2018年度秋季総合分科会(於:岡山大学)

SUMMARY: The classical hypergeometric functions here include Appell’s hypergeometric functions \(F_1, F_2, F_3, F_4\), Lauricella’s hypergeometric functions \(F_A, F_B, F_C, F_D\), and the generalized hypergeometric function \({}_{n+1}F_{n}\). The corresponding differential equations will be denoted by \(E_1, E_2, E_3, E_4, E_A, E_B, E_C, E_D\), and \({}_{n+1}E_{n}\).

We talk about the recent development of the study on the systems of differential equations associated with the classical hypergeometric functions and their related topics —monodromy representations, connection formulas, Wronskian formulas, reducibility and irreducibility, invariant Hermitian forms, rigid Fuchsian systems, integral representations of solutions, regularizations of twisted cycles, intersection forms, Even-Odd family, Yokoyama-list, Dotsenko–Fateev equation, Knizhnik–Zamolodchikov equations, twisted de Rham theory etc.

SUMMARY: We consider a semilinear elliptic equation with a dynamical boundary condition in a unbounded domain \(\Omega \). In this talk we first consider the case \(\Omega \) is the \(N\)-dimensional half-space and discuss results on existence, nonexistence and large time behavior of small positive solutions. Furthermore, we show that local solvability of this problem is equivalent to global solvability of this problem and solvability of the stationary problem. Next we consider the case \(\Omega \) is the exterior of the unit ball and discuss the effects of the change of the domain. Finally, as preparation for nonlinear problem of diffusion equations, we consider the large diffusion limit for the heat equation on a half-space with a dynamical boundary condition.
This talk is based on series of joint works with M. Fila and K. Ishige.

SUMMARY: In recent years, certain multilinear convolution estimates have arisen and played an important role in the development of the well-posedness theory of Zakharov systems. In 2008, Bejenaru, Herr and Tataru observed that if one is given three transversal hypersurfaces in \(\mathbb {R}^3\), then the convolution of two \(L^2\) functions supported on two of these hypersurfaces has a well-defined restriction to the third hypersurface as an \(L^2\) function. For three coordinate hyperplanes, this fact follows from the Loomis–Whitney inequality in \(\mathbb {R}^3\), which is not difficult to prove. However, curved hypersurfaces cause substantial difficulty, and one is led to the nonlinear Loomis–Whitney inequality first proved by Bennett, Carbery and Wright in 2004. In 2009, Bejenaru, Herr, Holmer and Tataru used quantitative forms of such singular convolution estimates in their work on the Zakharov system in two spatial dimensions, where a local well-posedness result was established for large data in very low regularity Sobolev spaces. Similar use of multilinear singular convolution estimates has appeared in work of Bejenaru and Herr on the three-dimensional Zakharov system (2010), and more recently in work of Kinoshita on a Klein–Gordon–Zakharov system (2016) and Hirayama and Kinoshita on a system of quadratic derivative nonlinear Schrödinger equations (2018). This talk will include discussion of recent progress on the nonlinear Brascamp–Lieb inequality which substantially extends earlier work on the nonlinear Loomis–Whitney inequality.

SUMMARY: The stochastic Gross–Pitaevskii equation is used as a model to describe Bose–Einstein condensation at positive temperature. The equation is a complex Ginzburg–Landau equation with a trapping potential and an additive space-time white noise. In this talk, we present an answer to two important questions for this system: the global existence of solutions in the support of the Gibbs measure, and the convergence of those solutions to the equilibrium for large time. In order to prove the convergence to equilibrium, we use the associated purely dissipative equation as an auxiliary equation, for which the convergence may be obtained using standard techniques. Global existence is obtained for all initial data, and not almost surely with respect to the invariant measure.

SUMMARY: Poincaré investigated relationships between analytic properties of a system of \(q\)-difference equations and that of its solutions. In this talk, we extend the relationships to a system of functional equations with more general operators called convergent triangular operators, which act on formal power series as infinite triangular matrices.

SUMMARY: We consider a system of linear linear partial differential equations with constant coefficients and commensurate time lags (D\(\Delta \)-system) and show that compatibility condition is the necessary and sufficient for the solvability of the system. Moreover, we show that solutions of a homogeneous D\(\Delta \)-system can be approximated by exponential polynomial solutions. The proof is essentially reduced to a problem in complex harmonic analysis: division with bounds. The key technique is the division formula of M. Andersson whose integration kernel is expressed by residue currents. Another important tool is the use of a non-Nötherian ring of D\(\Delta \)-operators introduced by H. Glüsing-Lürßen.

