30  17:30
sw w@w 3 251
u
iwj
u
Existence of recurrent traveling waves in a two-dimensional cylinder with undulating boundary --- the virtual pinning case
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In this talk we study traveling wave solutions for a curvature-driven motion of plane curves in a two-dimensional infinite cylinder with undulating boundary. Here a traveling wave in non-periodic inhomogeneous media is defined as a time-global solution whose shape is ''a continuous function of the current environment''. Under suitable conditions on the boundary undulation we show the existence of traveling waves which propagates over the entire cylinder with zero lower average speed. Such a peculiar situation called ''virtual pinning'' never occurs if the boundary undulation is periodic.

2011 N 6 17 ijj 15:30  17:30
sw w@w 3 251
u
iwK@wj
u
̎wHartree̒ݔg
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̎wHartree̒ݔgɂčl@B EʓIȏꍇɂݔg݂̑ɂĂ̌ʂqׂ̂A ȏꍇɂ͂ڂ͂ł̂ŁAp ̌Ǘg݂ɉ]ŜƂē^ȂǂЉB

2011 N 6 24 ijj 15:30  17:30
sw w@w 3 251
u
m iswj
u
The existence and non-existence of positive solutions of the nonlinear Schrodinger equations for one dimentional case
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2011 N 7 1 ijj 15:30  17:30
sw w@w 3 251
u
F} q isw͌j
u
Local exisetence of the analytic solution and the smoothing effect for the fifth oreder modified KdV equation
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We consider the initial value problem for the fifth order modified KdV equation. We show the existence of the local solution which is real analytic in both time and space variables, if the initial data $\phi\in H^{s}(\R)$ $(s\geq 3/4)$ satisfies the condition \begin{eqnarray*} \sum_{k=0}^{\infty}\frac{\large{A_0^k}}{\large{k!}}{\|}(x\pt_x)^k\phi{\|}_{H^s}<{\infty}, \end{eqnarray*} for some constant $A_0\;(0 The proof of our main result is based on the contraction principle and the bootstrap argument. In this talk we will present an outline of the proof for this result. 2011 N 7 8 ijj 15:30  17:30 sw w@w 3 251 u R{ @ ikwj u ǏIȊgUʂڗgỦ̎ԑ拓ɂ uv| We consider the initial value problem for the drift-diffusion equation arising in a semiconductor-devise simulations. This equation is a coupling system of the semilinear parabolic equation and the Poisson equation. For the initial value problem for the drift-diffusion equation, the well-posedness, the time-global existence and the decay of solutions are known. We study the large-time behavior of solutions to the drift-diffusion equation. 2011 N 7 15 ijj 15:30  17:30 sw w@w 3 251 u G iʑwj u Time local and global well-posedness for the Chern-Simons-Dirac System in one dimension uv| ʎqz[ʂ⍂dnłChern-Simons-Dirac(CSD) ͐wI1ōl@Bl̎ԋǏKؐюԑKؐ߂B ̍\ƂĔ܂Dirac-Klein-GordoniDKG)ƌ邱Ƃł邪A ̘_iMachihara-Nakanishi-Tsugawa, Kyoto J. Math. 2010j̐ёo̕] gBɎԑKؐɂĂDKGƂ͈قȂCSDL̐p]sB {BournaveasACandyƂ̋ɊÂB 2011 N 8 5 ijj 15:30  17:30 sw w@w 3 251 u Yung-fu Fang (National Cheng Kung University) u On Some Problems for Schroedinger Equations uv| First we discuss some bilinear estimates for some nonlinear Schroedinger equations. Next we discuss beriefly a global well-posedness problem and related scattering problem. Then we discuss a system of coupled Schroedinger equations and a defocusing cubic Schroedinger equation. Finally we sketch the proof of a bilinear estimate for homogeneous Schroedinger equations. 