2019年度年会(於:東京工業大学)

SUMMARY: A survey of some recent results for the estimates of multilinear pseudo-differential operators in Lebesgue, Hardy, and BMO spaces will be given.

SUMMARY: In this talk I briefly overview the history of harmonic analysis on semisimple Lie groups, especially the case of real rank one, and the Jacobi hypergroup \(({\bf R}_+, \Delta , *)\). Then I introduce recent topics of singular integrals on the Jacobi hypergroup. We would like to generalize the Calderón-Zygmund theory for the Jacobi hypergroup. Actually, we shall obtain a CZ class on \({\bf R}_+\) such that, if a function \(g\) belongs to the CZ class, the convolution operator \(g*\) is bounded from \(L^p(\Delta )\) to itself for \(1<p \leq 2\). However, we have some obstacles. The case of \(SU(1,1)\) is excluded and a restriction on \(p\) is required.

SUMMARY: The classical total variation flow is the \(L^2\) gradient flow of the total variation. The variation of a function is a singular energy at the place where the slope of the function equals zero. Because of this structure, its gradient flow is actually nonlocal in the sense that the speed of slope zero part (called a facet) is not determined by infinitesimal quantity. Thus, the definition of a solution itself is a nontrivial issue even for the classical total variation flow.

Recently, there need to study various types of such equations. A list of examples includes the total variation map flow as well as the classical total variation flow and its fourth order version in image denoising, crystalline mean curvature flow or fourth order total variation flow of exponential type in crystal growth problems which are special important problems in materials science.

In this talk, we survey recent progress on these equations with special emphasis on finite extinction property and a crystalline mean curvature flow whose solvability was left open more than ten years. We shall give a global-in-time unique solvability in the level-set sense.

SUMMARY: The author introduced the oriented and non-oriented volumes in the \begin{math} C^0 \end{math} class in 2016, and revised the non-oriented volume in 2018. A “counter example” has been found to violate the property “the non-oriented volume exceeds the oriented volume”, with respect to the former definition. This talk gives an example similar to the above “counter example”. which becomes clear not to violate the above property, with respect to the latter definition. It suggests that the “counter example” turns an affirmative example.

SUMMARY: For a function \(f: [a, b] \longrightarrow \mathbb {R}\) and a point \(c \in [a, b])\), \(\displaystyle F(x) = \int _{c}^{x} f(t)dt\) is called the indefinite integral of \(f\) and a function \(G\) satisfying \(G' = f\) is called the primitive function of \(f\).

As well-known, if the function \(f\) is continuous, then \(F' = f\) holds everywhere (fundamental theorem of calculus). Therefore, ignoring the difference in constants, \(G = F\) holds.

In this talk, we consider in the case where \(f\) is not necessarily continuous, moreover, \(G\) is not continuous.

SUMMARY: In this talk, we study \(L^p\) spaces over finitely additive measures. For a given finitely additive measure \(\mu \), we give a method of extending the space \(L^p(\mu )\). The completeness of such an extended \(L^p\) space is also discussed.

SUMMARY: The supremum increment of a nonadditive measure \(\mu \) on a \(\sigma \)-field \(\mathscr F\) is the nonadditive measure \(^\Delta \mu \) defined by \(^\Delta \mu (A) = \sup \{\mu (A\cup B)-\mu (B)\mid B\in \mathscr {F},\ \mu (B)<\infty \}\) for \(A\in \mathscr F\). If \(\mu \) is null-additive, then the null-continuity of \(\mu \) is equivalent to the null-continuity of \(^\Delta \mu \). Generally, even if \(\mu \) is null-continuous, \(^\Delta \mu \) is not necessarily null-continuous; even if \(^\Delta \mu \) is null-continuous, \(\mu \) is not necessarily null-continuous. If \(\mu \) is null-additive and has property (S), then \(^\Delta \mu \) has property (S). If \(\mu \) is uniformly autocontinuous, then \(\mu \) has property (S) iff \(^\Delta \mu \) has property (S). Generally, even if \(\mu \) has property (S), \(^\Delta \mu \) does not necessarily have property (S); even if \(^\Delta \mu \) has property (S), \(\mu \) does not necessarily have property (S).

