2018年度年会(於:東京大学)

SUMMARY: The exact WKB analysis is based on the Borel resummation method (or the Borel–Laplace method) and, as its consequence, the instanton-type expansions of Painlevé transcendents play an important role in the exact WKB analysis of Painlevé equations. However, such instanton-type expansions are very wild objects and we cannot expect their convergence in general. In this talk, I would like to propose a new approach to handle the instanton-type expansions of Painlevé transcendents. A key idea of this approach is to use the instanton-type expansions of elliptic functions. The structure of instanton-type expansions of elliptic functions will be also discussed in the talk.

SUMMARY: We first study the global and local behavior of bifurcation curves for elliptic nonlinear eigenvalue problems in \(L^q\)-framework (\(q > 1\)). We consider the case where \(\lambda \) is parameterized by the \(L^q\)-norm \(\alpha = \Vert u_\lambda \Vert _q\) of the solution \(u_\lambda \) corresponding to \(\lambda \) and is represented as a continuous function \(\lambda = \lambda (\alpha )\). Especially, we restrict our attention to the asymptotic behavior of \(\lambda (\alpha )\) as \(\alpha \to \infty \) and \(\alpha \to 0\). We establish several precise asymptotic expansion formulas for \(\lambda (\alpha )\) as \(\alpha \to \infty \) and \(\alpha \to 0\) to understand well the total structures of the bifurcation curves. We next consider the inverse bifurcation problems. Especially, we will show some results for inverse problems by using a variational method and asymptotic expansion formulas for \(\lambda (\alpha )\) for \(\alpha \gg 1\).

SUMMARY: When several stable states coexist, propagation phenomena are often observed in many fields including dissipative situations. To characterize the universal profiles of these phenomena, traveling wave solutions and entire solutions play important roles. Here traveling wave solution is meant by a solution of a partial differential equation that propagates with a constant speed, while it maintains its shape in space, and an entire solution is a solution defined for all space and time variables. In this talk we focus on the Allen–Cahn–Nagumo equation, which is a single reaction diffusion equation with bistable nonlinearity and explain how to construct entire solutions and the relation between traveling wave solutions and entire solutions.

SUMMARY: We consider the ill-posedness problem for the compressible Navier–Stokes system under the barotropic condition in the critical Besov spaces. It is known that the existence and the uniqueness of the solution hold in the homogeneous Besov spaces \(\dot B^{\frac {n}{p}}_{p,1}\times \dot B^{\frac {n}{p}-1}_{p,1}\) with \(1\leq p<2n\), where the density and the velocity belong to \(\dot B^{\frac {n}{p}}_{p,1}\) and \( \dot B^{\frac {n}{p}-1}_{p,1}\), respectively. On the other hand, if \(p > 2n\), the solution does not depend on initial data continuously in general. In this talk, we show that for the critical case \(p = 2n\) the system is ill-posed by showing the norm inflation.

SUMMARY: In this study, we investigate a mathematical model of hematopoietic stem cells. The model is described by a system of partial differential equations, which depend on space and age. By applying the method of characteristics, we reformulate the model into a reaction-diffusion equation with a nonlocal spatial term and time delay. We prove the existence, uniqueness and positivity of the solution, and obtain a threshold condition for the global asymptotic stability of the trivial equilibrium. In addition, we obtain sufficient conditions for the existence of nontrivial equilibrium and the uniform persistence of the system.

SUMMARY: In this talk, we introduce the notion of dimensioned numbers and extended dimensioned numbers. The notion of dimensioned numbers originates in geometric measurements, and they can be used to describe iterated exponential functions of a single variable. We apply the theory of dimensioned numbers to solving some functional equations of a single variable.

SUMMARY: The oscillation problem of a nonlinear delay difference equation is studied. Sufficient conditions for all solutions of the equation to be oscillatory and for the existence of a nonoscillatory solution are established. Our main results are proved by use of the phase plane analysis which is developed in a similar way to Sugie and Ono [4] in 2004.

SUMMARY: We consider a Hamilton–Jacobi flow staring from a pathological function. Here, a function on \(\bf R\) is said to be pathological, if it is everywhere continuous but nowhere differentiable.

SUMMARY: Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the \(p\)-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the \(p\)-Laplacian. However, few applications to differential equations unrelated to the \(p\)-Laplacian are known. We will apply GTFs with two parameters to a nonlinear nonlocal boundary value problem without \(p\)-Laplacian.

