Nagoya Differintial Equations Seminar 2016/2017

Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa

The discrete-time quantum walk is the quantum counterpart of the random walk. The asymptotic behavior of a random walker is obtained by the central limit theorem, whereas the asymptotic behavior of a quantum walker is obtained by the weak limit theorem. In this talk, we give the weak limit theorem for the quantum walk with a position dependent coin.

1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let {a_k }, k=0,1,2..., be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function f : T=[-π, π) → C, that is Σ|a_k|<∞. Under which conditions on {a_k } the re-expansion of f (t) (f (t)−f (0), respectively) in the cosine (sine) Fourier series will also be absolutely convergent?
 We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function f and its conjugate f^˜ are both of bounded variation, their Fourier series converge absolutely.
 We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.
 There are multidimensional generalizations of these results.

We study the existence and asymptotic stability of time-periodic solutions to the drift-diffusion model for semiconductors. If alternating-current voltage is applied to PN-junction diodes, a time-periodic current flow is observed. The main purpose of this talk is mathematical analysis on this periodic flow. We construct a time-periodic solution by utilizing the Galerkin method. The solution is unique in a neighborhood of a thermal equilibrium, and it is globally stable. Proofs of the uniqueness and the stability are based on the energy method employing an energy form.

Consider the fluid-structure (rigid body) interaction in the steady state regime, where a viscous incompressible fluid fills the domain exterior to the rigid body in 3D. Both the external force (such as gravity) and external torque are assumed to be absent, so that the body moves itself in the fluid only by a mechanism due to the interaction at the boundary. It is called the self-propelled motion. Given a small velocity (consisting of the translational and angular velocities) of the body, our theorem provides a physically relevant boundary control which can propel the body with the velocity specified above. This talk is based on a joint work with Ana Silvestre (Lisbon) and Takeo Takahashi (Nancy).

We will show a method of construction for a fundamental solution of the exact form for degenerate operators of Grushin type, using the modified Bessel functions. This method can be applied to the Kohn-Laplacian with parameter. We will discuss the poles and its residues of the meromorphic extension of the fundamental solution. This is a joint work with W. Bauer and K. Furutani.

We consider the stability estimates for the fractional Sobolev (FS) and Hardy-Littlewood-Sobolev (HLS) inequalities, which measure the deviation from the well-known equality cases in these inequalities using L ^p norms. Estimates for the FS inequality go back over twenty years while estimates for the HLS inequality were only proved very recently. I will show how to obtain the HLS stability estimate from the FS stability estimate in a simple manner, and in doing so connect these results to some other refinements of the FS inequality which were not known to be related. I will also discuss applications of our method to the Sobolev trace theorem on the sphere, and to the Strichartz estimates for the kinetic transport equation. This is joint work with Neal Bez (Saitama Univ.) and Tohru Ozawa (Waseda Univ.).

We consider the two-dimensional Navier-Stokes equations in an exterior domain, subject to the non-slip boundary condition. It is well known that there exit various non-trivial stationary solutions, which are asymptotically constant and with a finite Dirichlet integral. On the other hand, (even local) solvability was known for the non-stationary problem for such non-decaying initial data. In this talk, I report some global well-posedness result for bounded initial data with a finite Dirichlet integral, and existence of asymptotically constant solutions for arbitrary large Reynolds numbers.

We study the 3D wave equation with a non-autonomous nonlinearity which is effective only at a given point. Higher dimensional problems of this type arise in equivariant wave maps. For positive initial energy, we establish the global existence of unbounded solutions in the energy space using the Hardy inequality with additional terms. For negative initial energy, we construct solutions which blow up in finite time applying the method of T. Kobayashi and G. Nakamura, American Journal of Mathematics, Vol. 108, No. 6.

We consider the initial value problem for semilinear wave equations in two space dimensions. For the quadratic nonlinearity, Lindblad (1990) showed that the estimate of the lifespan is divided into two cases if the total integral of the initial speed is zero or not. Recently, Takamura (2015) have obtained the upper bound of the lifespan if the nonlinearity is strictly less than the quadratic, and positive initial speed or positive initial position(initial speed is zero). In this talk, we show the optimality of the result of Takamura if the total integral of the initial speed is zero or not. This is a joint work with T. Imai and Professor H. Takamura (Future Univ. Hakodate) and Professor M. Kato (Muroran Inst. Tech.).

We consider the rotating incompressible Navier-Stokes equations on a three-dimensional torus. In the periodic setting, Babin, Mahalov and Nicolaenko ('97, '99) proved existence of global smooth solutions for large Coriolis parameters. Their proof was based on analysis of the limit equation, in which nonlinear interactions are restricted onto the resonant frequencies. In this talk, we give an improved estimate on the resonant interactions by counting the total number of nontrivial resonant frequencies via combinatorial argument. As an application, we show global regularity for the rotating Navier-Stokes equations with fractional Laplacian. This talk is based on a joint work with Tsuyoshi Yoneda (University of Tokyo).

This talk is based on a joint work with Chunhua Li [arXiv:1603.04966]. We are interested in small data global existence issues for derivative nonlinear Schrödinger systems with multiple masses. The mass resonance case was investigated in the previous work [Li-Sunagawa: Nonlinearity (2016)]. The aim of this talk is to discuss a non-resonance counterpart of it.

We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded C ¹ convex three dimensional domains for gases with cutoff hard potential and cutoff Maxwellian gases. Suppose that a solution has a bounded weighted L ² norm in space and velocity with the weight of collision frequency, which is a typical norm for existence results for boundary value problems. We prove that this solution is Hölder continuous with order less than 1/2 away from the boundary provided the incoming data have the same regularity and uniformly bounded by a fixed function in velocity with finite weighted L ² norm with the weight of collision frequency. The key observation is that the regularity in velocity obtained by collision can be transferred to space by transport and collision. An interplaying between velocity and space is introduced in order to realize this observation.

The uniform Sobolev estimate due to Kato-Yajima and Kenig-Ruiz-Sogge is one of limiting absorption principles for the resolvent of the free Schrödinger operator and can be regarded as a generalization of the Hardy-Littlewood-Sobolev inequality to non-zero energies. Recently, this estimate has been used by a series of papers by R. L. Frank and his collaborators to study spectral properties of Schrödinger operators with complex-valued potentials, such as Keller and Lieb-Thirring type inequalities. In this talk, I will discuss recent progress [arxiv.org/abs/1607.01187, arxiv.org/abs/1607.01727] on uniform Sobolev estimates for Schrödinger operators with potentials exhibiting critical singularity, and their applications to (1) Keller type eigenvalue bounds for Schrödinger operators with complex-valued potentials, and (2) global-in-time Strichartz and smoothing estimates for the Schrödinger equation. A typical example of critical potentials we have in mind is the inverse square potential. A part of this talk is based on joint work with Jean-Marc Bouclet (Toulouse III).

The talk is composed of two parts. The first part of the talk concerns the relationship between the higher dimensional subcritical Hardy inequality and the lower dimensional critical Hardy inequality. This part is a joint work with Prof. F. Takahashi (Osaka City Univ.). The second part concerns the minimization problem associated with the best constant of the generalized critical Hardy inequality in different bounded domains.

We consider flows, called W ^u flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we show that W ^u flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C ². This generalises recent results on time changes of horocycle flows.

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