2019年度年会(於:東京工業大学)

SUMMARY: Resurgent analysis originates from the work “Les fonctions résurgentes” by J. Écalle in 1981. As it was revealed in his works, the framework of his theory has various of applications in the analysis of differential equations, vector fields, dynamical systems, multiple zeta values and so on. Resurgent analysis is built on the basis of the theory of resurgent functions. However, even fundamental properties of such functions are not understood well.

In this talk, we study the singularity structure of resurgent functions in the Borel plane through the analysis of convolution products of analytic functions. As an application, we discuss the resurgence of formal series solutions of nonlinear ordinary differential equations.

SUMMARY: I will present some recent results obtained in collaboration with Massimiliano Morini and Vesa Julin concerning the local in time existence and the asymptotic stability of the evolution equation \[ V_t=\Delta _{\Gamma _t}(H_t-W(E(u_t))). \] Here \(V_t\) denotes the normal velocity at time \(t\) of an evolving set \(F_t\) compactly contained in a given open set \(\Omega \subset \mathbb R^3\), \(\Delta _{\Gamma _t}\) is the Laplace–Beltrami operator on \(\Gamma _t=\partial F_t\), \(H_t\) is the scalar mean curvature of \(\Gamma _t\) and \(E(u_t)=(\nabla u_t+(\nabla u_t)^T)/2\) is the symmetric part of the gradient of the minimizer \(u_t:\Omega \setminus F_t\mapsto \mathbb R^3\) of the following linear elasticity problem \[ \min \bigg \{\int _{\Omega \setminus F_t} W(E(w(x)))\,dx:\,w\in H^1(\Omega \setminus F_t;\mathbb R^3),\,w=u_0\,\,{\rm on}\,\,\partial \Omega \bigg \}, \] where for a \(3\times 3\) symmetric matrix \(A\), we have set \(W(A)=\mu |A|^2+\lambda [Tr(A)]^2/2\) and \(\mu >0,\lambda +\mu >0\). The equation above has been proposed to model the evolution of a material void \(F_t\) inside an elastic material under the action of a chemical potential acting on the boundary of \(F_t\). The results I will present are new also in the special case \(u_0\equiv 0\), hence \(u_t\equiv 0\), when the equation reduces to the surface diffusion equation \[ V_t=\Delta _{\Gamma _t}H_t. \]

SUMMARY: We study the local and global in time solvability for a semilinear heat equation. In particular, we consider the case where the equation does not possess the self-similar structure. By focusing on some quasi-scaling property and its invariant integral, we develop a classification theory for the existence and nonexistence of local in time solutions, and then we discuss the existence of global in time solutions for small initial data. We also study the nonexistence of global in time solutions for nonnegative initial data. These results give a generalization of the Fujita exponent for a semilinear heat equation with general nonlinearity, and classify the existence and nonexistence of global in time solutions.

SUMMARY: We review the general theory on the stability analysis for hyperbolic systems of balance laws. The theory assumes two structural conditions for the system: One is the existence of a mathematical entropy and the other is the stability condition (called Shizuta–Kawashima Condition). Under these structural conditions we can show the global existence and optimal decay of solutions for small initial data. This general theory is valid for systems with symmetric relaxation. Recently, however, we found several interesting examples which have non-symmetric relaxation and hence the general theory is not applicable. Among such examples, we mention the Timoshenko system and the Euler–Maxwell system. We report recent works for these examples and explain their weak dissipativity of the regularity-loss type.

SUMMARY: We consider iterative functional equations of the type \(f^n(x)=g(x)\). We develop the theory of dimensioned numbers, which are suitable for representing iterated power functions or iterated exponential functions, and try to solve the equations.

SUMMARY: We introduce invariants of linear difference equations. Moreover, using these invariants, we develop a systematic method for constructing transformation formulae for hypergeometric functions. By applying this method to Gauss’s hypergeometric function, not only known formulae, such as algebraic transformation formulae and transformation formulae of special values, but also new-type transformation formulae are obtained.

