Time: Mar 10 (Wed) 4-5PM
Speaker: Kihoon Seong (KAIST)
Title: Quasi-invariance of Gaussian measures under the flow of the cubic fourth order NLS in negative Sobolev spaces
Abstract: Transport properties of Gaussian measures under different transformations have been studied in probability theory. In this talk, we discuss transport properties of Gaussian measures on periodic functions/distributions under nonlinear Hamiltonian PDEs, taking the cubic nonlinear Schr\"odinger equation with the fourth order dispersion (4NLS) as a primary example. Previously, Oh-Tzvetkov (2017) and Oh-Sosoe-Tzvetkov (2018) proved quasi-invariance of the Gaussian measures supported on L^2 under the 4NLS dynamics. In this talk, we extend this quasi-invariance result to the Gaussian measures supported on negative Sobolev spaces. This establishes the first quasi-invariance result for nonlinear Hamiltonian PDEs in negative Sobolev spaces.
This is based on a joint work with Tadahiro Oh (University of Edinburgh).
Time: Mar 24 (Wed) 4-5PM
Speaker: Jinyeop Lee (KIAS)
Title: Derivation of Vlasov-Poisson System from $N$-fermionic Schr\"odinger System
Abstract: : We consider the quantum dynamics of a large number $N$ of interacting fermions. The particles in the system interact with each other via repulsive Coulomb-like interaction force which is regularized with a polynomial cutoff with respect to $N$. From such a quantum system, we derived the Vlasov-Poisson system by considering simultaneously the semi-classical and mean-field limits in terms of Husimi measure.
Time: Apr 14 (Wed) TBA
Speaker: Claudio Munoz (Universidad de Chile)
Title: Long time asymptotics of large data in the Kadomtsev-Petviashvili models
Abstract: In this talk, I shall consider the Kadomtsev-Petviashvili (KP) equations posed on the plane. For both equations, I will describe sequential in time asymptotic descriptions of solutions, of arbitrarily large data, inside regions not containing lumps nor line solitons, and under minimal regularity assumptions. I will sketch the proof that involves the introduction of two virial identities adapted to the KP dynamics. We do not require the use of the integrability of KP (joint work with A. Mendez, F. Poblete and J. C. Pozo, see arXiv:2101.08921.)
Time: Apr 28 (Wed) 4-5PM
Speaker: Jin Woo Jang (Bonn)
Abstract: In this talk I will discuss three recent related results regarding the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, the change of variables from v \to v' is a crux of the widely used "cancellation lemma". In a first result, in collaboration with James Chapman and Robert M. Strain, in the special relativistic situation we calculate this very complex ten variable Jacobian determinant and illustrate some numerical results which show that it has a large number of distinct points where it is machine zero. In a second result, in collaboration with Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well known change of variables p \to p' is not well defined in the special relativistic context. The third result, also in collaboration with Strain, I will explain our recent proof of global-in-time existence, uniqueness and asymptotic stability for solutions near the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Jeżewska (in 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. This is the first global existence and stability result for the relativistic Boltzmann equation without angular cutoff.
Time: May 12 (Wed) 4-5PM
Speaker: Eunhee Jeong (Jeonbuk Nat'l Univ)
Title: Sharp $L^p$-$L^q$ estimate for the spectral projection associated with the twisted Laplacian
Abstract: In this talk, we are concerned with sharp estimate for the spectral projection $P_µ$ associated with the twisted Laplacian in the Lebesgue spaces. We provide a complete characterization of the sharp $L^p$–$L^q$ bound for $P_µ$, which is similar to that for the spectral projection associated with the Laplacian. As an application, we discuss the resolvent estimate for the twisted Laplacian. This talk is based on a joint work with Sanghyuk Lee and Jaehyeon Ryu.
Time: May 26 (Wed) 4-5PM
Speaker: Kyungkeun Kang (Yonsei Univ)
Title: Construction of solutions with singular behaviors for the Navier-Stokes equations near boundary
Abstract: In this talk, local boundary regularity is discussed for the Stokes system and the Navier-Stokes equations in a half-space. Unlike in the interior case, non-local effects can lead to a violation of local regularity in the spatial variables near the boundary. Some known results are summarized and recent developments are presented as well.
Time: June 9 (Wed) 4-5PM
Speaker: Ihyeok Seo (Sungkyunkwan Univ)
Title: On the radius of spatial analyticity for the Klein-Gordon-Schrödinger system
Abstract: In this talk I shall consider the Klein-Gordon-Schrodinger system. This system is a classical model which describes a system of complex scalar nucleon elds interacting with neutral real scalar meson elds. The well-posedness of the system with initial data in Sobolev spaces has been intensively studied. Once we have the well-posedness, it is often of great interest whether spatial analyticity of the initial data persists at later times. More precisely, if the initial data are real-analytic and have a uniform radius of analyticity, so there is a holomorphic extension of the data to a complex strip, then we may ask whether or not and up to what degree the solution at some later time preserves the initial analyticity; we would like to estimate the radius of analyticity of the solution at later time, which is possibly shrinking. This talk is based on the recent joint work with Jaeseop Ahn and Jimyeong Kim.
Time: June 23 (Wed) 4-5PM
Speaker: Youngran Lee (Sogang Univ)
Title: On the relation between the nonlinear Schrödinger equation with periodically varying coefficients and the Gabitov-Turitsyn equation
Abstract: We show the global well-posedness of the nonlinear Schr \"{o} dinger equation(NLS) with periodically varying coefficients in the strong dispersion regime with a small parameter $\epsilon>0$ and the Gabitov-Turitsyn equation equation or averaged dispersion managed NLS, which are used in optical fiber communications. We also prove that the solutions for the NLS converge to the solution for the Gabitov-Turitsyn equation, as $\epsilon>0$ tends to zero
Time: July 14 (Wed) 4-5PM
Speaker: Akansha Sanwal (Bielefeld Univ.)
Title: Local well-posedness for the Zakharov system in dimension $d \leqslant 3$
Abstract: We consider the Zakharov system in dimension $d \leqslant 3$ and show that it is locally well-posed in Sobolev spaces $H^s × H^l$. We construct new solution spaces by modifying the $X^{s,b}$ spaces and use the contraction mapping principle. The result obtained is sharp up to endpoints.