2017年度秋季総合分科会(於:山形大学)

SUMMARY: We are going to explain explicit construction of general solutions to isomonodromy equations, with the main focus on the Painlevé VI equation. We will start by deriving Fredholm determinant representation of the Painlevé VI tau function. The corresponding integral operator acts in the direct sum of two copies of \(L^2(S^1)\). Its kernel is expressed in terms of hypergeometric fundamental solutions of two auxiliary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant \(4\)-point system via a decomposition of \(\mathbb {C}\mathbb {P}^1\backslash \{ 4\,\,\mathrm {points}\}\) into two pairs of pants. In the Fourier basis, this kernel is given by an infinite Cauchy matrix. It will be shown that the principal minor expansion of the Fredholm determinant yields a combinatorial series representation for the general solution to Painlevé VI.

SUMMARY: We survey some recent results for the uniqueness and the non-degeneracy of positive solutions for a class of semilinear elliptic equations in \(\mathbb {R}^N\). Especially, we show the uniqueness and the non-degeneracy of positive solutions in the cases where nonlinear terms may have sublinear growth at infinity. As applications, we obtain the uniqueness and the non-degeneracy of positive solutions for some quasilinear elliptic equations.

SUMMARY: Theory of partial differential equations has been developed mainly in smooth domains. In non-smooth domains such as polyhedral or cracked domains, mathematical difficulties appear because domains have singular points. Then, it is important to analyze the precise behavior of the solution of the partial differential equations with boundary conditions near the singular points. This kind of analysis has possibility of application in various fields of science and engineering such as fracture problems, inverse problems (nondestructive evaluation) and so on.
In this talk, first we have an overview of crack problems from both sides of mathematics and mechanics. Second we introduce some convergent series expansions of solutions of a boundary value problem at a crack tip in a linearized elasticity model. Next, we note that the study of cracks within the context of the linearized theory of elasticity has a drawback due to an inconsistency with regard to the strain, namely the strain becomes infinite at the crack tip. Then we consider a boundary value problem in a nonlinear elastic body that exhibits limiting small strain, which does not suffer from the inconsistency. For this problem we introduce the concept of a non-smooth viscosity solution, called generalized solution, which is described as generalized variational inequalities and coincides with the weak solution in the smooth case. Lastly, we mention future and ongoing researches in crack problems and their applications.

SUMMARY: In this talk, we consider the initial value problems for the rotating Navier–Stokes equations and the stably stratified Boussinesq equations. We establish the sharp dispersive estimates for the linear propagators related to the rotation and the stable stratification. As applications, we give explicit relations between the size of initial data and the angular/buoyancy frequency which ensure the unique existence of global solutions to the above systems. Consequently, it is shown that the size of initial data can be taken large in proportion to the speed of rotation and the strength of stable stratification.

SUMMARY: We define a relative twisted homology group \(H_1(T,D;\mathcal {L})\) isomorphic to the space of local solutions to Lauricella’s hypergeometric system \(F_D\) for any parameters. We define a relative twisted cohomology group \(H^1(T,D;\mathcal {L})\) as its dual space. We show that \(H^1(T,D;\mathcal {L})\) is isomorphic to three kinds of twisted de Rham cohomology groups. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility.

SUMMARY: In this talk, we deal with recurrence relations of the form \[ a_{n+1}-a_{n}=P(a_n), \] where \(P\) is a polynomial consisting of deg \(\geq 2\) terms or more generally, \[ a_{n+k}-a_{n}=P(a_n,\ a_{n+1},\ a_{n+2},\ldots ,\ a_{n+k-1}) \] and find general terms which converge to \(0\).

SUMMARY: We consider a chatacteristic initial value problem of a class of second order linear partial differential equation with regular singular initial data in the complex domain. We express the solution by means of series of hypergeometric functions, and show that the solution has regular singularities on three intersecting hypersurfaces. We also clarify the structure of analytic continuation of the solution.

SUMMARY: Kang and Alouini gave a determinant formula for the cumulative distribution function of the largest root of complex non-central Wishart matrices in their study of a wireless communication system with multiple antennas. The entries of the determinant formula are expressed in terms of a hypergeometric function in 2 variables. We give an asymptotic formula of the hypergeometric function and give a stable numerical analysis scheme to evaluate the hypergeometric function.