SUMMARY: I study the monodromy group for Lauricella’s hypergeometric function \(F_C\). In this talk, I would like to give finiteness conditions of the monodromy group. It is a generalization of the finiteness conditions of the monodromy group for Appell’s \(F_4\), which are given by M. Kato (1997).

SUMMARY: Tsuda obtained a monodromy preserving deformation which has special solutions represented by a generalization of the Gauss hypergeometric function. Our purpose is to obtain a \(q\)-analog of the result. We define a series \(\mathcal {F}_{N,M}\) as a certain generalization of \(q\)-hypergeometric functions. We talk about a monodromy preserving deformation which admits particular solutions in terms of the function \(\mathcal {F}_{N,M}\).

SUMMARY: We explain an algorithm of computing cohomology intersection numbers associated to a hypergeometric type connection. We will see that the inverse of the cohomology intersection matrix is a rational function which satisfies a certain Pfaff system whose rational solutions are up to constant mmultiplication equal to the inverse of the cohomology matrix. Combining this with the theory of GKZ hypergeometric system, we can completely determine the intersection matrix.

SUMMARY: The Voros coefficients of the origin are defined and their explicit forms are given for the generalized hypergeometric differential equation of \({}_3F_2\) with a large parameter.

SUMMARY: We consider the Cauchy problem for linear hyperbolic systems with characteristic roots of constant multiplicities atmost double. We show the smooth continuation of eigenvectors of double characteristic roots along each characteristic strips (This is new.) By this, we can obtain the admissible data space corresponding to the regularity of the coefficients when the coefficients depend only on the time variable. (This is not new.)

SUMMARY: This talk deals with Hyers–Ulam stability of second-order linear difference equations \[ \Delta _h^2x(t)+\alpha \Delta _hx(t)+\beta x(t) = f(t), \quad t \in h\mathbb {Z}, \] where \(\Delta _hx(t) = (x(t+h)-x(t))/h\) and \(h\mathbb {Z} = \{hk|\,k\in \mathbb {Z}\}\). The purpose of this talk is to find an explicit HUS constant for the second-order linear difference equations.

SUMMARY: We consider the bifurcation problems of nonlinear ordinary differential equations with nonlinear diffusion term and the oscillatory nonlinearities. By using a generalized time-map method, \(\lambda \) is parameterize by the maximum norm \(\alpha = \Vert u_\lambda \Vert _\infty \) of the solution \(u_\lambda \) associated with \(\lambda \) as \(\lambda = \lambda (\alpha )\). Moreover, \(\lambda (\alpha )\) is continuous for \(\alpha > 0\). We are interested in the effect of nonlinear diffusion and oscillatory nonlinear term to the asymptotic behavior of \(\lambda (\alpha )\) as \(\alpha \to \infty \) and \(\alpha \to 0\).

SUMMARY: We report on the Rellich–Leray inequality with optimal constant for axisymmetric and solenoidal vector fields in \(\mathbb {R}^N\). This is a vector field version of the Rellich inequality (1953) with the best value of constant factor. We aim to see how the best constant changes if the unknown vector fields are constrained by the divergence-free condition with axial symmetry.

SUMMARY: We consider the inequality which has a dual relation with the logarithmic Sobolev inequality of Beckner type. By using the relative entropy, we identify the sharp constant and the extremal of this inequality. Moreover, we derive the logarithmic uncertainty principle like Beckner’s one.

SUMMARY: We consider a maximization problem on the Trudinger–Moser inequality with Lebesgue term. As known result, a maximizer of the maximization problem on the classical Trudinger–Moser inequality exists. In this talk, we show that the attainability changes depending on the condition of the Lebesgue term.

SUMMARY: We consider a bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(C^2\) boundary. Then we establish the extension of Hardy’s inequality with weights using the function of distance from the boundary of \(\Omega \). Also we consider the variational problem associated with the weighted Hardy’s inequality involving compact perturbation. Then we study the infimum of the variational problem and the existence or non-existence of minimizer for the variational problem.