2011 N 10 14 ijj 15:30  17:30 sw w@w 3 251 u sG (ICMAT) u Multiple peak aggregations for the Keller-Segel system uv| In this talk I will talk about aggregations of Dirac mass in the Keller-Segel system. It is common that blow-up takes place under the assumption that the mass is initially greater than$8\pi$and the distribution of initial mass is localized around a point of the domain under consideration. Every blow-up produces a Dirac mass at the blow-up time. The quantity of the Dirac mass is, in general, unknown. We show that it is possible to construct, by matched asymptotic expansion techniques, a formal solution having two peaks that aggregate at the blow-up time and thus producing$16\pi$as the quantity of Dirac mass. 2011 N 10 28 ijj 15:30  17:30 sw w@w 3 251 u Tristan Roy (sww@w) u A Weak form of The Soliton Resolution Conjecture For High-Dimensional BiSchrodinger Equations uv| The soliton resolution conjecture says that solutions of semilinear fourth-order Schrodinger equations that do not blow up in finite time should be divided as time goes to infinity into a radiative part and a nonradiative part. The radiative part corresponds to a free fourth-order Schrodinger solution. It is believed that the nonradiative part is made of a finite sum of stationary or traveling solitons in most of the cases. This statement is known to be very difficult to prove. In this talk, we show a weak form of this soliton resolution conjecture. More precisely, we show that the orbit of the nonradiative part approaches as time goes to infinity a compact set modulo translations. 2011 N 11 18 ijj 15:30  17:30 sw w@w 3 251 u Γc ˎq (ȑw) u Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type uv| {uł͏މ-^Keller-Segeln̑Ȋ݂ɂčl. ȉmFgŰ, qF, NFƂ. y藝Pz qm+2/N (Sub-critical case) ̂Ƃl̑傫Ɋ֌WȂI݂. ̒藝P͍܂Keller-Segelnɂ͎gĂȂő吳pƂؖ̍ő̃|Cgł. y藝Qz qm+2/N (Super-critical case) ̂Ƃ, l\ΑI݂. 藝Q͉L^r]𓾂邽߂ɍő吳pƂƁA L^\infty]𓾂邽߂ R. Suzuki (1998) ɂL^\infry-L^r]̏ؖ@𗘗pƂ|Cgł. {uł́AL̂Q̒藝̏ؖ̃|CgɂĐ. 2011 N 11 25 ijj 15:30  17:30 sw w@w 3 251 u rc O (w) u Small data blow-up of L^{2}-solution for the nonlinear Schr\"{o}dinger equation with a nongauge invariant power nonlinearity uv| {u͎ᐙE()Ƃ̋EɊÂełB Βlp(1p1+2/n)̔VfBK[(NLS)ɑ΂ Eԑ݂̑lBNLSɑ΂ĎԋǏL^{2}݂̑͗ǂmĂB A̎̔̓Q[WsϐȂ̂ŉL^{2}ۑsł邱ƂA ̏Uʂ邽ߑI̒oł茋ʂȂB (n,p)=(2,2)̏ꍇ͒ʏ̔gpf̔񑶍݂mĂB M̉̔̌ŗǂp ZhangɂeXg֐̕@ƃQ[Wsϐ̂ VfBK[ɑ΂H^{2}_Ƃgݍ킹邱Ƃɂ B 1p1+2/n̏ꍇAlĂIׂ΂̋ǏL^{2}͑Iɂ͉ł ̍ő呶ݎԂłL^{2}mB ͂̏ؖ̃|CgЉB 2011 N 12 2 ijj 15:30  17:30 sw w@w 3 251 u ēc (kCw) u lXȘfɑ݂я󗬂̏wɂ莮A yтɊ֘Abɂ uv| {uł́AlXȘfɑ݂я󗬂ɑ΂鏃wʂȂǂAIj oXŐB я󗬂莮邽߂Ɏgń̕AnK̗͂̂̐lvZ LgĂ ɂȂ̂łApɂ߂ďdvȕłB 悸́uȂn̕dvvƂ_A ̎ɁAn̂ʁiRI͂Ɩxwjɑ΂鏃wIʂ Ɋ֘A錋ʂA vȃACfA𕂂ɂȂ璚JɉB{u͖kC̐_ۋA s̎RcAIbrahimAKoniecznyAÏꎁ-MahalovƂ 5̋yі{u҂̒PƌɊÂB 2011 N 12 9 ijj 15:30  17:30 sw w@w 3 251 u Fs (w) u Weighted Lp estimates of the solutions to the Navier-Stokes equations in some unbounded domains uv| We consider the decay estimates for the solutions to the Navier-Stokes equations in some unbounded domains;$R^n, $R^n_+$, an exterior domain and a perturbed half space. We discuss the weighted Lp-Lq estimates for the Stokes semigroup and its applications to the Navier-Stokes equations. The result of this talk is a joint work with Professor Takayuki KUBO (University of Tsukuba).