SUMMARY: Voros coefficients are important objects in exact WKB analysis to study global behaviors of solutions of differential equations. In this talk we will report that the Voros coefficients for (confluent) hypergeometric differential equations are given by the generating functions of free energies defined in terms of the Eynard–Orantin topological recursion. From these results, we can give concrete forms of the free energies for algebraic equations related to these equations.

SUMMARY: We introduce a simultaneous ultradiscrete Painlevé II equation with parity variables, which is shown to be more suitable for studying two-parameter solutions than the single second-order ultradiscrete Painlevé II equation with parity variables. We investigate several types of two-parameter solutions and the solutions which are related with the ultradiscrete limit of determinant type solutions of \(q\)-Painlevé II.

SUMMARY: We consider the boundary value problem of the stationary transport equation, an intedro-differential equation, in the slab domain of general dimensions. In this talk, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the boundary value problem. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along the positive characteristic lines. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.

SUMMARY: In recent days the Feynman path integral has been constructed mathematically for the Dirac equation. In this talk we will show that this Feynman path integral is relativistically covariant, i.e. has the property of spinor under the Lorentz transformations. First we give the representation of the fundamental solution by Fourier transformation with respect to momentum variables \(p \in \mathbb {R}^4\) to the free Dirac equation, as of the Feynman propagator. Then, our proof can be completed by means of the theories of both the Dirac matrices and the Lie group.

SUMMARY: We study the existence of periodic solutions for a prescribed-energy problem of Hamiltonian systems whose potential function has a singularity at the origin like \(-1/|q|^{\alpha } (q \in \mathbb {R}^N)\). It is known that there exist generalized periodic solutions which may have collisions, and the number of possible collisions has been estimated. In this talk we provide a new estimation of the number of collisions. Especially we show that the obtained solutions have no collision if \(N \ge 2\) and \(\alpha >1\).

SUMMARY: Decay of solutions to generalized pendulum equations are considered. Explicit sufficient conditions are given for solutions of such equations to decay at the infinity.

SUMMARY: We consider the unfolding of the codimension-two fold-Hopf bifurcation and prove its meromorphic nonintegrability in the meaning of Bogoyavlenskij for almost all parameter values. Our proof is based on a generalized version of the Morales–Ramis–Simó theory for non-Hamiltonian systems and related variational equations up to second order are used.

SUMMARY: It is shown that each 2-soliton obtained by an inverse scattering method with respect to an energy dependent Schrödinger equation has a finite life span, in the reflectionless case where the transmission coefficient has two poles on the imaginary axis.

SUMMARY: The shape of the object is determined by materials and environment. The materials is described by the boundary value problem for partial differential equation with given functions (the environment) and the shape given by the boundary. We already constructed the theory of shape optimization of sets of singular points which determines the shape of the object and the method of numerical calculations exists. I will talk about that the strength of the singular point give the influence to the shape optimization process that the trace theorem of Sobolev spaces. On the contrary, we can observe the strength of a singular point with a shape optimization process.

SUMMARY: We consider the embedding related to Strauss’s compactness lemma. We study the sufficiently condition of compactness and non-compactness for the embedding from the radial Sobolev space to the Lebesgue space with variable exponent.

SUMMARY: We consider the unit ball \(\Omega \subset {\mathbb {R}}^N\) (\(N\ge 2\)) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal configurations with respect to the torsional rigidity functional. In particular we analyse the role played by the configuration obtained by putting a smaller concentric ball inside \(\Omega \). In this case the stress function admits an explicit form which is radially symmetric: this allows us to compute the sign of the second order shape derivative of the torsional rigidity functional with the aid of spherical harmonics. Depending on the ratio of the conductivities a symmetry breaking phenomenon occurs.

SUMMARY: This talk is concerned with the Dirichlet problem of a diffusive Lotka–Volterra prey-predator system with population flux by attractive transition. We study the global bifurcation structure of positive stationary solutions. Moreover, we discuss the asymptotic behavior of positive stationary solutions as the nonlinear diffusion coefficient approaches infinity.