SUMMARY: We consider the oscillation problem for nonlinear differential equation \((|x'|^{p(t)-2}x')'+(\lambda /t^{p(t)})|x|^{p(t)-2}x=0\), where \(\lambda \) is a positive constant and \(p(t)>1\) is a nondecreasing and smooth function. Using Riccati technique and function sequence technique, we obtain sufficient conditions for this equation to be (non)oscillatory. The obtained results show that there exists a critical value for this problem.

SUMMARY: We study the bifurcation of positive solutions for the one-dimensional \((p,q)\)-Laplace equation with nonlinear term \(u^{r-1}\). There are five types of order relations for \((p,q,r)\). We study the exact shape of the bifurcation curve in each type of the order relation. Furthermore, we investigate the asymptotic profile of the normalized solution \(u(x)/\|u\|_\infty \) as \(\|u\|_\infty \to 0\) or \(\|u\|_\infty \to \infty \), where \(\|u\|_\infty \) denotes the \(L^\infty \)-norm of \(u\).

SUMMARY: This talk deals with the two-dimensional linear differential system \[ x' = y, \quad y' = -x-h(t)y \] on \([t_0,\infty )\), where \(h \in C^1[t_0,\infty )\) and \(h(t)>0\) for \(t \ge t_0\). Criteria to obtain the box dimension of graphs of solution curves are established.

SUMMARY: The autonomous planar half-linear differential system is considered, which is a generalization of the autonomous planar linear system. It is well-known that the autonomous planar linear system can be solved by eigenvalues, that is, roots of the characteristic equation. In this talk, the characteristic equation for the autonomous planar half-linear differential system is introduced, and the asymptotic behavior of its solutions is established by roots of the characteristic equation.

SUMMARY: A global attractor is a notion for a topological semi-dynamical system whose phase space is a metric space. In particular, this notion is important when the phase space is an infinite-dimensional Banach space by the finiteness of its fractal dimension. However, it should be noticed that the notion of a global attractor depends on the specific choice of a metric. In this talk, we “define” global attractors in the context of the “non-existence of natural metrics” of the phase space and study those properties. This includes a case where the phase space is a Fréchet space, which is motivated by differential equations with unbounded delay. We obtain sufficient conditions for the existence, which will be applied to such equations.

SUMMARY: Voros coefficients are important objects in exact WKB analysis to study global behavior of solutions of differential equations. In this talk I will report that the Voros coefficients for hypergeometric differential equations with two variables are given by the generating functions of free energies defined in terms of Eynard and Orantin’s topological recursion.

SUMMARY: From the viewpoint of the division by zero (\(0/0=1/0=z/0=0\)) and the division by zero calculus (\(\tan (\pi /2) = 0\)), we will show some incompleteness of the theory of differential equations in an undergraduate level and we will propose fundamental open problems as the results.

Other topics in this talk: Differential equations with singularities; Continuation of solution; Singular solutions; Solutions with singularities; Solutions with an analytic parameter; Special reductions by division by zero of solutions; Partial differential equations; Introduction of \(\log 0 =\log \infty =0\); and Applications of \(\log 0=0\); \(e^0= 1,0\).

SUMMARY: Here, we will give the interpretation for the Hadamard finite part of singular integrals by means of the division by zero calculus - \(\log 0 =\log \infty =0\) (not as limiting values) in the meaning of the one point compactification of Aleksandrov.

SUMMARY: Let \(\Omega \) be a domain in \(\mathbb {R}^{n}\) with \(n \geq 2\), and let \(u\) be a nonnegative \(p\)-superharmonic function in \(\Omega \). Kilpeläinen and Malý proved that there exists a constant \(C > 0\) such that \begin{equation*} u(x) \leq C \left ( \inf _{B(x, R)} u + \mathbf {W}_{p}^{\mu }(x, 2R) \right ) \end{equation*} whenever \(B(x, 2R) \subset \Omega \), where \(\mu \) is the Riesz measure of \(u\) and \(\mathbf {W}_{p}^{\mu }(x, 2R)\) is the Wolff potential of \(\mu \). In this paper, we extend this inequality to near the boundary of \(\Omega \). More precisely, we give a pointwise estimate for \(p\)-superharmonic functions which vanish on the boundary and a global integrability estimate of \(p\)-superharmonic functions. Combining the two estimates, we give an analog of the Carleson estimate.