SUMMARY: We consider a system of coupled elliptic partial differential equations with critical growth in \(\mathbb {R}^d\) for \(d=3,4\) and study bifurcations of three families of radially symmetric, bounded solutions. We reduce the problems of the three families to those of three symmetric homoclinic orbits in a four-dimensional reversible system of ordinary differential equations and show that transcritical or pitchfork bifurcations of the three families occur at infinitely many parameter values.

SUMMARY: We will give exact solutions of a nonlocal boundary value problem with respect to the inviscid primitive equations in terms of generalized trigonometric functions. Moreover, we will show the asymmetry of the solutions by calculation corresponding to the evaluation of the median of the beta distribution.

SUMMARY: We investigate the structure of the nonnegative solutions set of an indefinite concave elliptic problem with Robin boundary conditions. We establish several qualitative properties of nontrivial nonnegative solutions of the problem. In particular, we prove a positivity property, which enables us to show that this problem has a subcontinuum of positive solutions. Furthermore, we prove an exact multiplicity result for nontrivial nonnegative solutions. Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds.

SUMMARY: In this talk, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.

SUMMARY: In this talk, we consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn–Vogelius functional.

SUMMARY: The restricted three-body problem is an important subject that deals with significant issues referring to scientific fields of celestial mechanics, such as analyzing asteroid movement behavior and orbit designing for space probes. The 2-center problem is its simplified model. The goal of this paper is to show the existence of brake orbits, which means orbits whose velocities are zero at some times, under some particular conditions in the planar \(2\)-center problem by using variational methods.

SUMMARY: We considered the random discretization of O’hara energy. O’hara energy is the energy defined for a knot, in the case of a specific index, it is called Moebius energy. Due to energy invariance under Moebius transformation, it is possible to show the existence of minimizer in the prime knot. Moreover, Kim and Kusner defined the discretization of Moebius energy for polygons in ’93, and S. Scholtes proved its Gamma convergence in ’14. We defined the random discretization of the weighted O’hara energy and show that the local uniform convergence and the compactness of our energy.

SUMMARY: We introduce a new discretization of the Möbius energy. The Möbius energy was named after its invariant property under Möbius transformations of the surrounding space. Known discretizations lose the Möbius invariance or \( \Gamma \)-convergence. This new discretization has the invariance, and converges to the original energy minus the energy of right circles in the sense of \(\Gamma \)-convergence under very natural assumptions. The starting point for this new discretization is the cosine formula of Doyle and Schramm.

SUMMARY: We consider both the minimization of a class of nonlocal interaction energies over non-negative measures with unit mass and a class of singular integral equations of the first kind of Fredholm type. Our setting covers applications to dislocation pile-ups, contact problems, fracture mechanics and 1D log gases. Our main result shows that both the minimization problems and the related singular integral equations have the same unique solution, from which we infer new regularity results on the minimizer of the energy and new positivity results on the solutions to singular integral equations.

SUMMARY: Recently, H. Mitake and H. V. Tran have provided a new and simple way to investigate the structure of viscosity solutions for a single ergodic problem of Hamilton–Jacobi equation. In this talk, as a generalization of this result, we address a uniqueness structure for a weakly coupled system. In particular, we study comparison principle with respect to a generalized Mather measure. To get the main result, it is important to construct Mather measures effectively. Nonlinear adjoint methods enable us to overcome this difficulty.

SUMMARY: The global equicontinuity estimate on \(L^p\)-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of \(L^p\)-viscosity solutions is established via approximation of given datum. The local Hölder continuity on the space derivatives of \(L^p\)-viscosity solutions is shown when the obstacles belong to \(C^{1, \beta }\), and \(p>n+2\).

SUMMARY: This talk is concerned with gradient estimates for the Dirichlet problem of heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

SUMMARY: We consider the heat diffusion in the whole space consisting of three layers with different constant conductivities, where initially the above two layers have temperature 0 and the below layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the above two layers and the below layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the below layer, then the two interfaces must be parallel hyperplanes.