SUMMARY: We study the global and local behavior of bifurcation curves for nonlinear eigenvalue problems which include some oscillatory nonlinear term \(g\). We consider the case where \(\lambda \) is parameterized by the maximum norm \(\alpha = \Vert u_\lambda \Vert _\infty \) of the solution \(u_\lambda \) corresponding to \(\lambda \) and is represented as \(\lambda = \lambda (g,\alpha )\). Especially, we restrict our attention to the case where \(\lambda (g,\alpha ) \to \pi ^2/4\) as \(\alpha \to \infty \). We establish several precise asymptotic formulas for \(\lambda (g,\alpha )\) as \(\alpha \to \infty \) and \(\alpha \to 0\) with the exact second terms to understand well the total structures of the bifurcation curves.

SUMMARY: We construct countably infinitely many nonradial singular solutions of the problem \[ \Delta u+e^u=0\ \ {in}\ \ \mathbb {R}^N\backslash \{0\},\ \ 4\le N\le 10 \] of the form \(u(r,\sigma )=-2\log r+\log 2(N-2)+v(\sigma )\), where \(v(\sigma )\) depends only on \(\sigma \in \mathbb {S}^{N-1}\). To this end we construct countably infinitely many solutions of \[ \Delta _{\mathbb {S}^{N-1}}v+2(N-2)(e^v-1)=0,\ \ 4\le N\le 10, \] using ODE techniques.

SUMMARY: In this talk, we study, with variational technique, the existence of positive solutions for quasilinear elliptic equations of Born–Infeld type. We obtain the existence result for a large class of nonlinearities.

SUMMARY: In this talk, we consider the Brezis–Nirenberg problem on the thin annulus of the standard sphere \({\Bbb S}^n\). By the consideration of Morse index of linearlized equation, we construct bifurcation solution from radially symmetric solution. To obtain the results, uniqueness of the positive radial solution plays an important role.

SUMMARY: In this lecture, we study the \(p\)-Laplace equation in a hollow symmetric domain. Let \(H\) and \(G\) be closed subgroups of the orthogonal group such that \(H \subset G\) and \(H\neq G\). Then we prove the existence of a positive solution which is \(H\) invariant and \(G\) non-invariant.

SUMMARY: In the previous paper in 2016, we considered a two-phase heat conductor in \(\mathbb R^N\) with \(N \geq 2\) consisting of a core and a shell with different constant conductivities. Among other things, when the medium outside the two-phase conductor has a possibly different conductivity, we treated the Cauchy problem for \(N \ge 3\) with the initial condition where the conductor has temperature 0 and the outside medium has temperature 1. It was shown that if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be spherical. Here we report that the same proposition holds true even when \(N=2\), and as by-products, we can give other proofs of all the previous results of that paper in \(N(\ge 2)\) dimensions and prove a symmetry theorem on their related two-phase elliptic overdetermined problems.

SUMMARY: “Knot energy” was proposed to answer the question of what is a “good” figure in a given knot type. A requirement of the knot energy is that the better the figure of a knot is, the lesser the value of its energy is. Jun O’Hara gave a definition of a family of knot energies satisfying such a property in 1991. One of his energies is invariant under Möbius transformations and is called the Möbius energy. To study the Möbius energy, several discrete versions have been proposed, for example, by Kim–Kusner in 1993 and by Simon in 1994. Rawdon–Simon in 2006, Rawdon–Worthington in 2010, and Scholtes in 2014 showed their convergence. In this talk, discrete energies not only of the Möbius energy but of the general energies are given, and their convergence is discussed.

SUMMARY: The Möbius energy is one of the knot energies defined by O’Hara. Blatt showed the global existence and convergence of the gradient flow of the Möbius energy near stationary points. The Łojasiewicz inequality played an important role for proving such results. On the other hand, by work of Ishizeki–Nagasawa, it is known that the Möbius energy can be decomposed into parts which keep the Möbius invariance. In this study, the Łojasiewicz inequality is proved for each decomposed part of the Möbius energy. In an appropriate function space setting, we can show that the 2nd variations of the decomposed Möbius energies have \(L^2\)-representations and, using a result of Chill, the energies satisfy the Łojasiewicz inequality.

