### Nagoya Differintial Equations Seminar

Date: Mondays, 16:00 ~ (90 min. ~ 120 min.)
Place: Rm. 453, Mathematics Bldg, Nagoya University
Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa

 2018 / 2019

April 16
Speaker: Takashi Furuya (Nagoya University)
Title: The factorization method for some inverse acoustic scattering problems

We consider some inverse acoustic scattering problems. For the purpose, we derive the factorization method, which is a sampling method for solving certain kinds of inverse problems where the shape and location of unknown objects have to be reconstructed. Here, we introduce new results related to the factorization method.

April 23
Speaker: Hideki Inoue (Nagoya University)
Title: Topological Levinson's theorem counts infinitely many bound states

May 7
Speaker: Yuta Wakasugi (Ehime University)
Title: $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the linear damped wave equation and give sharp $L^p$-$L^q$ estimates of the solution. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with slowly decaying initial data, and determine the critical exponent. In particular, we prove that the small data global existence holds in the critical case if the initial data does not belong to $L^1$. This talk is based on a joint work with Masahiro Ikeda (RIKEN), Mamoru Okamoto (Shinshu University), and Takahisa Inui (Osaka University).

May 14
Speaker: Hiroyuki Tsurumi (Waseda University)
Title: Solutions of the stationary Navier-Stokes equations in homogeneous Besov and Triebel-Lizorkin spaces

May 21
Speaker: Miho Murata (Kanagawa University)
Title: Global well-posedness for the Navier-Stokes-Korteweg system

May 28
Speaker: Masayuki Hayashi (Waseda University)
Title: Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that (DNLS) has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui, the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit.

June 4
Speaker: Tristan Roy (Nagoya University)
Title: Jensen-type inequalities and nonsmooth radial solutions of loglog supercritical Schrödinger equations

In this talk I will focus on the asymptotic behavior of nonsmooth radial solutions of semilinear Schrödinger equations with a barely supercritical nonlinearity (i.e a nonlinearity that grows faster than the critical power but not faster than a logarithm). It is known that we have scattering of smooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations. I will recall the techniques used to prove this result. Then I will explain how we can use Jensen-type inequalities to prove scattering of nonsmooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations.

June 11
Speaker: Yasunori Maekawa (Kyoto University)
Title: On stability of physically reasonable solutions to the two-dimensional Navier-Stokes equations in an exterior domain

June 18
Speaker: Hiroyuki Takamura (Tohoku University)
Title: Wave-like blow-up for semilinear damped wave equations

In this talk, I will discuss about a critical exponent for semilinear wave equations with time-dependent damping. When the damping is “effective,” it is Fujita exponent which is known to be the one for semilinear heat equations. Recently, by showing a sub-critical blow-up result, I have introduced a new conjecture that it is Strauss exponent which is known to be one for semilinear wave equations as far as the damping is “scattering.” I will also discuss about other nonlinearities and an intermediate situation, namely, the scaling invariant case. All the results in this talk are joint works with Ning-An Lai (Lishui University, China).

July 2
Speaker: Hiroyuki Hirayama (Miyazaki University)
Title: Well-posedness for the Zakharov-Kuznetsov-Burgers equation in two space dimensions

October 1
Speaker: Koichi Taniguchi (Chuo University)
Title: Boundedness of spectral multipliers for Schrödinger operators on open sets and its application to Besov spaces

October 15 [jointly organized with the Nagoya Probability Seminar]
Speaker: Yuzuru Inahama (Kyushu University)
Title: An introduction to para-controlled calculus

In this talk we give a simple introduction of Gubinelli-Imkeller-Perkowski's paracontrolled calculus. (This is basically a survey talk, but at the end we may present our own result a little bit.) This theory solves many formerly ill-defined, but physically important stochastic PDEs and is now competing with Hairer's regularity structure theory. Fortunately, paracontrolled calculus is based on existing theories and therefore not too big. It uses Besov space theory, in particular, Bony's paradifferential calculus. To make our presentation clear to non-experts, we give up generality and focus on the most important example, namely, the 3D dynamic $\Phi^4$-model (also known as the 3D stochastic quantization equation). It is a singular SPDE on $(0, \infty) \times T^3$ and looks like this: $$\partial_t u= \triangle_x u -u^3 +\xi \quad(\mbox{with u_0 given}).$$ Here, $\xi$ is a space-time white noise and $T^3$ is the 3 dimensional torus.