SUMMARY: Let \(\Omega \) be a smooth bounded domain of \({\bf R}^N\). In this paper, we study the equivalences among \(p\)-capacity, \(p\)-Laplace-capacities and Hausdorff measure. Firstly we present the equivalence between \(p\)-capacity \(C_{p}(K)\) and \(p\)-Laplace-capacitiy \(C_{\Delta _p}(K)\) relative to \(\Omega \) for a given compact set \(K \subset \Omega \). Secondly we establish the equivalence between \(p\)-Laplace capacity \(C_p(K,\partial \Omega )\) relative to \(\partial \Omega \) and Hausdorff measure \(\mathcal H^{N-1}(K)\) on \(\partial \Omega \) for a given compact set \(K\subset \partial \Omega \).

SUMMARY: The Lamé operator is a 2nd order linear elliptic differential operator frequently used to describe the oscillations that take place in a uniform isotropic elastic body. When the oscillations are time-periodic, the differential equation can be simplified to the spectral analysis of the Lamé operator. In our case, we study the case of a thin straight elastic body such that it has its ends fixed. In particular we provide results about the asymptotic behaviour of the eigenvalues and eigenfunctions as the domain gets thinner.

SUMMARY: We consider the inverse scattering problem of time-harmonic acoustic plane waves by multiple impenetrable obstacles. For the purpose, we derive the factorization method of Kirsch, which is a sampling method for solving certain kinds of inverse problems where the shape and location of a domain have to be reconstructed. We introduce new results to reconstruct the unknown obstacles by the factorization method. The main idea is to modify the original factorization method by using a priori known outer and inner estimations for a part of unknown obstacles. By our work, we can expand the application of the factorization method for some inverse acoustic scattering problems.

SUMMARY: We consider Lou–Nagylaki conjecture (2002) on a stationary problem of some reaction diffusion equation in population genetics. We deal with the case where spatial dimension is 1. In this case a stationary problem of the equation satisfies \(d u''+g(x)u^2(1-u)=0\) with Neumann zero boundary condition.

Under the condition \(\int _{\Omega }\,g(x)\,dx\geq 0\) and some additional condition on \(g(x)\), uniqueness of a nontrivial solution has been already shown. In this talk we construct many nontrivial solutions for some \(g(x)\) satisfying \(\int _{\Omega }\,g(x)\,dx<0\).

SUMMARY: This talk is concerned with the stationary problem of a diffusive Lotka–Volterra prey-predator model with population flux by attractive transition. We analyze two limiting systems as the nonlinear diffusion coefficient approaches infinity. A main result reveals the global bifurcation structure of positive solutions of one of the limiting systems.

SUMMARY: We study radial solutions of the semilinear elliptic equation \(\Delta u+f(u)=0\) under rather general growth conditions on \(f\). We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a certain transformation, all the limit equations can be reduced into two canonical cases, i.e., \(\Delta u+u^p=0\) and \(\Delta u+e^u=0\).

SUMMARY: It is known that every positive solution of a one-dimensional Gel’fand problem can be written explicitly. In this talk we give exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We study asymptotic behaviors of eigenvalues and eigenfunctions as the \(L^{\infty }\)-norm of the solution goes to the infinity. We also study the problem \(u''+\lambda e^{-u}=0\) and the associated linearized problem.

SUMMARY: In this talk, we discuss the existence of a loop component of nontrivial nonnegative solutions for a concave-convex elliptic problem with the Neumann boundary condition. Positivity for solutions on the loop is also discussed. Our approach relies on bifurcation analysis.

SUMMARY: In this talk we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one-sided Borel probability measure. If such a sequence blows up, we derive some estimate which is related to the behavior of solution near the blow-up point. Moreover, we study the two-intensities case to derive the sufficient condition for this estimate.

SUMMARY: In this talk we consider low energy sign-changing radial solutions to a elliptic problem related to the Trudinger–Moser inequality. We study the asymptotic behaviour of them. As a result, we show that when the solution has \(k\) interior zeros, it exhibits a multiple blow-up behaviour in the first \(k\) nodal sets while it converges to the least energy solution of a critical problem in the \((k+1)\)-th one. We also prove that in each concentration set, with an appropriate scaling, the solution converges to the solution of the classical Liouville problem in \(\mathbb {R}^2\).