2011 N 12 16 ijj 15:00  16:00
sw w@w 3 251
u
Keith Rogers (Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM)
u
On the fractal dimension of divergence sets for Schr\"odinger equations
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We will consider the Schr\"odinger equation, $i\partial_tu+\Delta u=0$, in $\mathbb{R}^{1+1}$, with initial data $u_0$ in potential spaces $H^s(\mathbb{R})$. Carleson proved that the solution converges, almost everywhere with respect to Lebesgue measure, to $u_0$ along the straight lines $t\to (x,t)$ as $t\to 0$ when $s\ge 1/4$. We improve this result in two ways. Firstly we show that the convergence holds everywhere apart from a set of Hausdorff dimension less than or equal to $1-2s$ when $1/4\le s\le 1/2$, and that this is sharp. Secondly we will prove that the convergence holds when the straight lines are replaced by continuously differentiable curves. This allows us to refine results of Sj\"ogren--Torrea and Yajima for the quantum harmonic oscillator. This is joint work with J.A. Barcel\'o, J. Bennett, A. Carbery and S. Lee.

2011 N 12 16 ijj 16:30  17:30
sw w@w 3 251
u
Juha Kinnunen (Helsinki University of Technology)
u
A reverse Holder inequality for a doubly nonlinear parabolic PDE
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In this talk we discuss a local higher integrability result for the weak gradient of a solution to a nonlinear parabolic partial differential equation of p-Laplacian type. In particular, we show that the weak gradient satisfies a uniform reverse Holder inequality in the natural geometry related to the equation.

2011 N 12 27 iΗjj 15:30  17:30
sw w@w 3 251
u
u
Invariant weighted Wiener measure for the derivative NLS
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In this talk, we consider the derivative NLS (DNLS) on T. In particular, we construct a weighted Wiener measure, globally defined flow almost surely with respect to this measure, and finally invariance of the measure. The basic idea is to follow Bourgain's argument ('94.) Due to the nature of nonlinearity and known local well-/ill-posedness results, we consider DNLS under the gauge transformation, which causes additional difficulty. In particular, the finite dimensional measures corresponding to the finite dimensional approximations are no longer invariant. In the end, we prove "almost" invariance of such finite dimensional measures via multilinear estimates to achieve our goal. This is a joint work with A. Nahmod (UMASS), L. Rey-Bellet (UMASS), and G. Sttafilani (MIT).

2012 N 1 6 ijj 15:30  17:30
sw w@w 3 251
u
{ aL (Lw)
u
The existence and uniqueness of a solution to the boundary blowup problem for $k$-curvature equation
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uEvƂāCEɋ߂ÂƊ֐̒lɔU ۂ́uEvƌĂ΂B{uł́C $k$-ȗƌĂ΂銮SΔɑ΂鋫Ẻ̑ шӐɊւ錋ʂqׂB

2012 N 1 10 iΗjj
sw w@w 3 552
u
Zihua Guo (Peking University)
u
Uniform well-posedness and Inviscid limit for the Benjamin-Ono-Burgers equation
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In this talk I will talk about the inviscid limit of Bejamin-Ono-Burgers (BOB) equation. We prove that the Cauchy problem for the BOB equation is uniformly (with respect to the viscid parameter) globally well-posed in $H^s$ ($s \geq 1$) for all. Moreover, we show that the solution converges to that of Benjamin-Ono equation in $C([0,T]:H^s)$ ($s \geq 1$) for any $T>0$ as $\ve\to 0$. Our results give a new proof without gauge transform for the global well-posedness of BO equation in $H1$ which was first obtained by Tao \cite{TaoBO}, and obtain the inviscid limit behavior in $H1$.

2012 N 1 10 iΗjj
sw w@w 3 552
u
Baoxiang Wang (Peking University)
u
Sharp global well posedness for the non-elliptic derivative Schrodinger equation
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2012 N 1 20 ijj 15:30  17:30
sw w@wE 3 251
u
a Iq (w|w)
u
Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane
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This talk is concerned with the existence of global solutions $(u,v)$ to Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, where $u$ and $v$ denote the density of cells and of chemical substance, respectively. There are a lot of papers of simplified chemotaxis system whose second equation is elliptic. In such a system, $8 \pi$ is a critical mass of $u$ that separates the blowup and the global existence. However the original system has not been quite studied. I show that if mass of $u$ is less than or equal to $8 \pi$, then the solution exists globally in time. Moreover the existence of forward self-similar solutions with mass of $u$ greater than $8 \pi$ is mentioned.

2012 N 1 27 ijj 15:30  17:30
sw w@w 3 251
u
j (cmw)
u
Introduction to Multi View Geometry and 3d-Reconstruction
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