SUMMARY: Regular GKZ hypergeometric functions are known to have Euler integral representations. In this talk, we give a basis of twisted cycles under a generic assumption of parameters with the aid of regular triangulations. Since this basis can easily be related to a basis consisting of \(\Gamma \)-series, we can also determine intersection matrix when the regular triangulation is unimodular.

SUMMARY: We study the eigenvalues of the self-adjoint Zakharov–Shabat operator corresponding to the defocusing nonlinear Schrödinger equation in the inverse scattering method. Real eigenvalues exist when the square of the potential has a simple well. We derive two types of quantization condition for the eigenvalues by using the exact WKB method, and show, moreover, that the eigenvalues stay real for a sufficiently small non-self-adjoint perturbation when the potential has some \(\mathcal {PT}\)-like symmetry.

SUMMARY: This talk deals with a two-dimensional rational difference system which is related to the Riccati difference equation. The purpose of this talk is to give a representation formula of solutions and to classify global behavior of solutions when no initial values belong to the forbidden set of the system.

SUMMARY: Let \(C\) be a given smooth planar curve, and let us roll a unit circle around it. Then a point on the perimeter of the circle traces a new curve \(G\). We call such \(G\) a generalized cycloid generated by \(C\). We consider the following inverse problems: (i) what kind of planar curves are realized as generalized cycloids? and (ii) for a given curve \(G\) how can we find the curve which generates \(G\) as a generalized cycloid?

SUMMARY: We consider minimization problems associated with best constants of the generalized critical Hardy inequalities. Especially, we consider an open problem mentioned by Horiuchi and Kumlin in 2012, and give a partial answer to it. Note that we can not apply the rearrangement technique since our potential functions are not radially decreasing in general.

SUMMARY: We report on the Hardy–Leray and the Rellich–Leray inequalities with best constants for curl-free vector fields. Costin–Maz’ya proved the sharp Hardy–Leray inequality for axisymmetric divergence-free vector-fields. In two dimensional case the optimal constant coincides with that for curl-free vector fields. Our aim is to see how the best constant changes in higher dimensions if the condition of divergence-free is replaced by that of curl-free.

SUMMARY: We study Hardy’s inequality in a limiting case on a bounded domain \(\Omega \) in \(\mathbb {R}^N\) with \(R = \sup _{x \in \Omega } |x|\). We study the attainability/non-attainability of the best constant in the inequality in several cases.

SUMMARY: The best constant of the Sobolev inequality in the whole space is attained by the Aubin–Talenti function, but not in bounded domains since the dilation invariance is breaking. We investigate a new scale invariant form of the Sobolev inequality in a ball, and show that its best constant is attained by Aubin–Talenti type functions.

SUMMARY: In this talk, we consider ground states for a class of quasilinear Schrödinger equation. We give the precise asymptotic behavior of ground states when the nonlinearity has \(H^1\)-critical growth.

SUMMARY: In this talk we study a functional associated with the mean filed limit of the point vortex distribution, that is, \begin{equation*} J_{\lambda }(v)=\frac {1}{2}\|\nabla v\|_2^2-\lambda \int _{I_+}\log \Big (\int _{\Omega } e^{\alpha v} dx \Big )\mathcal P(d\alpha ), \quad v \in H_0^1(\Omega ) \end{equation*} where \(\lambda >0\) is a constant, \(\Omega \subset \mathbb R^2\) is a smooth bounded domain and \(\mathcal P(d\alpha )\) is a Borel probability measure on \(I_+=[0, 1]\). We show the boundedness of \(J_{\lambda }\) from below with the extremal case for \(\lambda \) when \(\mathcal P(d\alpha )\) is continuous and satisfies the suitable assumptions.

SUMMARY: Generalized J-integral is the tool which is effective to study the shape optimization of singular points (containing boundary) with respect to various cost functions in boundary value problems for partial differential equations. I will talk shape sensitivity analysis of the energy using the sensitivity analysis of saddle point and propose Generalized J-integral in Stokes problem.