SUMMARY: We consider a free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. This problem may be applied to model the spreading of biological species, where unknown functions are population density and spreading front of the species.
Kawai and Yamada (2016) found multiple spreading phenomena to the problem. It is possible to show that the spreading speed and profiles of solutions to the free boundary problem are determined by Semi-wave problem (SWP) if it has a unique solution pair. Otherwise if (SWP) has no solutions, a terraced profile of a solution to the free boundary problem is observed by numerical simulations. We will show that, under a suitable condition, the solution converges to so called a propagating terrace as time tends to infinity.

SUMMARY: In this talk, we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.

SUMMARY: In this talk we obtain necessary conditions and sufficient conditions for the solvability of the problem \begin{equation*} {\rm (P)} \qquad \partial _t u=\Delta u,\quad x\in {\bf R}^N_+,\,\,\,t>0, \quad \partial _\nu u=u^p, \quad x\in \partial {\bf R}^N_+,\,\,\,t>0 \end{equation*} with the initial condition \begin{equation*} u(x,0)=\mu (x)\ge 0, \quad x\in D:=\overline {{\bf R}^N_+}, \end{equation*} where \(N\ge 1\), \(p>1\) and \(\mu \) is a nonnegative measurable function in \({\bf R}^N_+\) or a Radon measure in \({\bf R}^N\) with \(\mbox {supp}\,\mu \subset D\). Our sufficient conditions and necessary conditions enable us to identify the strongest singularity of the initial data for the solvability for problem \(({\rm P})\).

SUMMARY: We study existence and nonexistence of a local in time solution for the weakly coupled reaction-diffusion system \(\partial _t u=\Delta u+g(v)\), \(\partial _t v=\Delta v+f(u)\) in \(\mathbb {R}^N \times (0,T)\), where \(N\ge 1\), \(T>0\) and \(f\) and \(g\) grow rapidly. We mainly consider the case where \(f\) and \(g\) are exponential or superexponential. We show that if the nonnegative initial data satisfies a certain integrability condition, then the local in time solution exists. Moreover, we show that there exists an initial data not satisfying the integrability condition such that the solution does not exist.

SUMMARY: In this talk, we shall consider local and global existence of a nonlinear diffusion equation of porous medium type with a nonlinear source. We obtain the estimate of the blow-up time and blow-up rate of solutions.

SUMMARY: In this talk, we consider doubly nonlinear parabolic equations \(p\)-Sobolev flow type, which include the classical Yamabe flow concerning so-called Yamabe problem on a bounded domain in Euclidean space in the case \(p=2\). We present some regularity results for their equations.

SUMMARY: We consider a family of axisymmetric hypersurfaces evolving by its mean curvature with driving force. However, the initial hypersurface is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions in the plane into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.

SUMMARY: We study the surface diffusion flow governing a curve on a half line. Two boundary conditions are imposed on the boundary. The second boundary condition is nonlinear. This problem was initiated by W. W. Mullins in 1957. We establish existence and uniqueness of the self-similar solution to our problem. Moreover, we discuss a stability result of self-similar solution.

SUMMARY: A kinetic equation describing mutation process in bacteria is considered. Such equation has been studied in view of the probability generating functions, however, this formulation enables us to analyse the equation via the Fourier transform, which is known to be a strong tool for kinetic equations such as the Boltzmann equation. It is shown that the Fourier-transformed equation has a characteristic function as a solution, which in turn gives a probability measure solution to the original equation. A Paley–Wiener type theorem is also shown to reveal relation of these functions and measures.

SUMMARY: Microorganisms are known to form spatiotemporal patterns similar to those formed in the Rayleigh–Bénard model for thermal convection. Among such, Euglena gracilis form distinct patterns induced by phototaxes and sensitivity to the gradient of the light intensity.

This talk reports on microscopic light sensing in Euglena gracilis and resulting formation of macroscopic patterns of cells.