SUMMARY: Let \(([0, +\infty ], \oplus , \otimes )\) be the generalized ring. Let \((X, \mathcal {B})\) be a measurable space and \(\mu : \mathcal {B} \to [0, \infty ]\) be a set function satisfying \(\mu (\emptyset )=0\) (the non-additive measure). An extension \(E(f;\mu ) : \mathcal {F} \to [0, \infty ]\) of \(\mu \) is a mapping \(E(f;\mu ) : \mathcal {F} \to [0, \infty ]\) such that \(E(\chi _A;\mu ) = \mu (A), \ A \in \mathcal {B}\), where \(\chi _A\) is the characteristic function, \(\mathcal {F}\) is the set of all \([0, e]\) valued measurable functions and \(e\) is the left unit for \(\otimes (e \otimes e = a)\). We shall define two extensions of \(\mu \). Those are considered as the non-linear integrals of \(f \in \mathcal {F}\) with respect to \(\mu \).

SUMMARY: To investigate the geometry of normed space, geometric constants play important roles. Among them the James constant has been studied by a lot of mathematicians. We also treat the generalized notions of orthogonality in normed space. The generalized orthogonality in normed space have been studied in many papers. The usual orthogonality in inner product space is symmetric. By the definition Isosceles orthogonality is symmetric, too. However, Birkhoff orthogonality is not symmetric in general. The two-dimensional space in which Birkhoff orthogonality is symmetric is called Radon plane. We consider the value of James constant in such planes.

SUMMARY: We study the asymptotic behavior of two iterative sequences for approximating minimizers of convex functions in complete geodesic metric spaces with curvature bounded above.

SUMMARY: Let \(H\) be a real Hilbert space, let \(z\in H\) and let \(f,g:H\rightarrow (-\infty ,\infty ]\) be proper, lower semicontinuous and convex functions such that \(\mbox {dom}f\cap \mbox {dom}g\neq \emptyset \). We consider convergence rate of operator splitting methods for solving the following minimization problem: \begin{equation*} \mbox {minimize}~\frac {1}{2}\Vert x-z\Vert ^2+f(x)+g(x). \end{equation*}

SUMMARY: In this talk, we consider the best approximation problem in a Banach space and deal with some strong convergence theorems for the problem.

SUMMARY: In this talk, we study attractive points of normally \(2\)-generalized hybrid mappings and prove weak convergence theorems for the mappings. We study a broad class of sequences which covers nonexpansive sequences, generalized hybrid sequences, \(2\)-generalized hybrid sequences. Then, we prove nonlinear ergodic theorems for such sequence. We also prove weak and strong convergence theorems for the sequences. Further, we study fixed points theorems for some nonlinear mappings.

SUMMARY: Birkhoff(–James) orthogonality is one of the most important generalized orthogonality relation in Banach spaces. It is not globally symmetric in the most Banach spaces, but can be locally symmetric in some senses. In this talk, we clarify left (or right) symmetric points for Birkhoff orthogonality in von Neumann algebras.

SUMMARY: The notion of strong Birkhoff orthogonality is defined on Hilbert \(C^*\)-modules by using Birkhoff orthogonality and scalar products. We consider it in the setting of von Neumann algebras, and study its local symmetry. As an application, it is shown that if two von Neumann algebras have the same linear structure and strong Birkhoff orthogonality then they are \(*\)-isomorphic to each other.

SUMMARY: It is thought that some distance space which are not linear spaces have properties such that the generalization of properties in linear spaces. However, it remains to be elucidated what geometrical properties exist in the spaces. Here we report that the spaces have some geometrical properties given by triangles of three points and convex combinations.

SUMMARY: In 1978 Uchiyama gave a proof of the characterization of \(\mathrm {CMO}\) which is the closure of \(C^{\infty }_{\rm comp}\) in \(\mathrm {BMO}\). We extend the characterization to the closure of \(C^{\infty }_{\rm comp}\) in the Campanato space with variable growth condition. As an application we characterize compact commutators \([b,T]\) and \([b,I_{\alpha }]\) on Morrey spaces with variable growth condition, where \(T\) is the Calderón–Zygmund singular integral operator, \(I_{\alpha }\) is the fractional integral operator and \(b\) is a function in the Campanato space with variable growth condition.

