Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Takamori Kato
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2013 / 2014 |
In this talk we will review the recent work on the global quantization theory on compact Lie groups, and will present applications to partial differential equations and to harmonic analysis.
We establish the global existence and uniqueness of classical solutions to the three-dimensional fully compressible Navier-Stokes-Fourier system with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from vacuum, and are the first for global classical and weak solutions which may have large oscillations and can allow vacuum states.
We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equaion with the masses satisfying a resonance relation. The aim of this talk is two-fold: The first is to introduce a structural condition on the nonlinearity under which the solution is asymptotically free in the large time if the initial data is sufficiently small in an appropriate weighted Sobolev space. The proof relies on the commuting vector field method combined with the smoothing effect. The second is to present an example of small data blow-up. Our construction of the blowing-up solution is based on the Hopf-Cole transformation which allows us to reduce the problem to getting suitable growth estimates for a solution to another system. The first part of this talk is a joint work with Masahiro Ikeda and Soichiro Katayama. The second part is in collaboration with Tohru Ozawa.
In this talk, we will present recent results with M. D'Abbicco and S. Lucente for semi-linear damped wave models. We will distinguish between classical and structural damping. New strategies as higher order energies or non-classical energies basing on Lm, m ∈ [1,2) are presented. Some open problems complete the lecture.
During this seminar, we shall investigate the spectral and the scattering theory at low energy for relativistic Schrödinger operators. First of all, some striking properties at thresholds of this operator will be exhibited, as for example the absence of 0-energy resonance. Low energy behavior of the wave operators and of the scattering operator will then be studied, and stationary expressions in terms of generalized eigenfunctions will be provided for the former operators. Under slightly stronger conditions on the perturbation the absolute continuity of the spectrum on the positive semi axis will be demonstrated. During these investigations, the role of the dilations group will be emphasized.
The purpose of this talk is to investigate decay orders of the L2 energy of solutions to the incompressible homogeneous Navier-Stokes equations on the whole spaces by the aid of the theory of weighted Hardy spaces. The main estimates are two weighted inequalities for heat semigroup on weighted Hardy spaces and a weighted version of the div-curl lemma due to Coifman-Lions-Meyer-Semmes. It turns out that because of the use of weighted Hardy spaces, our decay orders of the energy can be close to the critical one of Wiegner.
We discuss the existence of the blow-up solution for multi-component parabolic-elliptic drift-diffusion model in two space dimensions. We show that the local existence, uniqueness and wellposedness of a solution in the weighted L2 spaces. Moreover we prove that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time.