2024 Spring (Abstracts)

Time: Mar 29 (Fri) 4:00-5:00 PM

Speaker: 김지회 (University of Cambridge)

Title: On self-Similar blow up for energy supercritical semilinear wave equation

Abstract: We analyse the energy supercritical semilinear wave equation

$$\Phi_{tt}-\Delta\Phi-|\Phi|^{p-1} \Phi=0$$

in $\mathbb R^d$ space. The aim of this talk is twofold.

1. Construction of globally smooth blow up profiles. This is done by matching the interior solution which bifurcates from the soliton and the exterior solution with suitable boundary condition at $x=\infty$.

2. We then prove the non-radial finite codimensional stability of these profiles by proving maximal accretivity of the linearized operator. This allows us to exhibit a well-localised smooth initial data which blows up in finite time.

Time: Apr 5 (Fri) 4:00-5:00 PM

Speaker: Van Tien Nguyen (NTU)

Title: Multiple-collapsing blowup solutions for the 2D Keller-Segel system. 

Abstract: It is well known that the 2D Keller-Segel system has finite time blowup solutions if the initial density has a total mass greater than $8\pi$ and finite second moment. We have several constructive examples showing that the solution blows up with only $8\pi$-mass concentration. We will exhibit a new blowup mechanism formed by a collision of two sub-collapses, resulting in a $16\pi$ mass-concentrating solution. A similar phenomenon appears in the 2D mass critical NLS by the work of Martel-Raphael in 2018, where the construction relies on two specific features: a conformal invariance and a minimal mass blowup constraint. These properties are not available for parabolic equations such as the Keller-Segel system. We bring here for the first time a directly rigorous construction of such a multiple-collapsing blowup solution.  

Time: Apr 26 (Fri) 4:00-5:00 PM

Speaker: 정의현 (KAIST)

Title: Quantized slow blow-up dynamics for the energy-critical corotational wave maps problem 

Abstract: We study the blow-up dynamics for the energy-critical 1-corotational wave maps problem with 2-sphere target. In the work of Raphaël-Rodnianski in 2012, the authors exhibited a stable finite time blow-up dynamics arising from smooth initial data. In this talk, we exhibit a sequence of new finite-time blow-up rates (quantized rates), which can still arise from well-localized smooth initial data. We closely follow the strategy of the paper by Raphaël-Schweyer in 2014, who exhibited a similar construction of the quantized blow-up rates for the harmonic map heat flow. The main difficulty in our wave maps setting stems from the lack of dissipation and its critical nature, which we overcome by a systematic identification of correction terms in higher-order energy estimates. 

Time: May 10 (Fri) 4:00-5:00 PM

Speaker: Shinya Kinoshita (Tokyo Insititute of Technology)

Title: Convolution estimates and its application to a system of quadratic derivative nonlinear Schr\"{o}dinger equations 

Abstract: In this talk, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"{o}dinger equations on the torus. We show the global well-posedness of the problem in the energy space. Key tool is the convolution estimates of functions whose supports are restricted to some thickened paraboloids. This talk is based on the joint work with Hiroyuki Hirayama (Miyazaki) and Mamoru Okamoto (Osaka). 

Time: May 24 (Fri) 4:00-5:00 PM

Speaker: Tomoyuki Tanaka (Doshisha University)

Title: Improved bilinear Strichartz estimates and generalized KdV type equations 

Abstract: We consider the Cauchy problem for generalized KdV type equations on the torus. In order to show the unconditional well-posedness, we introduce improved bilinear Strichartz estimates which are used to recover the derivative loss for resonant nonlinear interactions. Their proofs are based on counting estimates on a certain set. Since we work on the torus, we have an unfavorable term when we use a counting estimate. We overcome this difficulty by a kind of scaling argument, which is reminiscent of the uncertainty principle. This talk is based on the joint work with Luc Molinet (Tours, France). 

Time: Jun 14 (Fri) 10:00-11:00 AM

Speaker: Xueying Yu (Oregon State University)

Title: Modified scattering of cubic NLS on waveguide manifolds

Abstract: In this talk, we will consider the cubic nonlinear Schr\"odinger equation on rescaled waveguide manifolds, $\mathbb{R} \times \mathbb{T}^d$ for $d\geq 2$ and demonstrate boundedness of high Sobolev norms of small-data solutions. This is based on a joint work with Bobby Wilson.