SUMMARY: For a Young function \(\Phi \) and \(\varphi \in \mathcal {G}^{\rm dec}\), let \(L^{(\Phi ,\varphi )}(\mathbb {R}^n) = \left \{ f\in L^1_{\mathrm {loc}}(\mathbb {R}^n): \|f\|_{L^{(\Phi ,\varphi )}} <\!\infty \right \},\)
\(\|f\|_{L^{(\Phi ,\varphi )}} = \sup _{B} \|f\|_{\Phi ,\varphi ,B}.\) Then \(\|\cdot \|_{L^{(\Phi ,\varphi )}}\) is a norm and \(L^{(\Phi ,\varphi )}(\mathbb {R}^n)\) is a Banach space, since \(\|f\|_{\Phi ,\varphi ,B}=\|f\|_{L^{\Phi }(B,dx/(|B|\varphi (r)))},\) which is a norm on the Orlicz space \(L^{\Phi }(B,dx/(|B|\varphi (r)))\).

If \(\Phi (r)=r^p\) \((1\le p<\infty )\), then we denote \(L^{(\Phi ,\varphi )}(\mathbb {R}^n)\) by \(L^{(p,\varphi )}(\mathbb {R}^n)\) which is the generalized Morrey space.

We give a necessary and sufficient condition for the boundedness of \(M_{\rho }\) from \(L^{(\Phi ,\varphi )}(\mathbb {R}^n)\) to \(L^{(\Psi ,\varphi )}(\mathbb {R}^n)\).

SUMMARY: In this talk we give the characterization of pointwise multipliers on weak Morrey spaces. To do this we first prove a generalized Hölder’s inequality for the weak Morrey spaces. Next, to characterize the pointwise multipliers, we use the fact that all pointwise multipliers from a weak Morrey space to another weak Morrey space are bounded operators.

SUMMARY: The Kantrovitch operator is used to approximate functions. In particular, it is well known that this operator can be used to show the density of polynomials in \(C[0,1]\). Here we compare the Kantrovitch operator with the Hardy–Littlewood maximal operator. What is important here is the constant can be taken \(1\), which is optimal. This is a joint work with Burenkov in Moscow and Ghorbanalizadeh in Iran.

SUMMARY: We will disprove that the Morrey space \({\mathcal M}^p_{q_0}\), which is a subspace of \({\mathcal M}^p_{q_1}\), is dense in \({\mathcal M}^p_{q_1}\) when \(1 \le q_1<q_0 \le p<\infty \). The proof is based on a combinatoric observation done earlier by myself, Sugano and Tanaka.

SUMMARY: The aim of this talk is to propose to use uniformly locally square integrable function spaces to deal with the elliptic differential operators with non-smooth coefficients. We do not assume any other regularity condition. Although the singular integral operators fail to be bounded, we can handle the operators to some extent. This is a joint work with Mastylo in Poland.

SUMMARY: Mixed Morrey spaces are one of the extension of classical Morrey spaces and the generalization of some function spaces. In this talk, we give a necessary and sufficient condition for the boundedness of the commutators of fractional integral operators on mixed Morrey spaces.

SUMMARY: In this talk, we consider the boundedness of the linear and multilinear fractional maximal operator and the fractional integral operator within the framework of weighted Morrey spaces. By the observation of the endpoint cases, we obtain the results. The results recover the inequalities which is due to I. Sato, Sawano and Tanaka and the inequalities which is due to Sawano, Sugano and Tanaka.

SUMMARY: The properties of the generalized Orlicz spaces and the weak Orlicz spaves, with quasi-norms or F-norms constructed by \(\varphi \)-functions, are given.

SUMMARY: We consider the relation between multiple Fourier series and lattice point problems

SUMMARY: This talk considers a chemotaxis-Navier–Stokes system with nonlinear diffusion. In 2015 Zhang–Li showed existence of global weak solutions under some condition; however, the proofs of this result contained several essential gaps. Therefore the purpose of the present talk is to give existence of global weak solutions by correcting arguments in the previous result. The main result in this talk asserts that “strong” diffusion derives existence of global weak solutions.