SUMMARY: The uniqueness of solutions for a given initial value problem is one of the fundamental problems in the theory of differential equations. For ordinary differential equations without time lag, Okamura gave a necessary and sufficient condition for the uniqueness by using the so-called “Okamura’s distance function.” The important fact is the following: the function is continuously differentiable, and therefore, the condition for the uniqueness is given by information of the differential equation. In this talk, I give a condition for Okamura’s distance function for delay differential equations in view of a unified theory of well-posedness by prolongations given by the author. A concrete function is also given under the assumption of local Lipschitz about \(C^1\)-prolongations. We also consider the relationship with Winston’s uniqueness claim.

SUMMARY: In this talk, we deal with a system of linear differential equations with discrete delays. We derive the necessary and sufficient conditions concerned with the absolutely stable for the stability problem of the system.

SUMMARY: We study the eigenvalue problem of the second order elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Therefore, analyzing the properties of the eigenvalue we can retrieve information on the frequency of the oscillations. Particularly, we deal with a thin rod with non-uniform connected cross-section in several cases of boundary conditions. We see that there appear many small eigenvalues corresponding to the bending mode of vibrations of the thin body. We investigate the asymptotic behavior of these eigenvalues and obtain a characterization formula of the limit equation when the thinness parameter tends to 0.

SUMMARY: This talk is concerned with \(L^q\)-Lyapunov inequality for the one-dimensional \(p\)-Laplacian. We have known the best inequalities for \(q=1\) or \(p=2\), and we will deal with the case \(q>1\) and \(p \neq 2\). In particular, we will calculate the best constant of this inequality by the generalized trigonometric functions with two parameters.

SUMMARY: We study the asymptotic behavior of bifurcation diagrams of nonlinear ordinary differential equations, which contain some oscillatory nonlinearities. In our case, \(\lambda \) is a continuous function of the maximum norm \(\alpha = \Vert u_\lambda \Vert _\infty \) of the solution \(u_\lambda \) associated with \(\lambda \). So we write \(\lambda = \lambda (\alpha )\). In this talk, we consider the case where \(\lambda (\alpha ) \to \pi ^2/4\) as \(\alpha \to \infty \). Then the precise asymptotic formulas for \(\lambda (\alpha )\) as \(\alpha \to \infty \) and \(\alpha \to 0\) with the exact second terms are established. By these formulas, the global and local structures of the bifurcation curves are clearly understood.

SUMMARY: In this talk, we shall study intersection and separation properties of solutions to semilinear elliptic equations. In particular, in order to investigate the properties of solutions, we make use of the stability of solutions. Indeed, for unstable solutions, including non-radial solutions, we obtain an intersection property of solutions. Moreover, focusing stable radial solutions, we derive a separation property of radial solutions. Furthermore, applying the result on the intersection property of solutions, we shall show a Liouville-type result for positive solutions.

SUMMARY: The bifurcation problem of positive solutions for the Moore–Nehari differential equation \(u''+h(x,\lambda )u^p=0\) in \((-1,1)\) with \(u(-1)=u(1)=0\) is considered, where \(p>1\), \(h(x,\lambda )=0\) for \(|x| < \lambda \) and \(h(x,\lambda )=1\) for \(\lambda \le |x| \le 1\) and \(\lambda \in (0,1)\) is a bifurcation parameter. The problem has a unique even positive solution \(U(x,\lambda )\) for each \(\lambda \in (0,1)\). It is shown that there exists a unique \(\lambda _*\in (0,1)\) such that a non-even positive solution bifurcates at \(\lambda _*\) from the curve \((\lambda ,U(x,\lambda ))\), where \(\lambda _*\) is explicitly represented as a function of \(p\).

SUMMARY: We study positive singular solutions to the Dirichlet problem for the semilinear elliptic equation in the unit ball. We first show the uniqueness of the singular solution to the problem, and then study the existence of the singular extremal solution. In particular, we show a necessary and sufficient condition for the existence of the singular extremal solution in terms of the weak eigenvalue of the linearized problem.

SUMMARY: In this talk, we consider stationary solutions of the one-dimensional Schnackenberg model with heterogeneity. We are interested in the effect of the heterogeneity on the stability. First we construct symmetric one-peak stationary solutions. Furthermore, we give stability analysis of the solutions in details and reveal the effect of heterogeneity on the stability.

SUMMARY: The starting point is the gradient flow of non-locally interacting particles. Our main result is the rigorous limit passage as the number of particles goes to infinity. The limiting model is a non-local and nonlinear PDE. The proof relies on the theory of Wasserstein gradient flows and embedding results of Orlicz spaces.

