2019年度秋季総合分科会(於:金沢大学)

SUMMARY: We give a sparse bound for an integral operator with wave propagator by using a criterion due to Lerner and Nazarov. Our result is sharp with respect to the parameter. Since this operator dominates the maximal Riesz means, our result yields weighted bound for the maximal operator.

SUMMARY: Several partial differential equations are considered in homogeneous and isotropic spaces. The Cauchy problems for the equations are considered in Lebesgue spaces and Sobolev spaces. Dissipative and anti-dissipative effects from the spatial expansion and contraction on the problems are remarked.

SUMMARY: In this talk, we give a simple proof and some generalizations of results in [Falset, Llorens-Fuster, and Marino, Math. Model. Anal. 21 (2016)].

SUMMARY: In this talk, we establish the existence of absolute fixed points of normally \(2\)-generalized hybrid mappings in a Hilbert space. We prove some fixed point theorems in a Hilbert space. We also prove convergence theorems for iterative sequences.

SUMMARY: Let \(H\) be a real Hilbert space and let \(f\colon H \rightarrow (-\infty , \infty ]\) and \(g\colon H \rightarrow (-\infty , \infty ]\) be proper, lower semicontinuous and convex functions. We consider a problem of finding the resolvent \(J_{\partial (f+g)}\) of the subdifferential \(\partial (f + g)\). In particular, we obtain a strong convergence result of a splitting method.

SUMMARY: Convex minimization problem is one of the convex optimization problems. We study it by using many kinds of approximation methods in Hilbert spaces, Banach spaces and so on. In a complete CAT(0) space and a complete admissible CAT(1) space, the set of fixed points of the resolvent for the convex function coincides with the set of its minimizers. Therefore, we find a fixed point of the resolvent instead of a minimizer of the convex functions. In this talk, we consider some iteration methods for a finite family of resolvent operators.

SUMMARY: In this talk, we describe a methodology to derive the convergence theorems of nonlinear integrals of \(p\)-th order integrable functions converging in measure from the already established convergence theorems in nonadditive measure theory. We also discuss the completeness of the Lorentz space which is defined by a nonadditive measure.

SUMMARY: For a Young function \(\Phi :[0,\infty ]\to [0,\infty ]\) and a growth function \(\varphi :(0,\infty )\to (0,\infty )\), let \(L^{(\Phi ,\varphi )}(\mathbb {R}^n)\) and \(\mathcal {L}^{(\Phi ,\varphi )}(\mathbb {R}^n)\) be the Orlicz–Morrey and Orlicz–Campanato spaces, respectively. In this talk we give a relation between \(\|M^{\sharp } f\|_{L^{(\Phi ,\varphi )}}\) and \(\|f\|_{\mathcal {L}^{(\Phi ,\varphi )}}\).

SUMMARY: In this talk we give a characterization of pointwise multipliers on weak Morrey spaces \(\mathrm {w}{L}_{p,\phi }(\mathbb {R}^n)\). We denote by \(\mathrm {PWM}( \mathrm {w}{L}_{p_1,\phi _1}(\mathbb {R}^n), \mathrm {w}{L}_{p_2,\phi _2}(\mathbb {R}^n))\) the set of all pointwise multipliers from \(\mathrm {w}{L}_{p_1,\phi _1}(\mathbb {R}^n)\) to \(\mathrm {w}{L}_{p_2,\phi _2}(\mathbb {R}^n)\). We give a necessary condition for \( \mathrm {PWM}( \mathrm {w}{L}_{p_1,\phi _1}(\mathbb {R}^n), \mathrm {w}{L}_{p_2,\phi _2}(\mathbb {R}^n)) =\mathrm {w}{L}_{p_3,\phi _3}(\mathbb {R}^n). \)

SUMMARY: Our goal of this talk is to show that Ho’s vector-valued Morrey spaces can be realized as the special case of the pointwise multiplier space. This extends Lemarié-Rieusset’s theorem. One can not extend his theorem directly because we are handling Banach lattices instead of Lebesgue spaces. It turns out that mixed Morrey spaces, Lorentz–Morrey spaces and Orlicz–Morrey spaces fall under the scope of the framework.

SUMMARY: Let \(I_{\gamma }\) be a generalized fractional integral and \(b\) be a function in martingale Campanato spaces \(\mathcal {L}^-_{1,\phi }\). We show the boundedness and compactness of the commutator \([b,I_{\gamma }]\) from martingale Orlicz space \(L_{\Phi }\) to another martingale Orlicz space \(L_{\Psi }\) and from \(L_{\Phi }\) to a martingale Triebel–Lizorkin space \({F}^{\phi }_{L_{\Psi }}\).

SUMMARY: We give a proof of the Gagliardo–Nirenberg inequality (GN inequality) for Sobolev spaces using Muramatu’s integral formula with the Hardy–Littlewood maximal function. GN inequality has two main forms which correspond to cases where the parameter appearing in GN inequality takes the end values. In both cases we can derive GN inequality in a few lines from Muramatu’s integral formula if the integrability exponents are not 1. It is known that one of exceptional cases in GN inequality can be handled by BMO functions. We also consider such exceptional case.