October 22
Speaker: Yutaka Terasawa (Nagoya University)
Title: Weak Solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energy

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the fractional Sobolev norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization. This talk is based on a joint work with Helmut Abels (Regensburg).

October 29
Speaker: Ken Furukawa (University of Tokyo)
Title: Rigorous justification of the hydrostatic approximation for the primitive equations by the Navier-Stokes equations

The primitive equations is considered to be a fundamental model for geophysical flows, e.g. the ocean and the atmosphere. We can formally derive this equations from a scaled Navier-Stokes equations. In this talk we will give mathematically rigorous justification of its derivations in general settings.

November 5
Speaker: Goro Akagi (Tohoku University)
Title: Allen-Cahn equation with non-decreasing constraint in unbounded domains

In Damage and Fracture Mechanics, the degree of damage in a test specimen is often represented in terms of a phase-field, whose evolution is usually described by a gradient flow of a free energy. On the other hand, due to strongly irreversible characteristics of damaging phenomena, the phase-field (i.e., the degree of damage) is supposed to be monotone in time. Accordingly, evolution laws of such phase-fields are often given in terms of gradient flows with constraints. This talk is concerned with an Allen-Cahn type equation with the positive-part function, which is a typical example of constrained gradient flows and would be a good test bed to develop techniques for dealing with gradient flows with constraints and to investigate influence of such constraints upon properties and behaviors of solutions. In this talk, we shall discuss well-posedness of the Allen-Cahn type equation in (possibly) unbounded domains by developing (re)formulations of the equation as well as energy techniques based on subdifferential calculus. Moreover, we shall also overview qualitative properties and asymptotic behaviors of solutions and some of them appear to be peculiar and different from classical Allen-Cahn equations.

November 19 [jointly organized with the workshop "Partial Differential Equation and General Relativity"]
15:00 〜 16:00
Speaker: András Vasy (Stanford University)
Title: The stability of Kerr-de Sitter black holes

In this lecture, based on joint work with Peter Hintz, I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum. I will first discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In the last part of the talk I will discuss analytic aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.

16:10 〜 17:40
Speaker: Daisuke Sakoda (Osaka University)
Title: Small data global existence for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions

November 26
Speaker: Takashi Okaji (Kyoto University)
Title: Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for the $L^{2}$ restrictions of Fourier transforms onto spheres in ${\bf R}^{n}$ which are independent of the radius of the sphere. This talk is based on a joint work with Hubert Kalf (Munich Univ.) and Osanobu Yamada (Ritsumeikan University).

December 10
Speaker: Yuki Naito (Ehime University)
Title: Incomplete blow-up of solutions for semilinear heat equations with supercritical nonlinearity

We consider the semilinear heat equation with supercritical power nonlinearity, and show the existence of backward self-similar solutions by an ODE shooting method. As an application, we will construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state. This talk is based on a joint work with Takasi Senba (Fukuoka University).

December 17
Speaker: Kenta Oishi (Nagoya University)
Title: On the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition

In this talk, we develop the R-boundedness for the generalized Stokes resolvent problem in an infinite layer, with Neumann boundary condition on both upper and lower boundary. This has not been proved for such a boundary condition, while it has been proved for Neumann and Dirichlet boundary condition on upper and lower boundary, respectively. As an application, we also establish the local well-posedness for the incompressible Navier-Stokes equation in an infinite layer with a free surface for both upper and lower boundaries.

January 21
Speaker: Keiichi Watanabe (Waseda University)
Title: Free boundary problem of compressible and incompressible two-phase flows with surface tensions and phase transitions in bounded domains

We consider a free boundary problem of compressible and incompressible two-phase flows with surface tensions and phase transitions in bounded regions. The compressible and incompressible fluids are described by the Navier–Stokes–Korteweg equations and the Navier–Stokes equations, respectively. The purpose of this talk is to show that our model is consistent with the second law of thermodynamics and can be considered to be an extension of the Navier-Stokes-Fourier system. In addition, we show a local and global in time unique existence theorem for the free boundary problem under the assumption that the initial data are near the equilibrium.

 brank