SUMMARY: The global Hölder continuity estimates on \(L^p\)-viscosity solutions of bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of \(L^p\)-viscosity solutions is obtained for continuous obstacles. The local Hölder continuity estimates on the first derivatives of \(L^p\)-viscosity solutions is shown when the obstacles belong to \(C^{1,\alpha }\), and \(p>n\).

SUMMARY: We discuss a stationary diffusive logistic equation in an interval. In particular, this talk focuses on Ni’s conjecture that the supremum of the ratio of a total population to total resources is 3. Recently, Bai–He–Li settled the conjecture by finding a special sequence of diffusion coefficients and carrying functions. The main question is the following: What is a profile of the solution corresponding to the special sequences? We give a partial answer to this question.

SUMMARY: This talk is concerned with the Neumann problem of a shadow system for the Lotka–Volterra competition model with cross-diffusion (the so-called SKT model). We study the global bifurcation structure of positive solutions. Among other things, we show that a bifurcation curve blows up as a parameter approaches the least positive eigenvalue of the minus Laplacian with homogeneous Neumann boundary condition.

SUMMARY: We will study a free boundary problem of the nonlinear diffusion equations of the form \(u_t=u_{xx}+f(u)\), \(t>0\), \(ct<x<h(t)\), where \(f\) is \(C^1\) function satisfying \(f(0)=0\), \(c>0\) is a given constant and \(h(t)\) is a free boundary which is determined by a Stefan condition.

When \(f(u)=u(1-u)\) (logistic nonlinearity), this problem was considered by [Matsuzawa, 2018].

In this talk, we will extend the results in the previous work to general monostable, bistable and combustion types nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in the previous work.

SUMMARY: We consider a free boundary problem for a reaction-diffusion equation modeling the spreading of species, where unknown functions are population density and spreading front of species. Moreover, the dynamical behavior of the free boundary is determined by a Stefan-like condition. The purpose of this talk is to show sharp estimates of the asymptotic profile of solutions and the spreading speed of the free boundary.

SUMMARY: In this study, we formulate an SIR epidemic model with nonlocal diffusion and study the asymptotic behavior of it. We define the basic reproduction number \(R_0\) by the spectral radius of the next generation operator, and show that the trivial equilibrium is globally asymptotically stable if \(R_0<1\). Furthermore, under an additional assumption that only infective individuals can diffuse, we prove that a positive equilibrium is globally asymptotically stable if \(R_0 > 1\). This is a collaborative work with Dr. Jinliang Wang in Heilongjiang University.

SUMMARY: The purpose is to start understanding from a mathematical viewpoint a famous regularized structures with spatially distinct bands or rings of precipitated material are exhibited, with clearly visible scaling properties. Such patterns are known as Liesegang bands or rings. In this paper, we study a one-dimensional version of the Keller and Rubinow model and present conditions ensuring the existence of Liesegang bands.

SUMMARY: A mathematical standard structure of a binary digit of memory in a cell is presented. This is based on a kind of frequency model with scale effect. This model has ability of “on-off” switching property, and moreover, this is affected by scale effect to make the memorable ability be reinforced. This property is derived from multiple covalent modification cites inducing important enzyme reaction, which is represented by the Michaelis–Menten type nonlinearity.

[1] “Standard model of a binary digit of memory with multiple covalent modifications in a cell”, J. Pur Appl Math Vol 2 No 1 April (2018) [2] “Memory Reinforcement with Scale Effect and its Application to Mutual Symbiosis among Terrestrial Cyanobacteria of Nostochineae, Feather Mosses and Old Trees in Boreal Biome in Boreal Forests”, Global Science Chronicle, Vol 1(1): 1–7 (2017)

SUMMARY: We study the asymptotic behaviors and quenching of the solutions to the shadow system of a two-component reaction-diffusion system modeling prey-predator interactions in an insular environment. First, we give a global existence result for the solutions. Then, by constructing some suitable Lyapunov functionals, we characterize the asymptotic behaviors of global solutions. Also, we give a finite time quenching result.

SUMMARY: We study an initial boundary value problem for a reaction-diffusion system arising in the study of a singular predator-prey system. First, convergence to a stationary solution of global solution under some parameter condition is given. Next, under an assumption on the growth rates and initial data, we show that the unique co-existence state is a center for the kinetic system by the method of Darboux’s theory of algebraic integrability. Then we prove that solutions of the diffusion system with equal diffusion become spatially homogeneous and are subject to the kinetic part asymptotically.