SUMMARY: The Möbius energy, defined by O’Hara, is one of the knot energies, and named after the Möbius invariant property which was shown by Freedman–He–Wang. It is also known that the energy can be decomposed into three parts keeping the Möbius invariance, proved by Ishizeki–Nagasawa. Though the decomposed energies are Möbius invariant, their densities are not. In this talk, the authors announce that the decomposed energies have alternative representation with the Möbius invariant densities. Using the fact that the cross ratio is invariant under the Möbius transformation, we define new Möbius invariant energies whose densities can be written by the cross ratio. Furthermore we show that these coincide with the decomposed energies.

SUMMARY: The Möbius energy, defined by O’Hara, is named after the Möbius invariant property which was shown by Freedman–He–Wang. The energy can be decomposed into three parts, each of which is Möbius invariant, proved by Ishizeki–Nagasawa. Several discrete versions of Möbius energy, that is, corresponding energies for polygons, are known, and it showed that they converge to the continuum version as the number of vertices to infinity. However already-known discrete energies lost the property of Möbius invariance, nor the Möbius invariant decomposition. Here a new discretization of the Möbius energy is proposed. It has the Möbius invariant property, and can be decomposed into the Möbius invariant components which converge to the original components of decomposition in the continuum limit.

SUMMARY: The generalized mean curvature is usually represented by the first variation of the varifold. A limit of integral means of the discretization of the classical mean curvature vector suggests a new representation of the generalized mean curvature without using the variation. A similar idea applicable to varifolds satisfying some regularity conditions. As a result, a new geometric representation of the generalized mean curvature is obtained, which has geometric meaning. The approximate tangent space of the measure on Euclidean space is used, however the tangential element of the varifold is not needed for our representation. Consequently, it has the advantage of giving rise to a definition of the generalized mean curvature vector for general measure on Euclidean space.

SUMMARY: Let \(H\) be a norm of \({\bf R}^N\) and \(H_0\) the dual norm of \(H\). Denote by \(\Delta _H\) the Finsler–Laplace operator defined by \(\Delta _Hu:=\mbox {div}\,(H(\nabla u)\nabla _\xi H(\nabla u))\). In this paper we prove that the Finsler–Laplace operator \(\Delta _H\) acts as a linear operator to \(H_0\)-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution of the Cauchy problem to the Finsler heat equation \[ \partial _t u=\Delta _H u,\qquad x\in {\bf R}^N,\,\,t>0, \] where \(N \ge 1\) and \(\partial _t:=\partial /\partial t\).

SUMMARY: We are concerned with blow-up mechanisms in a semilinear heat equation \[ u_t = \Delta u + |u|^{p-1}u, \qquad x \in \textbf {R}^N ,\, t>0, \] where \(p>1\) is a constant. It is well known that type II blow-up does occur if \(N \geq 11\) and \(p > p_{JL}\), where \(p_{JL}\) stands for the Joseph–Lundgren exponent. I will report a recent result on type II blow-up for \(p = p_{JL}\).

SUMMARY: Reaction-diffusion systems are one of nonlinear parabolic systems and are often used as models described chemical reaction system, combustion system, and so on. If the number of unknown variables of a reaction-diffusion system increases, the dynamics of a reaction-diffusion system may become complicated. In order to study the complexity of the dynamics, we consider the following question: What kinds of systems can we approximate by reaction-diffusion systems? In this talk, we introduce a reaction-diffusion system which approximates a semilinear wave equation. The proof is based on the energy estimates.

SUMMARY: Traveling front solutions with pyramidal shapes are studied in the Allen–Cahn equation (Nagumo Equation) in the \(n\)-dimensional Euclidean space. Here \(n\) is any integer that is greater or equal to 3. The existence of pyramidal traveling fronts was shown by [1] and [3]. The uniqueness and stability was shown by [2] for \(n=3\). In this work, I report the uniqueness and stability of pyramidal traveling fronts for \(n\) that are greater or equal to 3.

SUMMARY: This talk deals with the fully parabolic 1D chemotaxis system with diffusion \(1/(1+u)\). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates global-in-time solution. In view of our theorem one sees that one-dimensional Keller–Segel system is essentially different than its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known old Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.

