Abstract : The aim of this mini-symposium is to present recent results in analysis and probability with applications to the study of nonlinear PDEs relating to mathematical physics, kinetic theory, and fluid mechanics. This includes questions of regularity and irregularity, stability, and geometric properties of solutions. We want to bring together young researchers and specialists to foster scientific exchange and explore new exciting developments in the fields.
Organizer(s) : Tatsuya Miura, Tobias Ried, Jonas Sauer
[04215] A mixed-norm estimate of two-particle reduced density matrix of many-body Schrödinger dynamics for deriving Vlasov equation
Format : Talk at Waseda University
Author(s) :
Li Chen (Universität Mannheim)
Abstract : We re-examine the combined semi-classical and mean-field limit in the N-body fermionic Schrödinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in the authors' previous work. Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Based on this estimate, we find that the mean-field residue is of higher order than the semiclassical residue. The talk is based on the joint work with Jinyeop Lee, Matthew Liew, and Yue Li.
[03911] Hydrodynamic limit equations derived from microscopic interacting particle systems
Format : Talk at Waseda University
Author(s) :
Makiko Sasada (University of Tokyo)
Abstract : Hydrodynamic limit provides a rigorous mathematical method to derive the deterministic partial differential equations describing the time evolution of macroscopic parameters, from the stochastic dynamics of a microscopic large scale interacting system. In this talk, by introducing the notion of a class of valid interacting particle systems, and we discuss what kind of equations can be derived from such interacting particle systems.
[01645] The Vicsek-BGK equation in collective dynamics
Format : Talk at Waseda University
Author(s) :
Raphael Winter (University of Vienna)
Abstract : The Vicsek-BGK equation describes the collective motion of agents with local alignment. It is known that the spatially homogeneous model undergoes a phase transition from disoriented motion to collective motion. In this contribution we give a prove the onset of a phase transition in the spatially inhomogeneous case. Joint work with Sara Merino Aceituno and Christian Schmeiser.
[03240] Quasilinear SPDEs with rough paths
Format : Talk at Waseda University
Author(s) :
Alexandra Neamtu (University of Konstanz)
Antoine Hocquet (Technical University of Berlin)
Abstract : We investigate quasilinear parabolic evolution equations driven by a $\gamma$-Hölder rough path,
where $\gamma\in(1/3,1/2]$. This includes the Brownian motion and a fractional Brownian motion
with Hurst index $H\in(1/3,1/2]$. We explore the mild formulation combining functional analytic
techniques with the controlled rough paths approach. We apply our results to the stochastic
Landau-Lifshitz-Gilbert equation for which we additionally prove the existence of stochastic flows.
This talk is based on a joint work with Antoine Hocquet.
[03798] A regularity structure for the quasilinear generalized KPZ equation
Format : Talk at Waseda University
Author(s) :
Masato Hoshino (Osaka University)
Ismael Bailleul (Universite Rennes 1)
Seiichiro Kusuoka (Kyoto University)
Abstract : We prove the local well-posedness of a regularity structure formulation of the quasilinear generalized KPZ equation and give an explicit form of the renormalized equation in the full subcritical regime.
[03415] Gradient continuity of weak solutions for perturbed one-Laplace problems
Format : Talk at Waseda University
Author(s) :
Shuntaro Tsubouchi (Graduate School of Mathematical Sciences, University of Tokyo)
Abstract : This talk is concerned with continuity of a spatial derivative of weak solutions to very singular problems that involve both one-Laplace and $p$-Laplace operators. The main difficulty is that one-Laplacian has both singular and degenerate ellipticity, which makes it difficult to prove Hölder continuity of a spatial gradient across a facet, the degenerate region of a gradient. In this talk, the speaker would like to talk about recent results on gradient continuity across the facet.
[02739] Asymptotic behavior of geometric flows with contact angle conditions
Format : Talk at Waseda University
Author(s) :
Takashi Kagaya (Muroran Institute of Technology)
Abstract : Several geometric flows were derived from interface phenomena. In this talk, contact angle conditions for the geometric flows are dealt with, motivated by surface tension problems. The asymptotic behavior of the geometric flows depends on the contact angle conditions. In particular, traveling waves have the asymptotic stability if we assume specific contact angle conditions. I will introduce my results related to the asymptotic behavior.
[05334] Weak-strong uniqueness for volume-preserving mean curvature flow
Format : Online Talk on Zoom
Author(s) :
Tim Laux (University of Bonn)
Abstract : I will discuss a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving mean curvature flow.
