名古屋微分方程式セミナー 2016年度

本講演では空間次元が 4 以上の Klein-Gordon-Zakharov 方程式の初期値問題の適切性について考える. そこでまず非線型項が f (1−Δ)^−1/2g となるように方程式を書き直す. その方程式を積分方程式に直し縮小写像の原理を用いて適切性を導く. ここでは Koch-Tataru によって導入された U ², V ² 型空間を適用することにより, 初期値が球対称かつ小さい場合に臨界空間における時間大域的適切性, 及び解の散乱を得ることができる. 一方初期値が非球対称の場合, 適切性の証明で核となる非線型項の評価式において, 非線型項の (−Δ)^−1/2 を十分に活用できない場合が存在するため, 臨界空間で適切性を導くことは難しい. 従って初期値が非球対称の場合には初期値の正則性に関して強い仮定が必要である.

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).

粘性気体に関連する方程式に例を取りながら, 時間大域解の構成とその漸近挙動の考察を可能とする種々のエネルギー法を紹介する.

1. バーガース方程式の定数解の大域的漸近安定性
2. バーガース方程式の定数解の大域的漸近安定性, 非線形粘性項の場合
3. バーガース方程式の粘性衝撃波の漸近安定性
4. 一次元粘性気体の方程式系の定数解の大域的漸近安定性
5. 一次元粘性気体の方程式系の粘性衝撃波の漸近安定性
6. 幾つかの注：リーマン問題, 希薄波への漸近性, 初期境界値問題, 未解決問題

本講演では空間二次元上の一般化 Zakharov-Kuznetsov 方程式の初期値問題について考察する. 解の適切性を縮小写像原理によって示す際, いくつかの時空間に関する線形評価式が必要となる. 今回は, とくに最大関数評価式について考えたい. この評価式をモジュレーション空間の枠組みで構成することで, 既存のものよりも滑らかさが弱くなることが分かった. その結果として, 通常のソボレフ空間では扱えないような特異な関数空間における適切性が得られることを示す.

本講演では非線形項に未知関数の導関数を含む波動方程式の爆発曲線を考え, 十分滑らかで大きな初期値をとった場合, 爆発曲線が連続微分可能になることを示す. 非線形項が u ^p である場合に, Caffarelli-Friedman により, 適切な初期値のもとで爆発曲線が滑らかになることが示されていた. しかし, 非線形項に微分を含む場合は, ある限られた状況に関するものしか知られていなかった. 本講演では, 非線形項が |u_t|^p である場合にも Caffarelli-Friedman と類似の結果が得られたので, それについて言及する. また, その数値例についても触れたい.

本講演では L ^p(R^N) (1< p <∞) において |x|^αΔ を主要項に持ち, スケール同次性をもつポテンシャル |x|^α-2 をもつ二階楕円型作用素 L による解析半群の生成について考察する. このような作用素の研究 Liskevich-Sobol-Vogt (2002) により 0≤α<2 の場合に半群の積分核については詳細に調べられているが, 半群の生成作用素である L の定義域の特徴づけは行われていなかった. そこで本講演では L ^p(R^N) において定義域の特徴づけに焦点をおいて α∈R のときの解析半群の生成可能性を述べる. また, 生成された半群に対する L ^p-L ^q 評価についても触れたい. 尚, 本講演は Salento 大学の Giorgio Metafune 氏, Chiara Spina 氏, 東京理科大学の岡沢登氏 との共同研究に基づく.

非線形波動方程式に対する初期値問題の理論の基礎について学ぶことを目的とする.

・非線形波動方程式の初期値問題の局所解の存在
・小さな初期値に対する解の大域解の存在条件
・解の有限時間での爆発
・大域解の漸近挙動

定常ナヴィエ・ストークス方程式の 2 次元外部問題に伴う困難の一つとして, 線形問題におけるストークスのパラドックスが挙げられる. 実際, 領域と外力に対称性等の特別な整合条件を仮定しない限り, この線形問題の非自明解は空間遠方で対数増大することが知られている. しかし, ごく最近 Hishida (2015) による新たな展開があり, 2 次元回転物体の周りを流れる時間周期的ストークス流の漸近挙動が明らかにされた. 漸近展開の主要項が空間遠方で一次減衰する軸対称旋回流の構造を含んでおり, これは回転の効果によるストークスのパラドックスの解消を意味する. 本講演では, Hishida によって得られたこの線形評価を拡張し応用することにより, 物体の回転角速度が小さい場合に, 2 次元回転物体の周りを流れる時間周期的ナヴィエ・ストークス流の存在定理及び, 解の空間遠方での漸近挙動が得られたことを報告する. 本研究は前川泰則氏 (京都大学), 中原悠氏 (東北大学) との共同研究に基づく.

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.

本講演では, 周期境界条件下において mKdV 階層の一つである5次 mKdV 方程式に対して初期値問題を取り扱い, 低い正則性での時間局所的適切性 (LWP) を考える. 津川光太郎氏 (名古屋大学) との共同研究において, 逐次近似法が適用できる臨界の Sobolev 指数である s=3/2 においてLWPが成立することを示した. この結果を拡張するためには, 最も特異性の強い共鳴項を評価することが重要になる. そこで, Takaoka-Tsutsumi (2004) による発展作用素を修正することで共鳴項の特異性を和らげるという手法と Bejenaru-Tao (2006) による強い非線形相互作用が集中する部分の重みを修正した Fourier 制限ノルムを構成するという手法を組み合わせることで, 解のアプリオリ評価を導き, s>5/4 で LWP が得られたことを述べる.

The first part of the talk will be devoted to background and motivation for the classical result (1966) of H. Fujita about finite time blowup and global existence of solutions to a nonlinear heat equation. After stating Fujita’s result, I will sketch part of the proof. I will then describe some more recent research which was motivated and inspired by Fujita’s result, concluding with some current results and perspectives for continued work. I emphasize that this talk is designed for a general mathematical audience.

Stein-Weiss (1958) は fractional integral に関する重みつき評価を得た. 重みに関する条件は最良である. De Naploi, Drerichman and Duran (2011) は radial function に限るとこれを改良できることを証明した. 彼らの証明は長く難しい. 彼らの結果に別証明を与え, その方法によると multilinear fractional integral にも適用可能で, その応用として Caffarelli-Kohn-Nirenberg inequality の bilinear 版を得た. これは Moen (2009) の結果を radial function の場合に改良している.

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.