| 日時 | 4月21日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | Priyanjana M.N. Dharmawardane 氏（Wayamba University of Sri Lanka） |
| 題目 | Decay property for the linearized systems of thermoviscoelasticity |
| 概要 | We consider the linearized hyperbolic-parabolic coupled systems, which describe heat-conductive materials having the properties of elasticity and viscosity. These systems contain the nonlinear viscoelastic systems and the total energy balance equation. In this talk, we present the results obtained for the decay property of the aforementioned systems. Appealing to the pointwise estimate in the Fourier space, we obtain decay estimates of solutions using the method developed by Prof. Kawashima, provided that the initial data are in \(L^2 \cap L^1\). Moreover, we obtain enegy estimate for linear solutions, decay property for solution operators and pointwise estimates of the fundamental solutions in the Fourier space. This talk is based on the joint research work with Professors S. Kawashima, T. Ogawa and J. Segata. |
| 日時 | 4月28日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 吉澤 研介 氏（九州大学） |
| 題目 | 一般化曲げエネルギーに対する障害物問題 |
| 概要 | グラフ曲線に, 障害物を表す既知函数を下回らないという外的束縛が加えられた条件の下, p-弾性エネルギーと呼ばれる高階幾何学的汎函数の最小化問題を考察する. 最小化問題の解は, (i) p-弾性エネルギーの Euler--Lagrange 方程式の特異性または退化性, (ii) 障害物の存在という, 正則性の損失が起きる二つの可能性を抱えている. そこで本講演では, (i), (ii) の二つの要因の内, どちらが影響として優位に働くかを見る. なお, 本講演の内容は Anna Dall'Acqua 氏 (Ulm University), Marius Müller 氏 (Freiburg University), 岡部真也氏 (東北大学) との共同研究に基づく. |
| 日時 | 5月12日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 坂本 祥太 氏（九州大学） |
| 題目 | Global solution to the Boltzmann equation without cutoff on the whole space in \((L^1\cap L^p)_k\) |
| 概要 | We consider a Cauchy problem of the Boltzmann equation without angular cutoff near the global Maxwellian. When a spatial domain is the torus, it is proved that the problem has a unique solution for small data in the Wiener space, the collection of functions whose Fourier coefficients are absolutely summable. Due to its Banach-algebra property, we do not need the Sobolev embedding in this space. Following this result, we consider the problem on the whole space. In this case, the control of the \(L^1\) norm on the Fourier side is not sufficient due to low-frequency terms. Therefore, inspired by Kawashima-Nishibata-Nishikawa's work on the viscous conservation laws, we employ the \(L^p\) norm estimates with respect to the frequency to control such parts. This \(L^1 \cap L^p\) strategy will close a priori estimates when combined with a time-weighted energy method. This work is based on a joint work with Renjun Duan (Chinese University of Hong Kong) and Yoshihiro Ueda (Kobe University). |
| 日時 | 5月19日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 平山 浩之 氏（宮崎大学） |
| 題目 | Existence and stability of the ground states to the system of nonlinear Schrödinger equations with derivative nonlinearity |
| 概要 | 本講演では, Colin-Colin (2004)によって導出された, 非線形項に１階の空間微分を含む非線形シュレディンガー方程式系(NLS系) の初期値問題について考える. NLS系の時間局所的適切性については先行研究によって詳しく調べられているが, 解の時間大域的な挙動についてはあまり得られていない. 特に孤立波の存在と安定性については, Colin-Colinにより未解決問題として指摘されている. 本講演では, NLS系の孤立波に関する定常問題に注目し, その基底状態の存在と安定性について得られた結果を紹介する. なお, 本講演は理化学研究所の池田正弘氏との共同研究に基づく. |
| 日時 | 5月26日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 千頭 昇 氏（名古屋工業大学） |
| 題目 |
藤田型方程式の符号変化解の無条件一意性に関する sharp threshold について
(Sharp threshold on the unconditional uniqueness of sign-changing solution for Fujita equation) |
| 概要 | 空間的に非一様な非線形項を持つ半線形放物型方程式である Hardy-Henon 方程式の符号変化解の無条件一意性について，新たに得られた結果を報告する．非線形項の冪を固定すると，無条件一意性が成立する Lebesgue 空間の臨界指数として，二つの重要な指数が現れる．これらは尺度臨界指数および Serrin 指数と呼ばれるものであり，それぞれ，解の局所存在のために初期値が許容できる局所特異性の限界と，非線形項が局所可積分になる限界を表す．単独の臨界指数以上では解の無条件一意性が成立し，これらの臨界指数が重なる二重臨界においては一意性が破綻する．本発表では，我々の結果を紹介した後，ポテンシャルがない藤田型方程式に焦点を絞り既存の結果を再解釈し，Lorentz 空間における補間指数が無条件一意性の sharp threshold として現れることを解説する．本発表は池田正弘氏 (理研)，谷口晃一氏 (東北大学)，Slim Tayachi 氏 (University of Tunis El Manar) との共同研究 (arXiv:2301.00506) に基づく． |
| 日時 | 6月2日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 上田 好寛 氏（神戸大学） |
| 題目 | Mathematical analysis for the viscous Burgers equation with time delay |
| 概要 | The viscous Burgers equation is well known as a simple equation describing fluid phenomena, and it is also known as a mathematical model of traffic flow. As a model of traffic flow, we consider the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay and analyze a delay effect. Indeed, because the viscous Burgers equation is a parabolic partial differential equation, its solution has an infinite speed of propagation. In terms of traffic flow, this means that drivers and their vehicles are assumed to react instantly changing the density and its gradient. To improve this troubling feature, we modify the term without a time delay to the one with a time delay. In this talk, we show the existence of the global-in-time solution when the product of the size of the delay parameter and the one of the initial history is suitably small. Moreover, we also prove some theorems concerning the regularity of the global-in-time solution. This result is joint research with Takayuki Kubo of Ochanomizu University. |