SUMMARY: In this talk, we consider the spatially periodic Fisher-KPP equation in \(\mathbb {R}\). We investigate a minimizing problem associated with the ‘spreading speed’ for this equation. Here the spreading speed is the asymptotic speed of an expanding front that starts from a compactly supported initial data. We introduce a condition under which equality holds in an inequality about the spreading speed derived by Nadin.

SUMMARY: We consider a variational problem for the spreading speed \(c^*(b)\) of the solutions of the equation \(u_t=u_{xx}+b(x)(1-u)u\), \(x\in {\mathbb R},\ t>0\), where the coefficient \(b(x)\) is nonnegative and periodic in \(x\in {\mathbb R}\) with a period \(L>0\). The solution of this equation with compactly supported, non-trivial, nonnegative initial data converges to \(1\) locally uniformly in \(\mathbb R\) as \(t\rightarrow \infty \). It is known that there is the spreading speed \(c^*(b)\) such that the observers who move slower than \(c^*(b)\) will see the solution converges to 1 and those who move faster than \(c^*(b)\) will see the solution converges to 0 locally uniformly in \(\mathbb R\) as \(t\rightarrow \infty \).

In this talk, we present our results of the study of the problem of maximizing \(c^*(b)\) by varying the coefficient \(b(x)\) under some constraints.

SUMMARY: This talk is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction-diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang-bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (2010). To this purpose, we perturb the distribution of resources, and compute the first and second variations of the total population.

SUMMARY: We investigate the behavior of positive solutions for the logarithmic diffusion equation. We are interested in the behavior of solutions which extinct in a finite time. More precisely, we prove that the re-scaled solution converges to a traveling pulse. In addition, we also discuss the behavior of solutions which converge to a monotone traveling wave.

SUMMARY: We study the initial boundary value problem for the reaction-diffusion equation with isochronous nonlinearity. We prove that small solutions become spatially homogeneous and is subject to the ODE part asymptotically. We also discuss blow-up of an parabolic system with quadratic nonlinearity having the origin as an uniform isochronous center.

SUMMARY: We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection-diffusion equation in a uniformly local function space, where the solution is not decaying at space infinity. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local Lebesgue spaces.

SUMMARY: This talk is concerned with asymptotic behavior of solutions to a fully parabolic Keller–Segel model with signal-dependent sensitivity. In the case that the signal-dependent function is given by \(\chi /v\), Fujie established global existence of bounded classical solutions under some smallness condition for \(\chi \) in 2015, and Winkler–Yokota showed asymptotic behavior of these solutions under some additional smallness condition for \(\chi \) in 2018. On the other hand, in the case that the signal-dependent function is given by \(\chi /(1+v)^k\) with some \(k>1\), global existence and boundedness of classical solutions were established under some smallness condition for \(\chi \) (M.–Yokota, 2017); however, asymptotic behavior of these solutions is still open. The purpose of the present talk is to discuss asymptotic behavior of classical solutions under some smallness condition for \(\chi \).

SUMMARY: In this talk we consider an effect of nonlinear diffusion on a lower bound for the blow-up time for solutions of a fully parabolic chemotaxis system. The case of linear diffusion was studied by Tao–Vernier Piro in 2016 and by Anderson–Deng in 2017. The purpose of this talk is to generalize these results to the case of nonlinear diffusion.

SUMMARY: We consider the non-existenceof a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the fast diffusion exponent. We show the solution for the fast diffusion cases with the diffusion exponent \(\frac {n}{n+2}<\alpha <1\) blows up in a finite time if the initial data satisfies certain condition involving the free energy. We also show the finite time blow up for radially symmetric case without finite moment condition. The key idea is to use the generlized version of Shannon’s inequality and apply the virial low of the system.

SUMMARY: We study the large time asymptotics of solutions to nonlinear dispersive equations which have a nonlocal and nonhomogeneous dispersive term. These equations with quadratic nonlinearities describe several models of nonlinear dispersive waves. In this talk we consider the equation with a cubic nonlinear interaction which is a critical nonlinearity for the existence of scattered states. We show that there exist modified scattered states.