SUMMARY: We consider the Cauchy problem of a parabolic-elliptic chemotaxis model in \({\mathbb R}^2\). Our purpose is to discuss the boundedness of nonnegative solutions to the Cauchy problem in the critical case where the total mass of the initial data is \(8\pi \).

SUMMARY: This talk deals with a 3D two-species chemotaxis-Stokes system with competitive kinetics. A single-species case was studied by e.g., Winkler (2012), Tao–Winkler (2015, 2016) and Lankeit (2016). However, there has not been rich results on coupled two-species-fluid systems. Recently, Hirata–K.–Mizukami–Yokota (2017) proved global existence of classical solutions for a 2D two-species chemotaxis-Navier–Stokes system. The present work asserts global existence and behaviour of classical solutions for the case of two species in 3D.

SUMMARY: This talk is concerned with asymptotic bahavior of solutions to a fully parabolic two-species chemotaxis-competition model. Bai and Winkler proved asymptotic behavior in the system under some conditions and special setting in 2016. Recently, the conditions assumed in the previous work were improved (M., 2017); however, this result did not give a complete improvement. The main result of this talk asserts complete improvement of the conditions assumed in Bai–Winkler (2016) and M. (2017) for asymptotic stability.

SUMMARY: In this talk, we consider the asymptotic behavior of solutions to Hamilton–Jacobi equations with large drift term in an open subset of two dimensional Euclidean space, where the set is determined through a Hamiltonian and the drift is given by the Hamiltonian vector field. Under some growth assumptions on the Hamiltonian, in the case where the Hamiltonian has degenerate critical points, we establish the convergence of solutions of the Hamilton–Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. The graph has many line segments more than four at a node.

SUMMARY: It is showed that the unique viscosity solutions of fully nonlinear elliptic partial differential equations under gradient constraint coincides with that of the equation with suitably selected bilateral obstacles. To this end, it is necessary to obtain the Lipschitz estimates on viscosity solutions of bilateral obstacle problems.

SUMMARY: We discuss a recent progress on uniform resolvent estimates for Schrödinger operators with scaling-critical potentials and their applications to global-in-time Strichartz estimates for the Cauchy problem of the Schrödinger equation and Keller type eigenvalue bounds for non-self-adjoint Schrödinger operators with complex-valued potentials.

SUMMARY: We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension \(N\ge 2\). We prove that if the nonlinearity is \(L^2\)-critical or supercritical in dimension \(N-1\), then any ground states are strongly unstable by blowup.

SUMMARY: We study analyticity of global solutions to NLS without gauge invariance.

SUMMARY: We study analytic smoothing effect for a system of Schrödinger equations.

SUMMARY: In this talk, we consider the Cauchy problem of the Hartree equation and we discuss the local existence when data are not characterized by any kind of square integrability.

SUMMARY: We consider the Cauchy problem of 2D and 3D Klein–Gordon–Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimatesare established by the Fourier restriction norm method. We utilize the nonlinearversion of the classical Loomis–Whitney inequality.

SUMMARY: Let \(T_{\varepsilon }\) be the lifespan for the solution to the Schrödinger equation on \(\mathbb {R}^{d}\) with a subcritical power nonlinearity and the initial data in the form \(\varepsilon \varphi (x)\). We provide a sharp lower bound estimate for \(T_{\varepsilon }\) as \(\varepsilon \rightarrow +0\) which can be written explicitly by the initial data and the nonlinearity. This is an improvement of the previous result by H. Sasaki [Adv. Diff. Eq. 14 (2009), 1021–1039].

SUMMARY: We consider the nonlinear Schrödinger equation with third order dispersion and intrapulse Raman scattering term. Without the Raman scattering term, the associated Cauchy problem is known to be locally well-posed in Sobolev spaces. We show that the Raman scattering term causes the ill-posedness of the Cauchy problem (nonexistence of local-in-time solutions) in Sobolev spaces. We also mention the unique solvability of the Cauchy problem in the analytic function space.

SUMMARY: We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS in \(\mathbb {R}^d\) (\(d=5,6\)) and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space.