SUMMARY: For a balanced bistable reaction-diffusion equation, axially symmetric traveling fronts have been recently studied. For this equation, the existence of axially non-symmetric traveling fronts with non-zero speed has been an interesting open problem. This work proves that axially non-symmetric traveling fronts with non-zero speed exist in balanced bistable reaction-diffusion equations.

SUMMARY: We consider the strong solutions of the Cahn–Hilliard equations in a bounded domain with permeable walls, which is a dynamic boundary conditions. From the maximal \(L_p\) regularity result of the linear equations with the dynamic boundary conditions, the fixed point theorem and a priori estimate, we prove that the solution exists uniquely and globally in time.

SUMMARY: We consider the sharp interface limit of the Allen–Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the assumption that the discrepancy measure vanishes on the boundary. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

SUMMARY: Several inequalities concerning the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. These hold only for functions representing closed plane curves, but give better estimates than Gadliardo–Nirenberg inequalities. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption. We do not assume convexity of the initial curve, instead we assume global existence of the flow. A global solution of non-local curvature flow considered by Jiang–Pan converges exponentially to a circle. The same result still holds for the area-preserving curvature flow and the length-preserving curvature flow.

SUMMARY: We consider critical dissipative estimates for a heat semigroup with the inverse square potential. It is proved by Metafune–Okazawa–Sobajima–Spina that the Schrödinger operator generates a semigroup under suitable restriction on its domain. Ioku–Metafune–Sobajima–Spina proved decay estimates of the semigroup in Lebesgue spaces. However the endpoint case is remained. In this talk we use Lorentz spaces instead of Lebesgue spaces, and show the endpoint decay estimate in a suitable Lorentz space.

SUMMARY: Let \(L:=-\Delta +V\) be a nonnegative Schrödinger operator on \(L^2({\bf R}^N)\), where \(N\ge 2\) and \(V\) is a radially symmetric inverse square potential. In this paper we assume either \(L\) is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of \(e^{-tL}\varphi \), where \(\varphi \in L^2({\bf R}^N,e^{|x|^2/4}\,dx)\).

SUMMARY: In my talk, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem \begin{equation*} \partial _t u+(-\Delta )^{\frac {\theta }{2}}u=u^p,\qquad x\in {\bf R}^N,\,\,t>0,\\ \qquad u(0)=\mu \ge 0\quad \mbox {in}\quad {\bf R}^N, \end{equation*} where \(N\ge 1\), \(0<\theta \le 2\), \(p>1\) and \(\mu \) is a Radon measure or a measurable function in \({\bf R}^N\). Our necessary conditions and sufficient conditions lead the strongest singularity of the initial data for the solvability.

SUMMARY: I will report recent results on blow-up of a semilinear heat equation \(u_t =\Delta u + u^p\) with \(p>1\), focusing on the supercritical nonlinearity in the Sobolev sense. Blow-up of solution is called of Type I if the blow-up rate is the same as of the self-similar solutions and of Type II otherwise. The previous results on classification of Type II blow-up solutions is restricted in the case where a certain linearized operator doesn’t have neutral eigenvalues due to the lack of a particular solution whose blow-up is driven by a neutral eigenvalue. Based on matched asymptotics, we prove the existence of particular solutions associated to a neutral eigenvalue in the most representative case where \(p = 1 +6/(N-10)\), so called, Lepin exponent.

SUMMARY: The initial value problem for the quasi-geostrophic equation is studied. Particularly, the critical and the super-critical dissipation is considered. For those problems, far field asymptotics of solutions are given.

SUMMARY: In this talk we consider a Keller–Segel-fluid system with singular sensitivity. In the system existence of global classical solutions was shown under some conditions in the previous work (Black–Lankeit–M., J. Evol. Equ.). In this talk we address the question, what conditions derives existence of global generalized solutions, which is one of weak concepts?

SUMMARY: We show the existence of global-in-time solutions depending on the initial data in the Morrey space to the Keller–Segel system of parabolic-parabolic type. We solve this system by the successive approximation based on the estimation of the heat semigroup in the Morrey space and on the fractional power of the Laplace operator.

