2018年度年会(於:東京大学)

SUMMARY: Mathematical developments are always based on problems. Challenges to difficulties generate new ideas. In this talk, we focus on a 30 years old open problem in Banach space theory, so-called Tingley’s problem, and present new geometric techniques (and results) derived from our challenges.

SUMMARY: We consider some equations describing double-diffusive convection phenomena of incompressible viscous fluid in a porous medium. Roughly speaking, this system consists of the Stokes equation and two advection-diffusion equations. Although the fluid equation in our model is linearized, the others still possess convection terms as non-monotone perturbations, which make it difficult to deal with this system.

Main topic of this talk is to show the global solvability of this double-diffusive convection system. In particular, we focus on the existence of time periodic solutions to the system in the whole space domain for large data, i.e., without any smallness conditions for given external forces. In previous results for periodic problems of parabolic type equations with non-monotone perturbation terms (e.g., incompressible Navier–Stokes equations and Boussinesq system), it seems that either of the smallness of given data or the boundedness of space domain is essential. However, in spite of the presence of non-monotone terms, the solvability of our problem in the whole space is shown for large external forces via the convergence of solutions to approximate equations in bounded domains.

SUMMARY: There are the wide Denjoy integral, the approximately continuous Perron integral, the approximately continuous Henstock integral, the approximately continuous Denjoy integral and the distributional denjoy integral as the wider integrals than Denjoy–Perron–Henstock–Kurzweil integral. These integrals are defined by replacing derivative with approximately derivative and distributuinal derivative. In this talk, we extend an integral by the Cauchy’s principal value and show the obtained results.

SUMMARY: The weak \(L_p\) space \(L^{p,\infty }(\mu )\) is introduced for the quasi-subadditive monotone measure \(\mu \). If \(\mu \) is continuous from below, them \(L^{p,\infty }(\mu )\) is a quasi-Banach space. As an application it is shown that there exists a real number \(\alpha \in (0,1]\) such that the power transformatin \(\mu ^{\alpha }\) is uniformly quasi-subadditive. Furthermore there exists a subadditive monotone measure \(\lambda \) satisfying \(\lambda \le \mu ^{\alpha } \le 2\lambda \).

SUMMARY: The consequent of the strong form of the Egorov theorem in non-additive measure theory is that strong almost everywhere convergence implies strong almost uniform convergence. This paper shows that the conjunction of the uniform subadditive continuity and the order continuity of the non-additive measure is a sufficient condition for the consequent of the strong form of the Egorov theorem, and that the monotone continuity is a necessary condition.

SUMMARY: Using the recently introduced resolvent of a convex function in a complete CAT(1) space, we obtain existence and convergence theorems for the proximal point algorithm in such a space.

SUMMARY: Let \(H\) be a real Hilbert space and let \(f:H\rightarrow (-\infty ,\infty ]\) and \(g:H\rightarrow (-\infty ,\infty ]\) be proper, lower semicontinuous and convex functions. This talk considers a problem of finding the resolvent \(J_{\partial (f+g)}\) of the subdifferential \(\partial (f+g)\). It is assumed that both the resolvents \(J_{\partial f}\) and \(J_{\partial g}\) of \(\partial f\) and \(\partial g\) can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily.

SUMMARY: In this talk, we introduce and study a quasi-nonexpansive mapping, a strictly quasi-nonexpansive mapping, and a strongly quasi-nonexpansive mapping in an abstract space. In particular, we give some basic properties of such mappings.

SUMMARY: In this talk, we study the relations among \(k\)-acute points, attractive points and fixed points. Further, we apply these to rearrange proofs of some known convergence theorems and to prove new convergence theorems for nonlinear mappings in Hilbert spaces. Using the ideas of attractive points, acute points and fixed points, we also prove convergence theorems for nonlinear mappings in Banach spaces.

SUMMARY: We will talk about two topologies on \(\nu \)-generalized metric spaces.

SUMMARY: In this talk, we study the von Neumann–Jordan constant of \(\pi /2\)-rotation invariant norms on \(R^2\). We know that any \(\pi /2\)-rotation invariant normed space is isometrically isomorphic to some Day–James space. Since the von Neumann–Jordan constant is invariant under isometrically isomorphic, for characterization of the constant, it is enough to consider the Day–James space. From this fact, we can give some estimations of the constant.

SUMMARY: We introduce a certain class of summability methods which are defined by the convolution operation in the group algebra \(L^1(\mathbb {R}^{\times })\) and study \(b\)-strongness and \(b\)-equivalence between them. In particular, this class contains an integral version of Cesàro summability method and we give a necessary and sufficient condition for a summability method in the class to equivalent to this one.