SUMMARY: O’Hara defined some knot energies for finding canonical configuration of a knot in a given knot type. One of them is known as the Möbius energy. The authors showed that the Möbius energy can be decomposed into three parts; the first one measures how the curve is bent, the second one does how the curve is twisted, and the third one is an absolute constant. The authors define generalized O’Hara’s energies, and announce that they can be decomposed in a similar way under suitable assumptions including the case of O’Hara’s \( ( \alpha , 1 ) \) energies with \( \alpha \in [ 2 , 3 ) \), which are self-repulsive and bounded for any smooth curves without self-intersections.

SUMMARY: We prove the existence of infinitely many solutions for \(- \Delta u + V(x) u = f(u)\) in \(\mathbb {R}^N\), \(u \in H^1(\mathbb {R}^N)\), where \(V(x)\) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions of \(f(u)\) only around \(0\) and at \(\infty \).

SUMMARY: The weak Harnack inequality for \(L^p\)-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is established. It is shown that Hölder continuity of \(L^p\)-viscosity solutions is derived from the weak Harnack inequality for \(L^p\)-viscosity supersolutions. Furthermore, the local maximum principle for \(L^p\)-viscosity subsolutions is shown. By these properties, the Harnack inequality for \(L^p\)-viscosity solutions is obtained. Several further remarks are presented.

SUMMARY: In 1957, A. Friedman proved the Phragmén–Lindelöf theorem for classical solutions of linear parabolic equations in cones, whose axis is the positive \(t\)-axis and whose vertex is the origin of \({\mathbb R}^{n+1}\). We establish the Phragmén–Lindelöf theorem for fully nonlinear uniformly parabolic equations with unbounded coefficients in same domains.

SUMMARY: We consider the time periodic abstract linear parabolic evolution equation \(\partial _t u + Au = f, t\in \mathbb {R}, u(t+T)=u(t)\). We construct the general theory on a real interpolation spaces \(D_A(\theta , p)\). It corresponds to time periodic version of the DaPrato–Grisvard maximal \(L^p-D_A(\theta , p)\) regularity theorem. Moreover as its application we prove that the nonlinear bidomain equations which is the electrophysiological model have unique time periodic solution near the stable solution by fixed point theorem if the periodic data is sufficiently small.

SUMMARY: We consider the bidomain equations with Fitzhugh–Nagumo type nonlinear term. We prove that there exists a periodic solution if the data is periodic. Here we do not assume the smallness of the data and the nonlinear term is super-linear. The proof is based on a weak-strong uniqueness argument. To construct the weak periodic solution, we use the Brouwer’s fixed point theorem for the Poincaré map.

SUMMARY: The subject of this work is to construct a new approach to a parabolic-elliptic Keller–Segel system from its parabolic-parabolic case, and to use the parabolic-parabolic case as a step to establish new results in the parabolic-elliptic case. Our aim is, by considering that the parabolic-elliptic case is as a limit of its parabolic-parabolic system, to establish a result such that, only dealing with the parabolic-parabolic Keller–Segel system is enough to obtain the new properties for solutions of its parabolic-elliptic case. In this talk we consider fast signal diffusion limit in a Keller–Segel system, which namely is convergence of a solution for the parabolic-parabolic Keller–Segel system to that for its parabolic-elliptic version.

SUMMARY: This talk deals with a 3D two-species Keller–Segel–Stokes system with competitive kinetics. Recently, in a 3D two-species chemotaxis-Stokes system Cao–K.–Mizukami proved global existence and asymptotic behaviour of classical solutions under some conditions. However, the same argument as in the previous work could not be applied to the present problem. The present work asserts global existence and asymptotic behaviour of classical solutions for the Keller–Segel setting.

SUMMARY: In this talk we will consider the initial boundary problem for degenerate Keller–Segel systems. For the non-degenerate systems, it is known that \(q=m+\frac {2}{N}\) (\(m\) denotes the intensity of diffusion, \(q\) denotes a nonlinearity and \(N\) is the space dimension) is the critical condition for boundedness and blow-up. It is expected that the case of degenerate diffusion has the same critical condition. However, for the blow-up results, the previous paper gave only the existence of unbounded solutions, which includes the blow-up in infinite time. This talk hence gives finite-time blow-up of energy solutions from the initial data with a large negative energy.