SUMMARY: We show the solutions to the drift-diffusion equation with initial data in homogeneous Besov space is in \(L^p\). Moreover it is analytic by use of Hölder condition of solutions

SUMMARY: We consider large time behavior of weak solutions to a degenerate drift-diffusion system related to Keller–Segel system with the mass supercritical cases under relaxed weight condition. It is known that the large time behavior of solutions is classified by the invariant norms of initial data. For the mass supercritical case, the sufficient condition for unboundedness is given by Kimijima–Nakagawa–Ogawa with the initial data decaying fast at spacial infinity. In this talk, we prove unboundedness of solutions to our problem with more slowly decaying initial data under same assumptions in Kimijima–Nakagawa–Ogawa.

SUMMARY: We study the large time asymptotics of solutions to the Cauchy problem for the generalized Korteweg–de Vries–Burgers–Kuramoto equation where the far field states are prescribed. Especially, we deal with the case when the convective flux is fully convex. Then the Cauchy problem has a unique global in time solution which tends toward a rarefaction wave as time goes to infinity. The proof is given by a technical energy method.

SUMMARY: We study the decay structure of solutions to the Cauchy problem for a one-dimensional scalar conservation law with nonlinear viscosity where the far field states are prescribed. Especially, we deal with the case when the convective flux is fully convex, and also the viscosity is a nonlinearly degenerate one. Then the Cauchy problem has a unique global in time solution which tends toward a rarefaction wave as time goes to infinity. We investigate that the decay rate in time of the corresponding solutions. The proof is given by a technical time-weighted energy method.

SUMMARY: We consider the energy estimate of the solution to the Cauchy problem of Klein–Gordon type equation with time dependent mass \(M(t)\), in particular \(M(t)\) has a singularity. The main purpose of this paper is to give sufficient conditions to \(M(t)\) for the energy to be asymptotically stable.

SUMMARY: We obtain asymptotic profiles of solutions to the linear strongly damped wave equations in \(\textbf {R}^n\) \((n\ge 1)\) by expanding evolution operators in the low frequency regions in the Fourier space.

SUMMARY: We are focusing on the upper bound of the lifespan of solutions of nonlinear wave equations with the scattering damping. All the result should be wave-like, namely the same as non-damped case. In this talk, I will introduce new results which cover most part of the expected one in semilinear case.

SUMMARY: In this talk, we consider sharp upper bounds of lifespan of solutions to the semilinear damped wave equation with nonlinearity \(|u|^p\). Here we introduce an “improved” test function method to derive sharp upper bounds for the critical case \(p=1+2/N\). Several applications of this technique will be also discussed.

SUMMARY: In this talk, we consider global existence and blowup of solutions to the semiliear heat equation with the nonlinearity \(u^p\) in two-dimensional exterior domain. We give sharp lifespan estimates of blowup solutions in the case of two-dimensional exterior domain, which is quite different from that in the case of \(N\geq 3\).

SUMMARY: We study one dimensional wave equation \(\partial _t^2 u - \epsilon ^2\partial _x^2 u = F(\partial _t u)\). The solution blows-up on a curve \(t = T_{\epsilon }(x).\) We define the blow-up curve corresponding to \(v''=F(v')\) by \(t = T(x)\). It is proven that \(|T_\epsilon (x) - T(x)|\rightarrow 0\) as \(\epsilon \rightarrow 0\).

SUMMARY: We prove global-in-time Strichartz estimates for the Schrödinger equation with positive real-valued potentials which decay slowly and satisfy a virial type condition at infinity. The repulsive Coulomb potential in three space dimensions is particularly included in our admissible class of potentials.

SUMMARY: We study the strong instability of standing waves \(e^{i\omega t}\phi _\omega (x)\) for nonlinear Schrödinger equations with an attractive inverse power potential, where \(\omega \in \mathbb {R}\) and \(\phi _\omega \in H^1(\mathbb {R}^N)\) is a ground state of the corresponding stationary equation. In this talk, we prove that if \(\partial _\lambda ^2S_\omega (\phi _\omega ^\lambda )|_{\lambda =1}\le 0\), then the standing wave is strongly unstable, where \(S_\omega \) is the action, and \(\phi _\omega ^\lambda (x):=\lambda ^{N/2}\phi _\omega (\lambda x)\) is the \(L^2\)-invariant scaling.

SUMMARY: We consider the Cauchy problem of the nonlinear fourth order Schrödinger equations with derivative nonlinearity, which contains second derivative with respect to the spatial variable. The well-posedness of this equation in the Sobolev space \(H^s\) is obtained by Hao, Hisao, and Wang (2006) for general settings. We improve this result by using the maximal function norm introduced by Pornnopparath (preprint). Our results contain the well-posedness of the model of the vortex filament treated by Segata (2003, 2004).