SUMMARY: This talk considers a chemotaxis-Navier–Stokes system with logistic-type damping. In the case that the logistic-type damping is given by \(+ \kappa n - \mu n^2\), a previous paper (Kurima–M.) obtained existence of global weak solutions under some conditions. However, the case that the logistic-type damping is given by \(+ \kappa n - \mu n^\alpha \) with some \(\alpha >1\) seems not to be studied. Therefore the purpose of the present talk is to obtain existence of global weak solutions in the case that the logistic-type damping is given by \(+ \kappa n - \mu n^\alpha \) with some \(\alpha >1\). The main result in this talk asserts that “strong” logistic-type damping derives existence of global weak solutions.

SUMMARY: We consider the asymptotic limits of the viscous Cahn–Hilliard equation. In 2014, Bui, et al. proved the existence of the solutions of the Cahn–Hilliard equation and the Allen–Cahn equation by considering the asymptotic limits of the viscous Cahn–Hilliard equation under the Sobolev subcritical growth condition on the nonlinear term. In this talk, we exclude this growth condition by decomposing the nonlinear function into the sum of a monotone function and a locally Lipschitz perturbation, and show the existence of the solutions of the Cahn–Hilliard equation and the Allen–Cahn equation and with some improved regularity.

SUMMARY: We are concerned with the comparison theorem of the initial-boundary value problem for nonlinear parabolic equations governed by the nonlinear boundary conditions. It is well known that classical comparison principle is a useful tool in the study of reaction diffusion equations. This classical result claims that if two initial data satisfy some order then the corresponding solutions keep the initial data order. In this talk, we clarify the relationship of two solutions of reaction diffusion equations governed by the different nonlinear boundary conditions.

SUMMARY: We consider the following complex Ginzburg–Landau equation (CGL) with linear term. \[ u_t - e^{i\theta }[\Delta u + |u|^{q-2}u]-\gamma u=0,\quad \textrm {on}\ [0, T) \times \Omega , \] where \(\theta \in (-\pi /2, \pi /2)\); \(i\) denotes the imaginary unit; \(\gamma \in \mathbb {R}\); \(\Omega \subset \mathbb {R}^N\) is a bounded domain; \(T > 0\). In this talk we investigate an asymptotic behavior of solutions of (CGL), especially finite time blow-up. It was shown by Cazenave et al, that finite time blow-up could occur for initial data which are negative in a certain energy. We would like to talk about the finite time blow-up for initial data with positive energy. In the proof, we apply the potential-well method.

SUMMARY: This study was made to observe the behavior of the solution by numerical analysis on the Cahn–Hilliard system coupled with viscoelasticity (CHV), which is one of nonlinear partial differential equations, and summarizes the results obtained by the solution and error estimate. CHV is a system of 4th order evolution equations for two unknowns describing a phenomenon in which a substance having a viscoelastic property such as a polymer arises phase separation. Although mathematical proofs of the existence and uniqueness of the solution are given, it has not been clarified the behavior of solution yet. Therefore, in this study, we demonstrate numerical simulations and error estimate. For the purpose, we use structure-preserving numerical methods that is expected to be stable and accurate.

SUMMARY: We propose a structure-preserving scheme for the Allen–Cahn equation with dynamic boundary conditions using the discrete variational derivative method (DVDM). In DVDM, how to discretize the energy which characterizes the equation, it is essential. Modifying the conventional manner and using another summation-by-parts formula, we can use the central difference operator as an approximation of a space derivative on the discrete boundary condition. In this talk, we mainly show the existence and uniqueness of the solution for the proposed scheme and the error estimate.

SUMMARY: In this talk, we consider a coupling system, consisting of Allen–Cahn type equation, and a PDE model of grain boundary motion, under the dynamic boundary condition. The motivation of this study is to develop the mathematical theories to deal with the more dynamical physical situations and to apply temperature optimal control problems for grain boundary motion. On this basis, we set the goal of this talk is to address the issues, concerned with the qualitative results of the system, such as the existence of solutions to the system, and the continuous dependence of the system, and so on.

SUMMARY: In this talk, we consider a new two-scale problem which is given as mathematical model for moisture transport in concrete materials. Our model consists of the diffusion equation for the relative humidity in the entire of concrete (macro domain) and the free boundary problems describing the relationship between the relative humidity and the degree of saturation in infinitely pores (micro domain). In our model, the structures of the micro domains are unknown, and this is a significant feature of our model to emphasize. In this talk, we discuss the existence and uniqueness of a solution to our model.