SUMMARY: In 1995, Perez introduced Bp-condition, which is necessary and sufficient condition for the boundedness of the Orlicz maximal operator on Lp spaces. After, necessary and sufficient condition of the Hardy–Littlewood–Sobolev type inequality for Orlicz-fractional maximal operator is derived by Cruz-Uribe and Moen in 2013. In this paper, we investigate the boundedness of Orlicz maximal operator, Orlicz-fractional maximal operator and fractional integral operator in Morrey and Orlicz–Morrey spaces on the assumption that each Young function satisfies these conditions, respectively. In particular, one of the main results is based on the Adams inequality in the framework of Morrey spaces.

SUMMARY: Let \(n\) be the spatial dimension. For \(d,0<d\le n\), let \(H^{d}\) be the \(d\)-dimensional Hausdorff content. The purpose of this talk is to investigate the region \((d,p)\) which guarantees the boundedness of the dyadic strong maximal operator on the Choquet space \(L^{p}(H^{d},{\mathbb R}^n)\).

SUMMARY: In this talk, we consider the sparse form bounds for Fourier integral operators associated with the symbol belonging to Hörmander class \({S}^{m}_{1,0}\). Furthermore, we study weighted \(L^p\) boundedness of Fourier integral operaors for Muckenhoupt weight class as an application of sparse form bounds.

SUMMARY: Young gave the example of the complete and minimal complex exponential system which is not a basis in \(L^2[-\pi ,\pi ]\). Bases on this result, we give another examples of the complete and minimal complex exponential systems which are not bases in \(L^2[-\pi ,\pi ]\).

SUMMARY: This work is concerned with the question that “how far does small chemotactic interaction perturb the Lotka–Volterra competition dynamics?”. A two-species chemotaxis-competition system was studied by e.g., Bai–Winkler (2016) and Lin–Mu–Wang (2015). However, there are still many open problems about the two-species chemotaxis-competition system. On the other hand, the Lotka–Volterra competition system has been studied extensively. Thus the development of this work will enable us to see new properties of solutions for the chemotaxis system. The main result of this talk gives convergence of solutions for the two-species chemotaxis-competition system to those for the Lotka–Volterra competition system on bounded convex domains.

SUMMARY: This talk deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase field system as an example which has studied by e.g., Grasselli–Petzeltová–Schimperna (2006) and Wu–Grasselli–Zheng (2007). Although a time discretization of an abstract evolution equation has been studied by e.g., Colli–Favini (1996), time discretizations of simultaneous abstract evolution equations seem to be not studied yet. In this talk we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.

SUMMARY: We consider the uniform boundedness for global solutions of nonlinear heat equations with nonlinear boundary conditions. As for the Dirichlet boundary conditions, there are many studies on the uniform bounds for global solutions by Ôtani, Cazenave–Lions, Giga, Quittner and so on. However, it does not work these methods for the nonlinear boundary condition case due to the nonlinearity on the boundary. In this talk, we modify the abstract theory on the asymptotic behavior for global solutions by Ôtani (1981) and show that global solutions are bounded uniformly in time in appropriate norm.

SUMMARY: In this talk, we consider the time-periodic problem for the complex Ginzburg–Landau equation (CGL): \[ \frac {du}{dt}(t,x)-(\lambda +i\alpha )\Delta u-(\kappa +i\beta )|u|^{q-2}u-\gamma u=f(t,x), \] where \(\lambda ,\kappa >0\) and \(\alpha ,\beta ,\gamma \in \mathbb {R}\) and \(f:(0,T)\times \Omega \to \mathbb {C}\) denotes an external force with a given period \(T>0\). We identify \(\mathbb {C}\) with \(\mathbb {R}^2\) and formulate (CGL) as an evolution equation governed by subdifferential operators in product Lebesgue space \(\mathbb {L}^2:=\mathrm {L}^2\times \mathrm {L}^2\), which is a Hilbert space. We show the existence of time-periodic solutions of (CGL) in bounded domains with assuming suitable smallness of \(f\) and \(\gamma \) by modifying the argument developped in Ôtani (1984).

SUMMARY: We consider the time periodic problem for the viscous Cahn–Hilliard equation with the homogeneous Dirichlet boundary condition. There are no results on this problem, except the work by Liu–Liu–Tang (2013) for the special case of Cahn–Hilliard equation. In this talk, we show the existence of the time periodic solutions by using Schauder fixed point theorem.

SUMMARY: In this talk, we consider a coupled system of the Kobayashi–Warren–Carter type, including the singular diffusion and dynamic boundary condition. The system is known as the mathematical model of grain boundary motion in a polycrystal, proposed by [Kobayashi et al., Physica D, 140 (2000), 141–150]. The objective of this study is to develop the mathematical theories which enable us to apply the mathematical observations for the grain boundary motion under various situations. Based on this, we set the goal to obtain the solvability of the system, including the representations of the solution.