SUMMARY: We consider the nonlinear Schrödinger system with quadratic interaction in five dimensions. We determine the long time behavior of the solutions to this system with data below the ground state. More precisely, we give conditions for the solutions scatter and conditions for the solutions blow-up or grow-up. Under the condition blowing-up or growing-up, if we additionally assume some conditions, we can prove that the solutions blow-up.

SUMMARY: In this talk, we prove that the third order Benjamin–Ono equation on the torus \[\partial _{t} u-4\partial _{x}^{3}u+3u^{2}\partial _{x} u-3H\partial _{x}(u\partial _{x} u)-3\partial _{x}(uH\partial _{x} u)=0, \quad x\in \mathbb {T}\] is locally well-posed in \(H^{s}(\mathbb {T})\) for \(s\in \mathbb {N}\) and \(s\ge 3\). Here, \(H\) is the Hilbert transform. The above equation arises from the Benjamin–Ono hierarchy. Nonlinearities in the equation may yeild two derivative losses, which is one difficulty of this problem. The proof is based on the energy method, adding a correction term into the energy so as to overcome the difficulty.

SUMMARY: We consider the Cauchy problem of a higher-order KdV-type equation with critical nonlinearity in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we show that the long-time behavior of the solution is determined by a self-similar solution.

SUMMARY: We consider the initial value problem for the generalized KdV equation with lower degree of nonlinearity than that of the KdV equation. The nonlinearity of our equation is non-Lipschitz continuous in the Sobolev spaces or the weighted Sobolev spaces which are basically considered as the class of solutions, so that the local well-posedness can not be established in these classes. In this talk, we prove the local well-posedness in an appropriate class, and give properties for the propagation of regularity in the right hand side of the data for positive times of real valued solutions in the class.

SUMMARY: We consider the Zakharov–Kuznetsov equation on a cylindrical space which is one of a high dimensional generalization of KdV equation. The orbital and asymptotic stability of the one soliton of KdV equation on the energy space was proved by Benjamin’72, Pego and Weinstein’92, Martel and Merle’01. We regard the one soliton as a line solitary wave of Zakharov–Kuznetsov equation on a two dimensional space. In the case of the cylindrical space, I showed the stability of the line solitary wave with the traveling speed less than the critical speed \(c_*\) and the instability with the traveling speed larger than \(c_*\). In this talk, we show the existence of a center stable manifolds around unstable line solitary wave.

SUMMARY: We consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension \(d=2\) and \(3\). This system was introduced by M. Colin and T. Colin (2004). The well-posedness of this system in the Sobolev space \(H^s\) was obtained by Hirayama (2014). But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using a nonlinear version of the classical Loomis–Whitney inequality and prove well-posedness in \(H^s\) for \(s \geq 1/2\) if \(d=2\), and \(s>1/2\) if \(d=3\).

SUMMARY: We consider the Cauchy problem of quasilinear thermoelastic Kirchhoff-type plate equations where the heat conduction is modeled by either the Cattaneo law or by the Fourier law. Additionally, we take into account possible inertial effects. The main task consists in proving sophisticated a priori estimates leading to obtaining the global existence of solutions for small data.

SUMMARY: To analyze the dissipative structure of the regularity-loss type, we need more concrete examples and have to study more detailed properties. To this end, we consider the Cauchy problem for a couple of wave and heat equations as one concrete example in this talk.

SUMMARY: This talk is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. We had already known the new dissipative structure called the regularity-loss type. Compared with the dissipative structure of the standard type, the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Under this situation, we introduce new concepts and extend our previous results to cover complicated models.

SUMMARY: In this talk, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition for the corresponding eigenvalue problem and derived the detailed relation between the coefficient matrices of the system and the eigenvalues. However, there are some complicated physical models which possess a non-symmetric viscosity or relaxation term and we can not apply this condition to these models. Under this situation, our purpose is to extend classical stability condition for complicated models.

SUMMARY: We will talk on the hydrostatic approximation on the Primitive equation and its justification from the Navier–Stokes equations. First we will introduce some previous results on the Primitive equation and the Navier–Stokes equations. After that we will show our results and essence of the proof.

SUMMARY: We consider the diffusion and heat systems on an evolving surface with boundaries from an energetic point of view. We employ an energetic variational approach to derive our diffusion and heat systems on the evolving surface. Moreover, we study the boundary conditions for our systems to investigate both conservation and energy laws of the systems.