SUMMARY: In this talk, we consider the dual inequality for the Hardy–Littlewood maximal operator \(M\) and the fractional integral operator \(I_{\alpha }\). Since the fractional integral operator \(I_{\alpha }\) has the property \(\int I_{\alpha } f g dx=\int f I_{\alpha }g dx\) for \(f\geq 0\) and \(g\geq 0\), the weight norm inequality \(I_{\alpha }: L^{p}(v^{p})\to L^{q}(u^{q})\) is equivalent to \(I_{\alpha }:L^{q'}(u^{-q'})\to L^{p'}(v^{-p'})\), where \(\frac {1}{p}+\frac {1}{p'}=\frac {1}{q}+\frac {1}{q'}=1\) (\(1<p<\infty \), \(1<q<\infty \)) and let \(u\) and \(v\) be weights. Analogously, we consider whether the dual inequality for the Hardy–Littlewood maximal operator holds on weighted Lebesgue spaces. In particular, we verify the dual inequalities of \(M:L^{p}(w)\to L^{p}(w)\) and \(M:L^{p}(Mw)\to L^{p}(w)\).

SUMMARY: We discuss the compactness of the commutators \([b,T]\) and \([b,I_{\rho }]\) on generalized Morrey spaces with variable growth condition, where \(T\) is a Calderón–Zygmund operator, \(I_{\rho }\) is a generalized fractional integral operator and \(b\) is a function in generalized Campanato spaces with variable growth condition.

SUMMARY: Let \(\mathbb {R}^n\) be the \(n\)-dimensional Euclidean space. Let \(b\in \mathrm {BMO}(\mathbb {R}^n)\) and \(T\) be a Calderón–Zygmund singular integral operator. In 1976 Coifman, Rochberg and Weiss proved that the commutator \([b,T]=bT-Tb\) is bounded on \(L^p(\mathbb {R}^n)\) (\(1<p<\infty \)), that is, \begin{equation*} \|[b,T]f\|_{L^p}=\|bTf-T(bf)\|_{L^p}\le C\|b\|_{\mathrm {BMO}}\|f\|_{L^p}, \end{equation*} where \(C\) is a positive constant independent of \(b\) and \(f\). For the fractional integral operator \(I_{\alpha }\), Chanillo proved the boundedness of \([b,I_{\alpha }]\) in 1982. These results were extended to Orlicz spaces by Fu, Yang and Yuan (2012, 2014). In this talk we discuss the boundedness of the commutator \([b,I_{\rho }]\) on Orlicz spaces, where \(I_{\rho }\) is a generalized fractional integral operator.

SUMMARY: We study the boundedness of the commutator of fractional indegrals on martingale Morrey spaces. We give a necessary and sufficient condition on the boundedness in terms of martingale Campanato spaces.

SUMMARY: We define the Besov spaces on an arbitrary open set of \(\mathbb R^d\). Based on the spectral theorem for the Dirichlet Laplacian, we introduce test function spaces and distributions to define the Besov spaces analogously to Peetre’s idea for the whole space case, which is by the dyadic decomposition of the spectrum. We will define the Besov spaces of the inhomogeneous type and the homogeneous type, and also show fundamental properties such as completeness, embedding, etc.

SUMMARY: The purpose of this talk is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.

SUMMARY: We define the Besov spaces on an arbitrary open set of \(\mathbb R^d\), based on the spectral theorem for Schrödinger operators with potential of which negative part is of the Kato class. The purpose of this talk is to show the isomorphism relations among the Besov spaces generated by the Dirichlet Laplacian and the Schrödinger operators.

SUMMARY: In this talk, we study the boundedness of the Schrödinger operator \(e^{i \Delta }\) on Wiener amalgam spaces and determine its optimal condition.

SUMMARY: This paper gives a first insight into making a mathematical bridge between the parabolic-parabolic signal-dependent chemotaxis system and its parabolic-elliptic version. To be more precise, this talk deals with convergence of a solution for the parabolic-parabolic chemotaxis system with strong signal sensitivity to that for the parabolic-elliptic chemotaxis system.

SUMMARY: This talk deals with vanishing viscosity for a Cahn–Hilliard type system on an unbounded domain with smooth bounded boundary. Colli–Gilardi–Rocca–Sprekels (2017) studied it in the case of a bounded domain using Aubin–Lions lemma. However, this lemma dose not work well in the case of unbounded domains. The present work asserts that we can discuss vanishing viscosity for the above system in an unbounded domain.

SUMMARY: We study an initial-boundary value problem for a reaction diffusion system, which consists of two real-valued unknown functions. This system describes diffusion phenomena of neutrons and heat in nuclear reactors, introduced by Kastenberg and Chambré. In this model, the unknwon functions represent the neutron density and the temperature in nuclear reactors. We proved that this equation has at least one positive stationary solution in last MSJ Autumn Meeting 2017. In this talk, we show that the positive stationary solution plays a role of threshold to classify initial data into two groups; corresponding solutions of the equation blow up in finite time and exist globally.

SUMMARY: We deal with initial-boundary problems for Vlasov–Poisson equations in a half-space with external magnetic force horizontal to a wall. In 2013, Skubachevskii gives local-in-time solvability to the system. Moreover, in 2017, global-in-time solutions were obtained by effectively using the magnetic force whose direction is horizontal to the wall. This talk provides an existence result for the system where the magnetic force has angle error in the vertical direction.