SUMMARY: We consider the nonlinear Schrödinger equation with a potential in three dimensions. We determine the long time behavior of the solutions to this equation with a data below the ground state. More precisely, we give sufficient conditions for the solutions scatter and sufficient 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 consider the local well-posedness for fourth order Benjamin–Ono type equations on the torus. The equation with specific coefficients is integrable and the third equation in the Benjamin–Ono hierarchy. The proof is based on the enery method with correction terms. Although correction terms can eliminate the worst term in the energy inequality, they may yield a different deriavtive loss, which is the main difficulty in our problem. In order to overcome the difficulty, we add a new correction term into the energy.

SUMMARY: We give the ill-posedness of the Cauchy problem for the Dirac–Klein–Gordon system in one dimension. This shows that the well-posedness result by Machihara et al. (2010) is optimal. Remark that our situation is different from the problems which can be applied the argument by Bejenaru–Tao (2006).

SUMMARY: We will talk about asymptotic behavior of complex valued solutions to Klein–Gordon equation with a critical gauge-invariant nonlinearity. The main result is the existence of a solution which asymptotically behaves as a linear solution with a logarithmic phase correction.

SUMMARY: We consider the instability for the standing wave solutions to the system of the quadratic Klein–Gordon equations. In the case of the nonlinear Klein–Gordon equation with power nonlinearity, stability and instability for the standing wave solutions have been extensively studied. On the other hand, in the case of our system, there is no result for the stability and instability for the standing wave solutions. In this talk, we prove the very strong instability for the standing wave solutions to our system. The proof is based on the techniques in Ohta and Todorova (2007). New ingredient is to need the mass resonance condition in two or three space dimensions whose cases are the mass sub-critical case.

SUMMARY: In this talk we consider upper bounds of lifespan of solutions to the semilinear wave equation \(\partial _t^2u-\Delta u = |u|^p\) in \(\mathbb {R}^N\). The main contribution of this work is to give an alternative proof of upper bounds of lifespan of solutions and then we found the way to prove it without the (pointwise) positivity assumption for initial data. The problem for weakly coupled systems are also discussed.

SUMMARY: The main purpose in this talk is to show the Strichartz estimates for magnetic Klein–Gordon equations in exterior domain. The fundamental tools are Strichartz estimates for free solutions and smoothing estimates for free and perturbed solutions. Moreover by using Strichartz estimates we will show the global existence and scattering theory for the solutions to nonlinear equations with small data.

SUMMARY: We consider the Cauchy problem of the semilinear wave equation with a damping term increasing near the spatial infinity. We determine the critical exponent, which is the threshold between the global existence and nonexistence for small initial data.

SUMMARY: The magnetohydrodynamic system (MHD) of damped wave type is considered as an intermediate system between the classical MHD and the Navier–Stokes–Maxwell system. We prove that the second one is obtained from the first one by taking limit in Bochner–Fourier–Lebesgue spaces. In this space we can calculate directly the symbol of the fundamental solution to damped wave and heat equations, and the results hold.

SUMMARY: We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.

SUMMARY: We prove that an analytic smoothing effect for a solution to the system of nonlinear Schrödinger equations for gauge invariant nonlinearities with mass resonant condition. It is shown that under rapidly decaying condition on the initial data, the solution shows a smoothing effect and is real analytic with respect to the space variable. Our theorem is an extension to the known results for the analytic smoothing effect to the nonlinear Schrödinger system with the gauge invariant setting.