| 日時 | 6月9日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 西畑 伸也 氏（東京工業大学） |
| 題目 | Asymptotic stability of spherically symmetric stationary solutions for the compressible Navier-Stokes equation for outflow/inflow problems |
| 概要 | In the present talk, we discuss an asymptotic behavior of a spherically symmetric solution on the exterior domain of an unit ball for the compressible Navier-Stokes equation, describing a motion of viscous barotropic gas. Especially we study outflow problem, that is, the fluid blows out a through boundary. Precisely we obtain the property of the stationary solution and the convergence rate as the spatial variable tends to infinity. Then we show the time global existence of the solution and it converges to the stationary solution as time tends to infinity. Here any smallness assumption of initial data is not imposed. We also make mention of the stability theorem for inflow problem. |
| 日時 | 6月23日(金) 15:30--17:00 |
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| 会場 | 福岡大学18号館1823教室 |
| 講師 | 勝呂 剛志 氏（大阪公立大学） |
| 題目 | ある Keller--Segel 系の初期値問題の Wiener のアマルガム空間における適切性について |
| 概要 | 放物-楕円型 Keller--Segel 系は走化性粘菌の運動を記述する 放物-放物型 Keller--Segel 系を単純化したものであり, 移流項が楕円型偏微分方程式で与えられる, 非線形の連立偏微分方程式である. 移流項である第二式の解は第一式の解とある積分核を用いて表され, 第一式は非線形干渉項を擁する非局所拡散方程式であることがわかる. 第二式に現れるパラメータによって, 遠方において減衰しない函数を取り扱う一様局所可積分空間において, 初期値問題の適切性の検証が困難となる. ここでは, 一様局所可積分空間を含む, Wiener のアマルガム空間における 放物-楕円型 Keller--Segel 方程式系の初期値問題の適切性を示す. |
| 日時 | 6月30日(金) 15:30--17:00 |
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| 会場 | 九州大学 伊都キャンパス ウエスト1号館 C504小講義室 |
| 講師 | 赤木 剛朗 氏（東北大学） |
| 題目 | Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion in bounded domains |
| 概要 | This talk is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded Lipschitz domains. It is well known that every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile, which is a nontrivial solution for a semilinear elliptic equation. Bonforte and Figalli (CPAM, 2021) first proved an exponential convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted L^2 norm, which is weaker than the L^2 norm, for smooth (at least C^2) bounded domains by developing the so-called nonlinear entropy method. On the other hand, the speaker (ARMA, 2023) developed an energy method along with a quantitative gradient estimate and also proved the same exponential convergence in the Sobolev norm for bounded C^{1,1} domains. The optimality of the exponential rate was conjectured in view of some formal linearized analysis; however, it has never been proved so far due to some difficulty arising from nontrivial stability nature of asymptotic profiles in the fast diffusion setting. In this talk, these results are extended to possibly sign-changing asymptotic profiles as well as bounded Lipschitz domains by improving the energy method as well as quantitative gradient inequality. Moreover, a (quantitative) exponential stability result for least-energy asymptotic profiles follows as a corollary. Finally, the optimality of the exponential rate will also be proved. This talk is based on a joint work with Yasunori Maekawa (Kyoto University). |
| 日時 | 7月28日(金) 16:00--17:00 |
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| 会場 | 福岡大学18号館1824教室 |
| 講師 | J. López-Gómez 氏（Complutense University of Madrid） |
| 題目 | New trends in Lotka-Volterra diffusive competition |
| 概要 | This talk discusses several recent findings on the dynamics of the spatially-heterogeneous diffusive Lotka-Volterra competing species model. First, it delivers a general (optimal) singular perturbation result generalizing, very substantially, the pioneering theorem of Hutson, López-Gómez, Mischaikow and Vickers (1994) for their mutant model, later analyzed, very sharply, by W. M. Ni and his collaborators. Then, it establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, \(\Omega\), any steady state of the spatial counterpart perturbing from it therein as the diffusion rates separate away from zero must be linearly unstable. From this feature one can derive a number of rather astonishing consequences, as the multiplicity of the coexistence steady states when the non-spatial model exhibits founder control competition somewhere in \(\Omega\), say \(\Omega_{bi}\), even if \(\Omega_{bi}\) is negligible empirically. Actually, this is the first existing multiplicity result for small diffusion rates. Finally, based on the Picone identity, we can establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models. This result generalizes, very substantially, those of W. M. Ni and collaborators for the autonomous model. |