SUMMARY: We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equations with super critical power nonlinearities. The class of the fourth-order nonlinear Schrödinger equations describe deep water wave dynamics. We prove global existence of small solutions in one or two space dimensions. This is an improvement of the result by N. Hayashi, J. A. Mendez-Navarro, and P. I. Naumkin. [Commun. Contemp. Math. 18(3), 1550035, 24 pp (2016)]

SUMMARY: We consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial in three dimensions. We give a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. This is the extension of the previous result in one and two dimensions. Moreover, we present a candidate of the second asymptotic profile to the solution.

SUMMARY: We study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity.

SUMMARY: In this talk we consider the wave equation with space-dependent damping coefficient \(a(x)=|x|^{-\alpha }\) \((\alpha \in [0,1))\) in an exterior domain \(\Omega \) having a smooth boundary. Weighted energy estimates with weight functions like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in \(\Omega \). The crucial idea is to use special solution of \(\partial _tu=|x|^\alpha \Delta u\) with a polynomial decay.

SUMMARY: In this talk, we consider the Cauchy problem for the dissipative nonlinear wave equation in one space dimension. In the work of the dissipative nonlinear wave equation, it is well-studied by using the energy estimates. The purpose of this talk is to show the point-wise estimates of solutions. Such kind of estimates describe the characteristics of the wave equation and precious information. Making use of these estimates, we can also get the result of the energy decay.

SUMMARY: We study the asymptotic decay of solutions toward a multiwave pattern (rarefaction wave and viscous contact wave) of the Cauchy problem for the one-dimensional viscous conservation law 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, and also the viscosity is a nonlinearly degenerate one. The proof is given by a technical time-weighted 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 the one-dimensional scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is a non-convex nonlinear function, and also the viscosity is a nonlinear function. The proof is given using a technical weighted energy method associated with the nonlinearity of the flux and the viscosity.

SUMMARY: We consider the existence and the nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations \(\partial _{t}^2 u = \partial _x(u^{2a} \partial _x u)\) with the nonnegative initial data \(u_{0}(x) \geq 0\) and \( a > 0\). This result is an extension of the results in the second author’s paper, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on \(\int _{-\infty }^{\infty } u_{1}(x) dx\) and the non-degeneracy condition \(u_{0}(x) \geq \delta _0 > 0\) on the initial data.

SUMMARY: We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration.

SUMMARY: We consider the Cauchy problem of systems of quasi-linear wave equations in 2-dimensional space. We assume that the propagation speeds are distinct and that the nonlinearities contain quadratic and cubic terms of the first and second order derivatives of the solution. We know that if the all quadratic and cubic terms of nonlinearities satisfy strong null-condition, then there exists a global solution for sufficiently small initial data. In this paper, we study about the lifespan of the smooth solution, when the cubic terms in the quasi-linear nonlinearities do not satisfy the strong null-condition.

SUMMARY: We study a blow-up curve for a system of nonlinear wave equations. The purpose of this talk is to show that the blow-up curve is a \(C^1\) curve if the initial values are large and smooth enough. To prove the result, we convert the system into a first order system, and then apply a modification of the method of Caffarelli and Friedman (1986).

SUMMARY: Satellites, International Space Station, orbital debris all move in highly rarefied atmosphere. This motivates the study of fluid-structure interaction in highly rarefied gas. Caprino et al. [Comm. Math. Phys., 264 (2006), 167–189] analyzed 1-D motion of a rigid body in a free molecular flow. They proved algebraic convergence of the body’s velocity \(V(t)\) to the terminal velocity \(V_{\infty }\). My question is whether the asymptotic behavior changes if there is a fixed wall behind the moving body. Caprino et al. considered the motion in the whole space \(\mathbb {R}^d\); I considered the motion in the half space \(\mathbb {R}_{+}^{d}\). I proved that the approach to the terminal velocity changes despite the fact that the body and the fixed wall go away infinitely.

SUMMARY: It is well-known that the critical exponent for semilinear damped wave equations is Fujita exponent when the damping is effective. Introducing a multiplier for the time-derivative of the spatial integral of unknown functions, we succeed to employ the analysis on semilinear wave equations and to prove a blow-up result for semilinear damped wave equations with sub-Strauss exponent when the damping is in the scattering range.