SUMMARY: We show unboundedness of solutions to a degenerate drift-diffusion equation with the mass critical exponent and estimates of the concentration quantity of radially symmetric solutions. When a given initial datum has finite second moment and the energy functional of it is initially negative, it is known that the corresponding solution blows up in finite time. We prove that solutions to our problem do not remain bounded in the energy space even if we do not impose such a weight condition under the negative energy condition. In particular, if the solution is radially symmetric and the energy functional of the initial datum is negative, then we can eliminate the possibility of growing up. Moreover, we give lower estimates of the mass for radially symmetric blow-up solutions.

SUMMARY: In this talk, we study a doubly nonlinear parabolic equation, called the \(p\)-Sobolev flow, which is the classical Yamabe flow on a bounded domain in Euclidean space. We show the existence of a weak solution to the \(p\)-Sobolev flow without geometrical assumption and present properties of its solution.

SUMMARY: We study how the presence of a surface of the constant flow property influences the shape of a two-phase heat conductor. The existence of a surface that satisfies the constant flow property at every moment in time is a very strong requirement: we show that this condition implies the radial symmetry of our heat conductor. In addition, we study the difference behaviour of two-phase heat conductors satisfying an analogous overdetermined elliptic problem. In this case we are able to construct a family of non radially symmetric solutions.

SUMMARY: We study a singular limit problem of the Allen–Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke’s mean curvature flow with a generalized right angle condition on the boundary of the domain.

SUMMARY: We consider a dynamic boundary value problem for singular degenerate parabolic equations in a half space. In the context of viscosity solutions, we establish a comparison principle and prove existence of solutions together with Lipschitz regularity of the unique solution. A relationship with a Dirichlet or Neumann condition is also studied.

SUMMARY: In this talk we will discuss blow-up of a harmonic map heat flow from \(R^d\) to \(S^d\), where \(S^d\) denotes a unit sphere in \(R^d\). Our main result yields a constructive examples of Type II blow-up solutions for \(d \geq 7\). These blow-up solutions satisfy various point-wise estimates in some space-time regions.

SUMMARY: In this talk, we consider a free boundary problem of Fisher-KPP equation \(u_t=u_{xx}+u(1-u)\), \(t>0\), \(ct<x<h(t)\). The number \(c>0\) is a given constant, \(h(t)\) is a free boundary which is determined by the Stefan-like condition. This model may be used to describe the spreading of a non-native species over a one dimensional habitat. The free boundary \(x=h(t)\) represents the spreading front. In this model, we impose zero Dirichlet condition at left moving boundary \(x=ct\). This means that the left boundary of the habitat is a very hostile environment and that the habitat is eroded away by the left moving boundary at constant speed \(c\). In this talk, I will give a trichotomy result, that is, for any initial data, exactly one of the three behaviours, vanishing, spreading and transition, happens.

SUMMARY: We consider the Cauchy problem for semilinear heat equation in \(R^N\). We study the case where initial data have polynomial decay rate at the spatial infinity, and investigate the convergence property of the global solutions to the forward self-similar solutions.

SUMMARY: For repulsive Hamiltonians we obtained Rellich’s theorem, the radiation condition and the limiting absorption principle. Our setting include the case of the inverted harmonic oscillator. In the proofs, we mainly use a commutator argument. This argument simple and elementary, and does not employ energy cut-offs or the microlocal analysis.

SUMMARY: My talk is concerning on the the mathematical theory of quantum measurements. Especially, we will consider the continuous quantum measurements of the position of particles during a finite time. The probability amplitude of particles just after the measurement is given in the form of the weighted Feynman path integrals, WFPI, or the restricted Feynman path integrals, according to Feynman and Mensky’s theory of quantum measurements. We will show that WFPI for it are defined mathematically in \(L^2\) and the weighted Sobolev spaces, and satisfy the non-self-adjoint Schrödinger equations.

SUMMARY: We obtain higher order expansions of evolution operators corresponding to the Cauchy problem of the linear damped wave equation in \(\textbf {R}^n\). Established hyperbolic part of expansion seems to be new in the sense that the order of the expansion of the hyperbolic part depends on the spatial dimension.