SUMMARY: We consider the following nonlinear Schrödinger equation of derivative type:
(1) \( i \partial _t u + \partial _x^2 u +i |u|^{2} \partial _x u +b|u|^4u=0 , \ (t,x) \in \mathbb {R} \times \mathbb {R}, \ b \in \mathbb {R} . \)
If \(b=0\), this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS). For DNLS it is known that if the initial data \(u_0\in H^1(\mathbb {R})\) satisfies \(\| u_0\|_{L^2}^2 <4\pi \), the corresponding \(H^1(\mathbb {R})\)-solution is global. The main aim of this talk is to investigate global well-posedness in the energy space \(H^1(\mathbb {R})\) for the equation (1) from the viewpoints of the solitons. We extend the global results for DNLS to the equation (1) by variational approach. Interestingly, if \(b<0\), \(4\pi \)-mass condition in DNLS is improved due to the defocusing effect from the quintic term.

SUMMARY: We consider the following nonlinear Schrödinger equation of derivative type:
\( i \partial _t u + \partial _x^2 u +i |u|^{2} \partial _x u +b|u|^4u=0 , \ (t,x) \in \mathbb {R} \times \mathbb {R}, \ b \in \mathbb {R} . \)
This equation has a two-parameter family of solitons. The value \(b=-\frac {3}{16}\) gives the turning point where the structure of the solitons changes. Especially algebraic solitons exist only for the case \(b>-\frac {3}{16}\). In previous results the orbital stability and instability of these solitons have been studied in the case \(b\geq 0\). In this talk, by variational approach we prove the orbital stability of the solitons including the algebraic solitons in the case \(-\frac {3}{16}<b<0\). We see that the effect of the momentum plays an essential role in the arguments on the stability of the solitons.

SUMMARY: In this talk, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma iteraction. Some well-posedness results in the Sobolev space \(H^{s}(\mathbb {R}^d)\) was obtained in the previous works by H. Hirayama (2014) and H. Hirayama and S. Kinoshita (2019). We improve these results for conditional radial initial data by rewriting the system into radial form.

SUMMARY: We consider mass-subcritical nonlinear Schrödinger equation. It is known that under the radial symmetry, the Cauchy problem is well-posed in the scale critical Sobolev space. In this talk, we consider long time behavior of solutions. In particular, we study a minimization problem with respect to non-scattering solutions.

SUMMARY: We discuss the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical \(BV\) solution, instead a \(\delta \)-shock appears, which can be viewed as a generalized measure–valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exists infinitely many bounded admissible weak solutions starting from the same initial data.

SUMMARY: We consider the motion of a point mass in a 1D viscous compressible fluid. Our main theorem shows that the velocity of the fluid \(u(x,t)\) decays time asymptotically as \(||u(\cdot ,t)||_{L^{\infty }}\approx t^{-1/2}\), while the velocity of the point mass \(V(t)\) decays at least as \(|V(t)|\approx t^{-3/2}\). This is in contrast with the result for the Burgers fluid (Vázquez and Zuazua, Comm. Partial Differential Equations, 28:1705–1738, 2003).

The result above is proven as a corollary of more detailed pointwise decay estimates of the fluid variables, which are shown by using the pointwise decay estimates of Green’s function for the corresponding Cauchy problem (Liu and Zeng, Mem. Amer. Math. Soc., 125(599), 1997). Our result shows that the Green’s function approach is useful in the analysis of fluid-structure interaction problems.

SUMMARY: Global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state is studied. This system describes liquid-vapor two phase flow with phase transition as diffuse interface model. In previous works they assume that the pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is accurately non-monotone function and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. It is shown that in the critical case for small data whose momentum has derivative form there exist global \(L^2\) solutions and the parabolic type decay rate of the solutions is obtained. The proof is based on decomposition method for solutions to a low frequency part and a high frequency part.

SUMMARY: We consider the compressible and incompressible two-phase flows with phase transitions and surface tensions in bounded regions. We assume that the two fluids are separated by a sharp free boundary. We show the existence of the global unique strong solution supposing that the initial data are small in their natural norms.

SUMMARY: We consider a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions and prove existence of weak solutions for it. In contrast to previous works, we study a model with a singular non-local free energy, which controls the fractional \(L^2\)-Sobolev norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.