SUMMARY: We consider compressible fluid flow on an evolving surface with boundaries from an energetic point of view. We employ both an energetic variational approach and the first law of thermodynamics to derive our compressible fluid systems on the evolving surface. Moreover, we investigate the boundary conditions in co-normal direction for our system to study conservation and energy laws of the system.

SUMMARY: We consider the dominant equations for the motion of the non-Newtonian fluid in a domain from an energetic point of view. A non-Newtonian fluid is a fluid that does not follow Newton’s law of viscosity. In this lecture, we focus on several energies dissipation due to the viscosities of the non-Newtonian fluid to study the dominant equations for the motion of the fluid. We apply our energetic variational approaches to make a mathematical model of the non-Newtonian fluid flow. Although the system is abstract, our results make it possible to give some previous models of non-Newtonian fluid flow to a mathematical validity.

SUMMARY: Density-dependent magnetohydrodynamics system describes the coupling between the density-dependent Navier–Stokes equation and the Maxwell equation. For the Euler equation, Beale–Kato–Majda proved the regularity criterion for the local solutions. That is a smooth solution of the Euler equation \(u\) in \(\mathbb {R}^3\) on \([0,T)\) is regular after \(t \ge T\), provided that \({\rm rot}\;u \in L^1(0,T;L^{\infty }(\mathbb {R}^3))\). In this talk, we study the Beale–Kato–Majda type criterion for the solution of density-dependent magnetohydrodynamics system.

SUMMARY: We report some result on existence of global weak solutions of the Euler equations in an infinite cylinder.

SUMMARY: We shall find the largest normed space (or the weakest norm) that satisfies the Brezis–Gallouet–Wainger type inequality. As an application of such an inequality, we shall establish the new a priori estimate of strong solutions to Navier–Stokes equations which has an almost single exponential growth form with respect to the scaling invariant quantity of the vorticity. Our method is based on the double logarithmic type inequality by means of the Morrey space.

SUMMARY: We study the two-dimensional stationary Navier–Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described by exhibiting a boundary layer structure and an axisymmetrization of the flow.

SUMMARY: We consider the stationary Navier–Stokes equations in \(\mathbb {R}^n\) for \(n\geq 3\) in the scaling invariant Besov spaces. It is proved that if \(n<p\leq \infty \) and \(1\leq q\leq \infty \), or \(p=n\) and \(2<q\leq \infty \), then some sequence of external forces converging to zero in \(\dot B^{-3+\frac {n}{p}}_{p,q}\) can admit a sequence of solutions which never converges to zero in \(\dot B^{-1}_{\infty ,\infty }\), especially in \(\dot B^{-1+\frac {n}{p}}_{p,q}\). Our result may be regarded as showing the borderline case between ill-posedness and well-posedness, the latter of which Kaneko–Kozono–Shimizu proved when \(1\leq p<n\) and \(1\leq q\leq \infty \).

SUMMARY: We consider the eigenvalue problem of the Stokes equations with slip boundary conditions. More precisely, under a smooth domain perturbation, we analyze the domain dependency of the multiple eigenvalue of the Stokes equations with such a boundary condition. We succeeded to prove the differentiability of the eigenvalues for the perturbation parameter, and to clarify the representation for that.

SUMMARY: In this talk, we would like to consider the Navier–Stokes–Korteweg equations on general domains under the non-slip boundary condition. We prove the local solvability of the Navier–Stokes–Korteweg equations in the maximal regularity class.

SUMMARY: In this talk, we consider the compressible fluid model of Korteweg type which was introduced by J. E. Dunn and J. Serrin in 1985. It is shown that the system admits a unique, global strong solution for small initial data in \(\mathbb R^N\), \(N \geq 2\). For the purpose, the main tools are the maximal \(L_p\)-\(L_q\) regularities and \(L_p\)-\(L_q\) decay properties to the linearized equations. This talk is based on a joint work with Professor Yoshihiro Shibata in Waseda University.

SUMMARY: We consider the initial value problem of the 3D inviscid Boussinesq equations for stably stratified fluids. We prove the long time existence of classical solutions for large initial data when the buoyancy frequency is sufficiently high. Furthermore, we consider the singular limit of the strong stratification, and show that the long time classical solution converges to that of 2D incompressible Euler equations in some space-time Strichartz norms.