SUMMARY: We consider the new structural conditions to show the decay property of the linear symmetric hyperbolic systems from the viewpoint of the dissipative structure. Especially, we are concerned with the model systems which have a non-symmetric relaxation and therefore their decay estimate is of regularity-loss. Recently, since the Shizuta–Kawashima stability theory cannot be applicable to such a system, the structural conditions to show the decay property of regularity-loss have been investigated. However, we have the last question that the structural condition which can be applied to the Timoshenko–Cattaneo system has not been developed yet. In this talk, we introduce the new structural condition which will be the first condition applicable to the Timoshenko–Cattaneo system and the other systems of the same weakest dissipative mechanism.

SUMMARY: In this talk, we addressed with rarefaction waves for a hyperbolic system of balance laws in the whole space or half space. We shall prove a priori estimate of a solution and the asymptotic stability of rarefaction waves by using \(L^2\)-energy method and standard calculus.

SUMMARY: We consider the semilinear wave equation with space-dependent critical damping term in \(\mathbb {R}^N (N\geq 3)\). The equation is of the form \(\partial _t^2u-\Delta u+V_0|x|^{-1}\partial _t u=|u|^p\). If \(V_0=0\), then small data blowup for \(p\leq p_0(N)\) and small data global existence for \(p>p_0(N)\) are proved with well-known Strauss exponent \(p_0(N)\). In this talk we will show that a similar blowup phenomenon occurs for \(\frac {N}{N-1}<p<p_0(N+V_0)\).

SUMMARY: We will study the initial-boundary value problem for the Kirchhoff type plate equation with rotational inertia. In particular, we consider the long-time behavior of the solution of the equation and show the existence of attractors and clarify their properties. Furthermore, we will also cover the influences of the rotational inertia on the long-time dynamics.

SUMMARY: We study the Cauchy problem of the damped wave equation \begin{align*} \partial _{t}^2 u - \Delta u + \partial _t u = 0 \end{align*}

and give \(L^p\)-\(L^q\) estimates of the solution for \(1\le q \le p < \infty \ (p\neq 1)\) with derivative loss. We apply this estimate to the nonlinear problem

SUMMARY: We consider the Cauchy problem of the nonlinear wave equation with a scaling critical time-dependent damping \(\mu (1+t)^{-1}u_t\). Here \(\mu \) is non-zero constant. When \(\mu =0\), the equation becomes the usual wave equation and the critical power dividing small data global existence and blow-up is given by Strauss exponent Strauss exponent \(p_0(N)\) (\(N\) means spatial dimension). We give a small data blow-up result in the case \(1+2/N<p\le p_0(N+\mu )\) if \(\mu \) is near \(0\).

SUMMARY: We study the large time asymptotics of solutions to the Cauchy problem for the one-dimensional dissipative wave equation where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. Then the Cauchy problem has a unique global in time solution which tends toward a multiwave pattern consists of rarefaction and viscous contact waves as time goes to infinity. The proof is given by a technical energy method and the careful estimates for the interactions between the nonlinear waves.

SUMMARY: We study the large time asymptotics of solutions to the Cauchy problem for a one-dimensional scalar conservation law with nonlinear viscositywhere the far field states are prescribed. Especially, we deal with the case when the flux function is fully convex, and also the viscosity is a nonlinearly degenerate one. Then the Cauchy problem has a unique global in time solution which tends toward a rarefaction wave as time goes to infinity. The proof is given by a technical energy method, and a Sobolev type inequality motivated by an idea of Kanel’.

SUMMARY: We consider the fourth order nonlinear Schrödinger equation \begin{equation*} i{\partial }_{t}u-\frac {1}{4}\partial _{x}^{4}u=f\left ( u\right ) ,\quad (t,x)\in {\mathbf {R}}\times \mathbf {R}, \end{equation*} where \(f\left ( u\right ) \) is the power nonlinearity of order \(p>5.\) We show scattering operators are well defined in the neighborhood of the origin of a suitable weighted Sobolev space.

SUMMARY: We consider initial data problem for 1d cubic nonlinear Schrödinger equation with repulsive delta potential. We will show that if a data belongs to a weighted Sobolev space and is sufficiently small then solution decays in time in the same order as a free equation and asymptotically behaves like free solution with a logarithmic phase correction. Recently, long range scattering for nonlinear Schrödinger equation with a potential is extensively studied. In the previous results, a class of smooth and decaying potential is considered. In our case, we fully use explicit formulas which are available in the delta potential case.

