Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa
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2014 / 2015 |
We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher. The talk is based on a joint work with Helmut Abels (Regensburg) and Lars Diening (Munich).
We show some a priori estimates for the resolvent Stokes equations and prove that the Stokes operator generates an analytic semigroup on spaces of bounded functions for bounded and exterior domains with smooth boundaries. The proof is based on a localization technique for elliptic operator due to K. Masuda and H. B. Stewart. This talk is based on a joint work with Y. Giga (U. Tokyo) and M. Hieber (TU Darmstadt).
We consider the eigenvalues of the Laplace-Beltrami operator on a spherical cap under the Neumann condition. All the eigenvalues of the Laplace-Beltrami operator on the unit sphere are well-known. However, if we remove a small region from the unit sphere, then how are the eigenvalues affected by such a domain perturbation? We give an answer to this question and show how the eigenvalues behave as the region becomes smaller and smaller. Also, as related topics, we discuss nonlinear problem on a spherical cap from the bifurcation theoretic point of view. The ingredients of the talk is based on the joint works with Professors C. Bandle (Univ. Basel), T. Kawakami (Osaka Prefecture Univ.), A. Kosaka (Osaka City Univ.) and H. Ninomiya (Meiji Univ.).
This talk is based on my recent joint work with N. Visciglia, Pisa University. We revisit the classical approach by Brézis-Gallouët to prove global well-posedness for nonlinear evolution equations.
We consider the initial value problem for nonlinear wave equation with weighted functions in one space dimension. Kubo & Osaka & Yazici (2013) showed that the solution exists globally in time if the initial data are odd functions. On the other hand, they showed that the solution blows up in finite time if the initial data are not odd functions. Howerver, the lifespan, which is the maximal existence time of solutions was not clarified in their result. Our aim in this talk is to show the sharp upper and lower bounds of the lifespan in such case.
The Dirac operator on Riemannian spin manifolds was introduced by Atiyah and Singer. In this talk, I would like to explain some results on resonances (poles of the resolvent operator) of the Dirac operator on hyperbolic and parabolic cylinders. Some parts of the contents are common in the previous talk at Himeji, February 20. However I expect to explain some preliminaries on spin geometry, and how to derive the expression of the Dirac operator this time.
We consider the derivation of the conservation law for nonlinear Schrödinger equations with non-vanishing boundary conditions at spatial infinity. For usual nonlinear Schrödinger equations with power nonlinearities, Ozawa (2006) derives conservation laws for a time local solution without approximating procedure. In this talk, combining the idea of Ozawa (2006) with decomposing the nonlinearity by applying the method for the decomposition of Schrödinger operator in Gérard (2006), we show that the conservation law is derived in a way independent of approximating procedure. As an application for the result, we remove some of the technical assumptions for the nonlinearity necessary to construct a time global solution.
In this talk we discuss the space of solenoidal vector fields in an unbounded domain Ω⊂R n, n≥2, whose boundary is given as a Lipschitz graph. It is shown that, under suitable functional setting, the space of solenoidal vector fields is isomorphic to the n−1 product space of the space of scalar functions. This result reveals a generic structure for the Stokes operator and the associated semigroup. Our result also covers the whole space case Ω=R n. This talk is based on a joint work with Yasunori Maekawa (Tohoku University).
In this talk we are interested in the radial solutions of semilinear wave equations with a supercritical nonlinearity, in dimension 3, and with initial data lying in the critical Sobolev space. It is well-known that one can construct solutions on a short time interval. Then next question is: what is the asymptotic behavior of these solutions? In other words do the solutions behave like free solutions or is there blow-up? In the latter case, the standard blow-up criterion says that a Strichartz-type norm must explode. In this talk we upgrade the standard blow-up criterion. We prove that if there is blow-up, then the critical Sobolev norm must also explode. (in collaboration with Thomas Duyckerts.)
We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in the whole space. We focus on the so-called critical Besov regularity framework. After recasting the whole system in Lagrangian coordinates, and working with ``the total energy along the flow’’ rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is greater than two. Back to Eulerian coordinates, this allows to improve the range of the Lebesgue exponent for which the system is locally well-posed, compared to previous results. This is a joint work with Raphaël Danchin (UPEC, LAMA).
We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential. We prove that for a given small asymptotic profile, there exists a solution to the cubic nonlinear Schrödinger equation with delta potential which converges to given asymptotic profile in L2 as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.
We consider the initial-boundary value problem of the Navier-Stokes equations in a space of bounded functions. The L∞-theory of the Navier-Stokes equations is important in order to investigate behavior of solutions near a blow-up time. So far, L∞-type results are available for the whole space and a half space. In this talk, we consider the initial-boundary problem subject to the non-slip boundary condition and establish a blow-up rate for a certain class of domains including bounded and exterior domains.
We study the k-summability of divergent formal solutions for the Cauchy problem of linear partial differential equations of first order with respect to t whose coefficients are polynomials in t. Our problem is reduced to k-summability of formal solutions for linear ordinary differential equations associated with the Cauchy problem. We employ the method of successive approximation in order to construct the formal solutions and to obtain the k-summability property. The talk is based on a joint work with Masatake Miyake (Professor Emeritus, Nagoya University).
The boundary triples provide a powerful tool for the study of self-adjoint extensions of symmetric operators. This approach allows one to present the self-adjoint extensions as abstract boundary value problems and to reduce their study to the analysis of some operator-valued Herglotz functions which generalize the notion of the Dirichlet-to-Neumann map. The talk will give a review of the machinery and present some applications to the study of special classes of differential operators appearing in various domains of mathematical physics. A special attention will be given to recent applications such as quantum graphs.