京都大学 NLPDE セミナー

今後のセミナーの予定
Schedules of future seminars

● 2016 年 10 月 14 日 16:00 〜 18:00
講演者
Piotr Rybka 氏 (University of Warsaw)
講演題目
Viscosity solutions to singular parabolic problems
講演要旨
A prototype problem we study is
   u_t = (sgn u_x)_x,  u(x,0) = u_0(x),  x is in (0,L),  t>0,
augmented with boundary conditions. This equation may be equivalently written as
   u_t = (dW/dp(u_x))_x,  u(x,0) = u_0(x)   (*)
where W(p) = |p|. Hence, an easy generalization is to consider a piecewise linear and convex function W. We define the notion of viscosity solutions to equations like (*). We show that weak solution to (*), where u_0 belongs to BV(0,L) are in fact viscosity solutions. We recall that a comparison principle holds for viscosity solutions. We show that this tool is very useful to deduce properties of solution. We will show examples.
(Joint project with Y.Giga, M.-H.Giga, M.Matusik, P.B.Mucha, A.Nakayasu)


2016 年 10 月 25 日 16:00 〜 17:30
講演者
Nadia Ansini 氏 (University of Rome 'Sapienza')
講演題目
Gradient flows with wiggly potential: a variational approach to the dynamics
講演要旨
Variational techniques and global minimisation have been proven to be very successful in many applications in materials science. The notion of Γ-convergence has been introduced to study the asymptotic behaviour of (global) minimizers of energy functionals in the limit when the parameters (related to the multiscale nature of the problem) get small. Even if Γ-convergence may fail in giving the correct description of the effect of local minimizers, variational techniques can be still applied to follow the pattern of the local minimizers of energy functionals. In this seminar I will present some recent results on microstructure evolution in materials undergoing martensitic phase transition (gradient flows with wiggly potentials).
The results are obtained in collaboration with Andrea Braides (Dept. of Mathematics, University of Rome 'Tor Vergata', Italy) and Johannes Zimmer (Dept. of Mathematical Sciences, University of Bath, UK).



● 2016 年 10 月 28 日 15:30 〜 17:30
講演者
渡辺 達也 氏 (京都産業大学理学部)
Tatsuya Watanabe (Kyoto Sangyo University)
講演題目
Orbital stability of standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation
講演要旨
In this talk, we consider the orbital stability of standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation. Applying the variational argument due to Cazenave-Lions, we show the standing wave is orbitally stable when the nonlinearity is quadratic and the coupling constant is small. This is a joint work with Mathieu Colin (University of Bordeaux).