SUMMARY: We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We handle the case in which the nonlinearity contains non-oscillating factor \(|u|^{1+2/d}\). It turns out that there is no solution which behaves like a free solution with or without any logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a small data finite time blow-up result.

SUMMARY: In this talk we shall treat the Strichartz estimates for magnetic Schrödinger equations and its application to global existence of the solutions and scattering theory with power type nonlinear term in an exterior domain. The proof relies on similar argument to the case of whole space.

SUMMARY: We consider the initial value problem for Schrödinger map equation. We provide some supplemental arguments for the work by Gustafson et al. (Duke Math. J. 145(3), 537–583, 2008), in which local well-posedness near the family of harmonic maps is asserted.

SUMMARY: We consider ill-posedness of the Cauchy problem for the fractional Schrödinger equation. More precisely, we prove norm inflation with general initial data. This argument with minor modifications also shows the ill-posedness for the generalized Boussinesq equations.

SUMMARY: We consider the initial value problem for a system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying a suitable resonance relation. We give a structural condition on the nonlinearity under which small data global existence holds. This is an extension of previous results by Hayashi–Li–Naumkin and Ikeda–Katayama–Sunagawa.

SUMMARY: We consider the Cauchy problem of the Zakharov–Kuznetsov–Burgers equation (ZKB for short) in two space dimensions. By using the Fourier restriction norm with the effect of the dissipative term, we prove the well-posedness in the Sobolev space \(H^s\) for \(s>-1/2\). It is interesting that ZKB has dissipative effect only \(x\)-direction, but the result for the regularity is better than the well-posedness of the Zakharov–Kuznetsov equation for both \(x\) and \(y\)-directions.

SUMMARY: We study the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation. One can expect that the solution to this equation converges to a self-similar solution to the Burgers equation, due to earlier works related to this problem. Actually, we obtain the optimal asymptotic rate similar to those results and the second asymptotic profile for the generalized KdV-Burgers equation.

SUMMARY: We study the Boltzmann equation near global Maxwellians in the \(d\)-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin–Lerner type space with respect to the phase variable \((x,v)\). Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the \(s\)-order Besov space with \(s \ge d/2\) including the spatially critical regularity.

SUMMARY: Axisymmetric solutions to the Einstein equations with the energy-momentum tensor of barotropic perfect fluid can be constructed mathematically as a post Newtonian approximation to slowly rotating axisymmetric solutions to the Euler–Poisson equation of gaseous stars, provided that the adiabatic exponent near the vacuum belongs to the interval ]6/5, 3/2[.

SUMMARY: We concern with the asymptotic behaviors of radially symmetric solutions for multi-dimensional Burgers equation on the exterior domain in \(\mathbb {R}^n\), where the boundary and far field conditions are prescribed. In a case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, we show the solution tends toward a superposition of stationary wave and rarefaction wave as time goes to infinity. We also show the decay rate estimate. Furthermore, for \(n = 3\), we give the complete classification of the asymptotic states, which includes even a superposition of stationary wave and viscous shock wave.

SUMMARY: A system of equations for compressible viscoelastic fluid is considered in an infinite layer. When the external force has a suitable form, the system has a solution of parallel flow type. It is shown that the solution of the system exists globally in time if the initial data is sufficiently close to the one of the parallel flow, provided that the initial data for the parallel flow is sufficiently small and the viscosity coefficient and the shear wave speed are sufficiently large.

SUMMARY: In this talk, we consider the free boundary problem for compressible-incompressible two-phase flows with phase transitions in isothermal case. Two fluids are separated by a sharp interface and a surface tension is taken into account. We use the Navier–Stokes–Korteweg equations for the compressible fluid and the Navier–Stokes equations for the incompressible fluid, whose model is thermodynamically correct. We show the maximal \(L_p\)-\(L_q\) regularity theorem with the help of the \(\mathcal {R}\)-bounded solution operators of the corresponding generalized resolvent problem and Weis’s operator-valued Fourier multiplier theorem.

SUMMARY: It is presented the local well-posedness of free boundary problem for the Navier–Stokes equations with surface tension without any restriction of the size of initial data. Hanzawa transform is used to represent the free surface. Since the standard linearized procedure requires a smallness restriction for the initial data, to avoid such smallness assumption, I used a modified linearized problem, which was first proposed by V. A. Solonnikov.