SUMMARY: We propose a structure-preserving scheme for the Cahn–Hilliard equation with a dynamic boundary condition by using the discrete variational derivative method (DVDM). In this method, 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 an outward normal derivative on the discrete boundary condition of the proposed structure-preserving scheme. In this talk, we focus on the existence and uniqueness of the solution for the scheme.

SUMMARY: We deal with initial-boundary problems for Vlasov–Poisson systems in a half-space. In 2013, Skubachevskii gives local-in-time solvability to the system. Moreover, in 2017, existence result with weaker condition were also obtained by effectively using the magnetic force whose direction is horizontal to the wall. This talk provides an existence result for the equation where the magnetic force has angle error in the vertical direction and depending on the first element of the spatial variable.

SUMMARY: Recently, we established the abstract theory of singular optimal control problems for nonlinear evolution equations governed by double time-dependent subdifferentials. Note that the corresponding state system has multiple solutions, in general. The non-uniqueness situation of state problem makes the numerical approach to singular optimal control problems quite difficult. Therefore, in this talk, we establish an approximation procedure to singular optimal control problems from the viewpoint of numerical analysis.

SUMMARY: One of the famous classical inequalities regarding the Ornstein–Uhlenbeck semigroup in quantum physics, Nelson’s hypercontractivity inequality, has been studied from many different perspectives. We will give a new approach by identifying a quantity which is monotone under a certain diffusion flow. Our approach is effective in a substantially more general setting of Markov semigroups.

SUMMARY: We consider the semilinear abstract evolution equations with countable time delays under local Lipschitz condition.

SUMMARY: We consider the Cauchy problem for a single conservation law and prove that the approximate solutions constructed by the wave-front tracking methods are Cauchy sequence.

SUMMARY: We consider one-dimensional Cauchy problems (CP) for scalar parabolic-hyperbolic conservation laws. The equation is regarded as a linear combination of the hyperbolic conservation laws and the porous medium type equations. Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. Accordingly, it is difficult to investigate the behavior of solutions to (CP). In this talk, we focus our attention on traveling wave solutions to (CP). More precisely, we construct concrete discontinuous traveling wave solutions and discuss the properties of its. Moreover, we show the qualitative properties for entropy solutions to (CP) using the modified traveling wave solutions.

SUMMARY: In the previous works, we proved the existence of a locally-in-time solution for a multiscale model which is given as a mathematical model describing moisture transport in porous materials. Our model consists of a diffusion equation of the relative humidity in a macro domain and the free boundary problems describing a wetting and drying process in infinite micro domains. In this talk, under the improvement of the diffusion equation of the relative humidity based on the experimental result, we discuss the global existence of a solution for our multiscale model.

SUMMARY: In this talk, we discuss the parabolic problem for the hardening phenomena. The unknown functions \(u\) and \(\sigma \) describe the displacement and stress, respectively in the one-dimensional interval. Our problem means the hardening problem that the materials are harden by plasticity. That is derived from the hardening model by Visintin (2006), and the perfect plasticity model by Duvaut–Lions (1976). In the perfect plasticity model, the function that is threshold value in the plastic deformation, is a constant. In this talk, we discuss the solvability for the above model with the threshold function depending upon time or unknown function, based on the idea of Duvaut–Lions (1976).

SUMMARY: In this talk, we discuss the well-posedness of a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type. This problem is a sort of Cahn–Hilliard system with dynamic boundary condition, which is one of the current topics. Volume conservations in the bulk and on the boundary are the point of emphasis. For this transmission problem, the well-posedness is discussing under the prototype settings of double well potentials, recently. In this study we extend the result for wider setting of maximal monotone graphs. Based on the time-discretization and suitable approximate problem, we can find the approximate solution and discuss the convergence to the target problem.

SUMMARY: In this talk, we consider a class of optimal control problems for state problems of one-dimensional semi-discrete systems. Each state problem is denoted by \( \mathrm {(S)}_\varepsilon \), with \( \varepsilon > 0 \), and is associated with the phase-field model of grain boundary motion, proposed by [Kobayashi et al.; Phys. D, 140 (2000), 141–150]. In this regard, each optimal control problem is denoted by \( \mathrm {(OCP)}_\varepsilon \), with \( \varepsilon > 0 \), and it is prescribed as a minimization problem of a cost. Additionally, the problems \( \mathrm {(S)}_\varepsilon \) and \( \mathrm {(OCP)}_\varepsilon \) are supposed to admit limiting profiles as \( \varepsilon \downarrow 0 \), and then, the limiting problems are supposed to contain no little singularityies In this talk, the main interest is in the case when \( \varepsilon > 0 \) (regular case), and the mathematical results concerned with the existence of the optimal control when \( \varepsilon > 0 \); (b) the necessary condition for the regular optimal control; (c) limiting observation as \( \varepsilon \downarrow 0 \); will be reported as the main theorems of this talk.

SUMMARY: The Cauchy problem for the Navier–Stokes equations is considered in homogeneous and isotropic spaces. Local and small global solutions are constructed in the spaces, which extend the results by T. Kato. The effects of the spatial expansion and contraction are studied through the problem.