SUMMARY: In this talk, I will talk about the global well-posedness for the two phase problem of incompressible viscous fluid flows separated by a sharp interface in the whole space. I consider the case where the surface tension is taken into account. The key is to use the Hanzawa transform whose vertex is at the barycenter point of the unknown time dependent domain, maximal \(L_p\)-\(L_q\) regularity results and \(L_p\)-\(L_q\) decay properties of linearized equations. Since the domain is unbounded, we have to choose different exponents \(q_1\) and \(q_2\) in space.

SUMMARY: In this talk, I will talk about the \(L_p\)-\(L_q\) decay properties of the Stokes semigroup which arises in the study of two phase problem for the viscous incompressible fluid flows separated by a sharp interface with surface tension in the whole space. This is a key step to prove the global well-posedness.

SUMMARY: The functional analytic property of the linearized operator in the equation of perturbations around spherically symmetric equilibria of gaseous stars governed by the Euler–Poisson equations has been studied. In spite of often used supposition, the spectrum of the self-adjoint realization is not of the Sturm–Liouville type generally, but its structure can be clarified with sufficient concreteness.

SUMMARY: We show the Brezis–Gallouet–Wainger inequalities by means of the Vishik type spaces which can be wider than \(\dot {B}^{0}_{\infty ,\infty }\). As an application of those inequalities. We prove that the strong solutions to Navier–Stokes equations can be extended if the scaling invariant quantity of vorticity is bounded. Namely, the Beale–Kato–Majda type regularity criteria are improved in the terms of the Vishik type space.

SUMMARY: Stability of the stationary solution of the Burgers equation in exterior domains in n-D is concerned. We consider the asymptotic stability of radially symmetric stationary solutions for multi-dimensional Burgers equation from the n-D perturbed fluid motion. For this purpose, we apply the result by Kozono and Ogawa which showed the asymptotic stability of stationary solutions for the incompressible Navier–Stokes equation on multi-dimensional spaces.

SUMMARY: In this talk, we concern the Cauchy problem of the nonhomogeneous incompressible Navier–Stokes–Korteweg equations on the two-dimensional space with vacuum as the far field density. We establish the local existence and uniqueness of strong solutions to the 2D Cauchy problem of the nonhomogeneous incompressible Navier–Stokes–Korteweg equations provided the initial density decay not too slow at infinity. Our analysis is based on some weighted energy estimates.

SUMMARY: In this talk, we consider a steady flow of an incompressible viscous fluid for a 3-D aperture domain. As is well known, J. G. Heywood[1] introduces an aperture domain. In the aperture domain he obtains a steady solution of the Navier–Stokes equations with the restricted flux condition. We define a generalized aperture domain. In such a domain, we consider the steady flow of an incompressible viscous fluid with the flux condition. We obtain a solution of such a problem with the restricted flux condition.

SUMMARY: We consider the stationary Navier–Stokes equations in 2D. Under the assumption that \(\nabla v \in L^q\) with some \(q \in (2,\infty )\), we give asymptotic properties of solutions. As its application, we also show the Liouville-type theorem.

SUMMARY: We study the temporal decay estimate of the Oseen semigroup in a two-dimensional exterior domain. We establish the local energy decay estimate with a suitable dependence on the small translation speed, which is a significant extension of Hishida’s result in 2016. As an application, we prove the \(L^q\)-\(L^r\) estimates of the Oseen semigroup uniformly in the small translation speed.

SUMMARY: We investigate the Navier–Stokes initial boundary value problem in the half-plane with non decaying initial data. We introduce a technique that allows to solve the two-dimesional problem, further, but not least, it can be also employed to obtain weak solutions, as regards the non decaying initial data, to the three-dimensional Navier–Stokes IBVP.

SUMMARY: A model from electro-magneto-hydrodynamics describing a completely ionized gas, a plasma, is studied. Local well-posedness of the problem in time weighted \(L_p\) space is obtained by means of maximal regularity of the linearized problem, and the induced local semiflow in the proper state space is constructed. Based on the principle of linearized stability, it is shown that the trivial solution of the problem is exponentially stable.