SUMMARY: We consider the global existence and blow-up of small data solutions for the wave equation with weighted nonlinear term in three space dimensions. We obtain upper and lower bound of the lifespan of solutions to the problem.

SUMMARY: In this talk, we report the sharp upper and lower bound of the lifespan of solutions to semilinear damped wave equations in the scale invariant case with a special constant in two space dimensions. The result is divided into two cases up to the total integral of the sum of the initial data.

SUMMARY: We consider the Couette–Taylor problem, a flow between two concentric cylinders, whose inner cylinder is rotating with uniform speed and the outer one is at rest. If the rotating speed is sufficiently small, the Couette flow (laminar flow) is stable. When the rotating speed increases, beyond a certain value of the rotating speed, a vortex flow pattern (Taylor vortex) appears. Mathematically, this phenomena can be formulated as a bifurcation problem. In this talk the Couette–Taylor problem is considered for the compressible Navier–Stokes equation and the bifurcation of the Taylor vortex is proved when the Mach number is sufficiently small.

SUMMARY: We consider a nonlinear model equation, known as the localized induction equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. The talk will report on the global-in-time unique solvability of an initial-boundary value problem describing the motion of a vortex filament on a slanted plane. The proof relies on the careful analysis of the shape of the filament near the boundary, and this will be the main focus of this talk.

SUMMARY: In this talk, we consider an initial boundary value problem for the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the initial vorticity and current density are both suitably small in some Sobolev space with arbitrary large initial mass density, and the vacuum of initial density is also allowed.

SUMMARY: We consider a blow-up problem for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariant.

SUMMARY: We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a parameterized (Young) measure and a dissipation defect in the total energy balance. A dissipative measure-valued solution can be seen as the most general concept of solution to the Euler system retaining its structural stability. In particular, we show that a dissipative measure-valued solution necessarily coincides with a classical one on its life span provided they share the same initial data.

SUMMARY: In this talk, we show the existence of \(\mathcal {R}\)-bounded solution operator families for a resolvent problem arising from the motion, where one fluid is a capillary compressible viscous flow and the other is an incompressible viscous flow. Moreover, we show that the regularity of density is \(W^3_q\) with respect to the space variable, although it is \(W^1_q\) in the usual case.

SUMMARY: We study the dynamics of vesicle membranes in incompressible viscous fluids. We prove existence and uniqueness of the local strong solution for this model coupling of the Navier–Stokes equations with a phase field equation in an \(L_p\)-\(L_q\) setting via maximal regularity. Moreover we have that the solution is real analytic in time and space. It is also shown that the variational strict stable solution is exponentially stable, provided the product of the viscosity coefficient and the mobility constant is large.

SUMMARY: In this talk, we discuss a local energy decay estimate of solutions to the initial-boundary value problem for the hyperbolic type Stokes equations of incompressible fluid flow.

SUMMARY: In this talk, we would like to consider a linearized system on general domains \(\Omega \) arising from a compressible fluid model of Korteweg type. The boundary of \(\Omega \) consists of two parts \(S\), \(\Gamma \) with \({\rm dist}(S,\Gamma )>0\). One imposes the free boundary condition on \(\Gamma \), while the non-slip condition on \(S\). It is admissible that \(S=\emptyset \) or \(\Gamma =\emptyset \) in this talk. We show the maximal \(L_p\)-\(L_q\) regularity for the linearized system.

SUMMARY: In this talk, we construct three dimensional Oseen type Navier–Stokes flows in Euclidean space and vertically periodic space and show their asymptotic stability in vertically periodic space under large initial perturbation.

SUMMARY: We consider the extension criterion of strong solutions to the Navier–Stokes equations in \(\mathbb {R}^N\). It is proved that among \(\frac {N(N-1)}{2}\) components of the vorticity, \(\left [\frac {N}{2}\right ]\) components are negligible for the criterion whether the time local solutions can be extended beyond the critical time. Our result may be regarded as generalization to the higher dimensional case of Chae–Choe and Kozono–Yatsu in the 3D case which showed that only two components in \(L^q\), \(\frac {3}{2}<q\leq \infty \), of the vorticity contribute to such an extension criterion. Furthermore, the critical case \(q=\infty \) originally treated by Kato–Ponce in \(\mathbb {R}^N\) is also generalized in such a way that \(\left [\frac {N}{2}\right ]\) components of vorticity are redundant for the extension criterion.

