Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa
2015 / 2016 |
Marcel Riesz constructed a fundamental solution of the wave equation as the analytic continuation of powers of the Lorentzian quadratic form, and explained the Huygens principle of the differential equation in connection with the poles of some gamma factors. Gindikin developed Riesz's idea in the theory of homogeneous cones, and constructed fundamental solutions of various differential operators with large symmetry. In this talk, we discuss a further generalization of Riesz-Gindikin theory, starting from new Laplace transform formula on a convex cone inspired by statistics.
In this talk we discuss the L 2 asymptotic stability of the time-global solutions to the Navier-Stokes equations in a two-dimensional exterior domain, when the global solutions possess at most scale-critical decays in space and time. In such a situation we are faced with a serious difficulty mainly coming from the lack of the Hardy-type inequality, which is specific to the two-dimensional case. We will overcome this difficulty by firstly establishing a logarithmic growth estimate for the L 2 norm of the disturbances. Our approach provides a global L 2 stability for small global solutions obtained by Kozono and Yamazaki (1995) in the weak L 2 space. In the latter part of the talk we will also discuss the stability of a special stationary solution decaying in a scale-critical order.
We introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution of convex analysis. This operation allow us to compare solutions of different parabolic problems in different domains. As examples of applications of our main result, we study the parabolic concavity of solutions to parabolic boundary value problems, analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term. We also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time, proving some optimality of the ball.
P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan established method of solution of homogenization problem for Hamilton-Jacobi equations whose Hamiltonians is coercive. However, their method is not valid for non-coercive Hamiltonians. We will treat a certain non-coercive Hamiltonians which is motivated by crystal growth, and explain that a result for non-coercive Hamiltonian is essentially different from coercive Hamiltonians.
In this talk we will give an overview of recent research on pseudo-differential operators on spaces with addition structures: for example on compact or nilpotent Lie groups, or spaces equipped with a fixed system of eigenfunctions for a given operator. We will make an overview of results and applications in related areas such as harmonic analysis or the theory of partial differential equations.
Wiener amalgam spaces are modern spaces introduced by H.G. Feichtinger in time-frequency analysis. In this talk, we will discuss the inclusion properties of these spaces with L p-Sobolev spaces. We will also show sharpness of the estimates arising in this inclusion. For applications, we have boundedness of unimodular Fourier multipliers, some Littlewood-Paley type estimates, and L p boundedness of pseudo-differential operators with symbols coming from modulation spaces M ∞,1(R 2n).
In this talk we consider Schrödinger operators on periodic graphs. We show that the spectra of the Laplacian is absolutely continuous via the locally conjugated operator theory and that it's stable under multiplicative perturbations with a certain decay at infinity.
We consider the convergence rate in time to solutions of the Cauchy problem for the viscous conservation law with a nonlinear viscosity where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. The proof is given by time-weighted energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves.
Combining the WKB (Wentzel-Kramers-Brillouin) method and the Borel resummation method, A. Voros initiated the exact WKB analysis. Now the method is known as a powerful tool to study global properties (monodromy / Stokes structures) of solutions of differential equations with a large parameter. The aim of this talk is to give an introduction to this theory. I’ll also explain some connections to other topics such as cluster algebras, spectral networks, topological recursion, and so on.
We will consider the large time behavior of the strong solutions of the compressible Navier-Stokes equation in the whole space around the motionless state. It was shown by Kawashima-Matsumura-Nishida ('79) and Hoff-Zumbrun ('95) that the perturbation of the motionless state is time-asymptotic to the solution of the linearized problem. In this talk we will give the second-order asymptotics of strong solution.
This talk is a survey of our joint work with Dr. Xiangdi Huang from Chinese Academy of Science on a blowup criterion of spherically symmetric classical solutions to an initial boundary value problem on a ball for an isentropic model of viscous gas. For smooth initial data away from vacuum, it is proved that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the concentration of mass forms around the center in Lagrangian coordinate system.
The purpose of this talk is to consider estimates for the term (u·∇)v on Hardy spaces with power weights. Critical weights, we will treat, are related to the optimal L 2-decay rate (n + 2)/4 for the incompressible Navier-Stokes equations on the whole space R n. Such rate was showed by Wiegner, and then the sharpness was proved by Miyakawa & Schonbek. Main tools for the proof of our result are Bogovski's formula for the divergence equation and a restricted weak type vector-valued inequality for the Hardy-Littlewood maximal operator.
I will talk about one phase problem for the incompressible viscous fluid flow with surface tension. The problem is formulated by the free boundary problem for the Navier-Stokes equations. The local well-posedness is proved by the maximal L p-L q regularity theorem for the linearized problem and the global well-posedness is proved by using some decay properties of the C0 semigroup associated with the linearized problem, which is analytic. I also would like to touch the problem for the global well-posedness in the exterior domain without surface tension.
The scale-function of the space is constructed based on the Einstein equation in a uniform and isotropic space. The Cauchy problem of semilinear partial differential equations are considered in Sobolev spaces. The effect of spatial variance is studied, and some dissipative properties are remarked.
The Hadamard variational formula is known as a functional differential equation of the first order for perturbation of domains. The typical example is such a variation as the Green function or eigenvalues of boundary value problems on elliptic PDEs. In this talk, we address to the Stokes equations describing the motion of viscous incompressible fluids. Our purpose is to determine the topological type of 3D domains with closed surfaces as the boundary. We first introduce an explicit representation formula of the variation for eigenvalues of the Stokes equations with multiplicity given by Jimbo-Ushikoshi. Then, it turns out that if the variation of some eigenvalue vanishes for all volume preserving perturbations of domains, then the original domain is necessarily diffeomorphic to the 2-dimensional torus. This is based on the joint work with Profs S. Jimbo, Y. Teramoto and E. Ushikoshi.