SUMMARY: I will talk about the global well-posedness for the Cauchy problem of a \(\mathbb Q\) tensor model of Incompressible Nematic Liquid Crystals in the \(N\)-dimensional Euclidean space. This is a joint work with Maria Schonbeck (Univ. California Santa Cruz). The proof is done by combination of \(L_p\)-\(L_q\) decay estimations and \(L_p\)-\(L_q\) maximal regularity for the heat equations and Stokes equations.

SUMMARY: Time decay estimate of a solution to the compressible Navier–Stokes–Korteweg system is studied. Concerning the linearized problem, the decay estimates with diffusive property for initial date are derived. As an application, the time decay estimates of a solution to the nonlinear problem are given. In contrast to the compressible Navier–Stokes system, for linear system regularities of initial dates are lower and independent of the order of derivative of the solution owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusive properties are obtained with initial dates having lower regularity than that of studies of the compressible Navier–Stokes system.

SUMMARY: Our purpose is to clarify the energy cascade mechanism for the incompressible Navier–Stokes equations. For the first step, we study the NS equations with partial hyperviscosity (dissipation is removed from some of the low Fourier modes) with DNS. This study direction is highly related to the previous result by T. Elgindi, W. Hu, V. Šverák (2017). This is a joint work with Professors Kishimoto, Saiki and Yoneda.

SUMMARY: We consider the incompressible Navier–Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional closed evolving surface as the width of the thin domain tends to zero, we give a formal derivation of limit equations on the degenerate evolving surface of the Navier–Stokes equations. We also compare our limit system with the Navier–Stokes equations on a stationary manifold, which is described in terms of the Levi–Civita connection.

SUMMARY: The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 R. Finn and D. R. Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. In this talk we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

SUMMARY: We shall establish a Beale–Kato–Majda type extension criterion of smooth solutions to the Navier–Stokes equations. It is known that if a smooth solution \(u\) to the Navier–Stokes equations on \((0,T)\) satisfies \(\int _{0}^{T}\|{\mathrm {rot}}\,u(\tau )\|_{L^\infty } d\tau <\infty \), then \(u\) can be continued to the smooth solution on \((0,T')\) for some \(T'>T\). In this talk, we shall slightly relax this condition for extension of smooth solutions to the 3D Navier–Stokes equations in not only the whole space but also the half space, bounded domains and exterior domains with smooth boundary.

SUMMARY: We consider the initial-boundary value problem for the Navier–Stokes equations in two dimensional domains. Under a certain condition on the asymptotic behavior of the vorticity at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

SUMMARY: There are mild solutions to the Navier–Stokes equtations in Serrin class with initial data in scale invariant homogeneous Besov spaces. We show the solution is uniformly analytic in \(x \in \mathbb {R}^n\) whose convegence radius is in proportion to \(\sqrt {t}\).

SUMMARY: We consider incompressible Navier–Stokes equations in the whole space \(\mathbb {R}^n\) under the non-trivial faces. In particular, we construct a strong solution to the Naiver–Stokes equations in weak Lebesgue space. Firstly we introduce a maximal subspace where the Stokes semigroup is strongly continuous. Then we construct a local in time weak mild solution of the Naiver–Stokes equations in \(L^{n,\infty }\). Then uniqueness criterion is discussed.

SUMMARY: We consider the stationary Navier–Stokes equations in \(\mathbb {R}^n\) for \(n\ge 3\). We show the existence and uniqueness of solutions in the homogeneous Triebel–Lizorkin space \(\dot F^{-1+\frac {n}{p}}_{p.q}\) with \(p\leq n\) for small external forces in \(\dot F^{-3+\frac {n}{p}}_{p.q}\). These are shown by the boundedness of the Riesz transform, the para-product formula, and the embedding theorem in homogeneous Triebel–Lizorkin spaces. Moreover, it is proved that under some additional assumption on external forces, our solutions have more regularity.

SUMMARY: The solutions of the stationary Navier–Stokes equations in \(\mathbb {R}^n\) for \(n\geq 3\) in the scaling invariant Besov spaces are investigated. It is proved that bounded smooth external forces whose \(\dot B^{-3}_{\infty ,1}\) norms are arbitrary small can produce bounded smooth solutions whose \(\dot B^{-1}_{\infty ,\infty }\) norms are arbitrary large. Such norm inflation phenomena are shown by constructing the sequence of external forces, as similar to those of initial data proposed by Bourgain–Pavlović in the non-stationary problem.