SUMMARY: In this talk, we discuss the long time behavior of Cahn–Hilliard system with dynamic boundary condition of GMS type. In this model, a characteristic property of conservation holds which is related to the sum of the volume in the bulk and on the boundary. By virtue of the effective usage of this property, the well posedness is discussed. Moreover, by applying the energy estimate the characterization of the omega limit set is obtained.

SUMMARY: We prove a continuous dependence on the initial conditions and flux functions of solutions for a single conservation law. We can derive from the result that approximate solutions constructed by the wave-front tracking methods are Cauchy sequence. The proofs rely on the well-posedness theory introduced by Liu and Yang.

SUMMARY: The Cauchy problem for the semilinear diffusion equation is considered in the de Sitter spacetime with the spatial zero-curvature. Global solutions and their asymptotic behaviors for small initial data are shown for positive and negative Hubble constants. The effects of the spatial expansion and contraction are studied on the problem.

SUMMARY: The Navier–Stokes equations are considered in homogeneous and isotropic spacetimes. The Cauchy problem for the Navier–Stokes equations is considered under the constant density.

SUMMARY: We show maximal regularity for the Cauchy problem of the heat equation in the class of bounded mean oscillation (BMO). It is known that the large class of the parabolic equation has maximal regularity in the UMD Banach space X. If the power is not the end-point case, the necessary and sufficient condition on maximal regularity is so called R-boundedness. Since UMD Banach space is necessarily reflexive, non-reflexive Banach space such as BMO is not the subject to the general theory. We show maximal regularity and sharp trace estimate of the solution of heat equations.

SUMMARY: We consider an initial-boundary value problem of a phase field model of Fix-Caginalp type whose boundary condition for the relative temperature has a quasi-variational structure. This structure gives one for inner products of the suitable real Hilbert space. So, we can apply the theory of evolution inclusions on a real Hilbert space which has a quasi-variational structure for inner products which was established in the paper, Evolution inclusion on a real Hilbert space with quasi-variational structure for inner products, JOCA 1981. Actually, applying the results obtained in the above paper, we can show that this initial-boundary value problem has at least one solution in the quasi-variational sense.

SUMMARY: We consider doubly nonlinear evolution equations governed by double time-dependent subdifferentials in uniformly convex Banach spaces. Note that our equations has multiple solutions, in general, and therefore the optimal control problem associated with our state equation is singular. Thus, in this talk, we investigate the singular optimal control problem formulated for our non-well-posed state systems.

SUMMARY: In this talk, we consider a one-dimensional version of “Kobayashi–Warren–Carter type system”, which is based on a phase-field model of grain boundary motion, proposed in [Kobayashi–Warren–Carter, Phys. D, 140 (2000), 141–150]. The interest of this talk is in the concrete behavior of special kinds of solutions, which reproduce the typical structures, with facets, observed in polycrystalline bodies. On this basis, we define a new class of solutions, named crystalline solutions. Under suitable assumptions, the sufficient conditions for the existence and uniqueness of crystalline solutions, and conditions for non-degeneracy of mobilities, will be shown as the Main Theorems of this talk.

SUMMARY: The life cycle of jellyfish is so complicated that we simplify it as follows: Jellyfishes lay eggs. The eggs become polyp or ephyra. Polyps live on tetrapod, increase with asexual reproduction which has three kinds of growth process. Moreover, ephyra becomes a jellyfish after growing up. Here, we assume that the region is a one-dimensional interval (0,1) and that there is tetrapod at \(x = 1\) and propose a system of diffusion equations with dynamic boundary condition having age structure. In this talk we show the modeling process, and existence and uniqueness of a solution to the model.

SUMMARY: The Soret effect is a substance flow due to the force of temperature gradient in mixed solution. There are several studies on molecular transport utilizing this phenomenon, and as one of the research methods, there is an experiment in which heat sources(metal) are periodically arranged. In the present study we consider the experiment related to the Soret effect from the standpoint of mathematical model. Here, we will show existence of a weak solution of the model.