SUMMARY: The existence problem for Cahn–Hilliard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point theorem in the level of approximate problem. One of the key point is the assumption for maximal monotone graphs with respect to their domains. Thanks to this, we obtain the existence result of the weak solution by using the passage to the limit.

SUMMARY: In this talk, we consider a coupled system of two parabolic type initial-boundary value problems, called the Kobayashi–Warren–Carter model of grain boundary motion in a polycrystal. The systems are denoted by (S)\(_{\varepsilon }\) with arguments \( \varepsilon \geq 0 \). The characteristic point of our systems is to assume the dynamic boundary conditions in one problem. Now, the focus of this talk is to address the three assignment concerned with the qualitative results of the systems. The first is the existence of solutions to the systems, including the representation of solutions. The second is the continuous dependence of the systems to (S)\(_\varepsilon \) for the variations of \( \varepsilon \geq 0 \). The third is the large-time behavior of solutions.

SUMMARY: We consider the initial value problem (CP) for degenerate parabolic-parabolic systems with variable coefficients. The systems are coupled with strongly degenerate parabolic equations and nonhomogeneous heat equations. Strongly degenerate parabolic equations are regarded as a linear combination of the time-dependent conservation laws (quasilinear hyperbolic equations) and the porous medium type equations (nonlinear degenerate parabolic equations). Thus, the equation has both properties of hyperbolic equations and those of parabolic equations. In this talk, we discuss the solvability for (CP).

SUMMARY: We consider the Cauchy problem for a semilinear diffusion equation in an expanding space, and we show global solutions for small initial data.

SUMMARY: We consider the extension of the Navier–Stokes equations and the elastic wave equations in the Minkowski spacetime to the equations in uniform and isotropic spacetimes.

SUMMARY: We consider the Stokes semigroup in a large class of domains including bounded domains, the half-space and exterior domains. We will prove that the Stokes semigroup is analytic in a certain type of solenoidal subspaces of BMO.

SUMMARY: The equation and dynamic boundary condition of Cahn–Hilliard type was introduced by Goldstein–Miranville–Schimperna (2011), this problem is called GMS model and it is similar to the general Cahn–Hilliard system. In this talk, the double well potential of logarithmic type is employed. A strict separation property from pure phases is considered.

SUMMARY: In this talk we introduce a new class of doubly nonlinear quasi-variational evolution equation governed by double time-dependent subdifferentials. The main aim of this talk is to show the existence of a solution to our equations.

SUMMARY: Throughout our recent researches, we showed that the tumor invasion model with quasi-variational structure can be rewritten into an evolution inclusion on a suitable Hilbert space. Moreover, its norms has a quasi-variational structure in general. That is, they depend upon an unknown function, which is one of the functions (components) of a unique solution to the tumor invasion model. In this talk, an evolution inclusion on a real Hilbert space with quasi-variational structure for inner products is considered. And a main purpose is to give the continuity property of proper l.s.c. convex functions with quasi-variational structure which appear as subdifferentials operators in the evolution inclusions.

SUMMARY: In this talk, we consider a gradient flow of a non-convex functional, which was proposed by [Berkels et -al, pp. 293–301, Vision Modeling and Visualization 2006 (2006)] as a possible governing energy for an anisotropic image processing on a bounded spatial domain \( \Omega \subset \mathbb {R}^2 \). Our gradient flow is descried in a nonstandard form of partial differential inclusions, which contains a composition \( \partial \gamma \circ R \) of: a (possibly) set-valued subdifferential \( \partial \gamma \) of an anisotropic metrix \( \gamma \in W^{1, \infty }(\mathbb {R}^2) \); and a rotation matric \( R \in C^\infty (\mathbb {R}; \mathbb {R}^{2 \times 2}) \). Under appropriate settings, some mathematical observations for the gradient system will be provided on the basis of the time-discretization approach.

SUMMARY: In this talk, we propose a mathematical model for moisture swelling process in concrete materials. Moisture swelling process appear in, for instance, frost damage in concrete materials which is a nonlinear phenomenon to give rise to crack inside of concrete. Our model consists of a diffusion equation for moisture in a one microscopic hole of concrete and a free boundary problem for the front of the moisture region. In this talk, we discuss the existence and uniqueness of a time global solution for this problem, and moreover, some results of the behavior of the free boundary as time goes to infinity.

SUMMARY: Recently, we investigated a free boundary problem describing a adsorption process in a porous media. In the global existence result to this model we require the smallness condition \(h < 1\) for the boundary function \(h\) to prevent that the free boundary touches the fixed boundary. This is a big obstacle to consider a multi-scale model consisting of nonlinear diffusion equation and the free boundary problem. To overcome this difficulty, we propose a new weak formulation to the free boundary problem. Also, we show existence of a weak solution without the smallness condition to the boundary data.