SUMMARY: The primitive equations with linearly growing initial data in the horizontal direction are concerned. The existence theory of mild solutions is established in certain interpolation spaces. A semi-group of Ornstein–Uhlenbeck type is investigated in Lebesgue spaces, including its smoothing property. Constructing mild solutions, the fixed point arguments of Fujita–Kato type are used.

SUMMARY: We give an asymptotic stability result for the incompressible Navier–Stokes equations in Besov spaces with sub/super critical smoothness. Following the argument by Kozono–Yamazaki (1995), we establish smoothing estimates for the semigroup generated by the Laplacian with a perturbation. Applying that, a critical estimate is proved, which is the main ingredient for the proof of the stability result.

SUMMARY: Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space \(\mathbb {R}^n\). Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity.

SUMMARY: Consider the motion of a viscous incompressible fluid in a 3D exterior domain when a rigid body moves with a prescribed time-dependent translational and angular velocities. We develop decay estimates of the evolution operator, which provides a solution of the linearized non-autonomous system, and then apply them to the Navier–Stokes initial value problem.

SUMMARY: Consider the Navier–Stokes flow in 3D exterior domains, where the translational velocity of the body becomes \(-u_\infty \) after some finite time. For the starting problem raised by Finn, we study some generalized situation in which unsteady solutions start from large motions being in \(L^3\). We then conclude that the steady solutions are still attainable as limits of evolution of those fluid motions provided \(u_\infty \) is small enough.

SUMMARY: In this talk, I would like to talk about local well-posedness for the magnetohydrodynamic equations in the two different liquids case. Since two divergence free conditions are over determined, I consider the jump condition for the divergence of magnetic field on the interface. This is a new aspect in treating the MHD. After transforming the time dependent unknown domain to the reference domain by Lagrange transformation, using the maximal \(L_p\)-\(L_q\) regularity for the linearized equations we prove a local in time unique existence theorem.

SUMMARY: In this talk, I will present the maximal \(L_p\)-\(L_q\) regularity theorem for the linearized electro-magnetic field equations. In solving MHD equations, linearized equations are decoupled, because the coupling terms are semi-linear. Thus, as linearized equations, we treat Stokes equations with interface conditions and linear electro-magnetic equations with interface conditions. Stokes equations with interface conditions have been studied by Pruess and Shimonett, and Maryani and Saito, so that I treat only the linear electro-magnetic equations with interface conditions.

SUMMARY: In this talk, I would like to talk about local in time unique existence theorem for the two components flow. I used the modelling due to Vicent Giovangigli: Multiccomponeent flow modeling, Birkhäuser. After transforming the equations to one component case by using the Giovangigli transfomation, I proved the local well-posedness for the resultant system of equations by using Lagrange transformation and the maximal \(L_p\)-\(L_q\) regularity theorem for the linearliezed equations. This is a joint work with Ewelina Zatorska (Imperial College of London).

SUMMARY: In this talk, I would like to talk about a global in time unique existence theorem for the reduced system of equations from the two component flow. A key is to prove the exponential decay estimate in some quotient space of the linearized equations. Some special structure of equations guarantees this exponential decay property.

SUMMARY: Global mild solution \(u\) to the Navier–Stokes equtation with the small initial data \(u(0) \in L_n(\mathbb {R}^n)\) is constructed by Kato. We show \(u\) becomes infinitely differentiable with respect to space having the decay property \(\|A^mu(t)\|_{L_p}=O(t^{-\frac n2(\frac 1n - \frac 1p)-m})\) as \(t\to \infty \) for all \(n\le p<\infty \).

SUMMARY: We consider the stationary problem of the Navier–Stokes equations in \(\mathbb {R}^{n}\) for \(n \ge 3\). We show existence, uniqueness and regularity of solutions in the homogeneous Besov space \(\dot {B}^{-1+\frac {n}{p}}_{p,q}\) which is the scaling invariant one. As a corollary of our results, a self-similar solution is obtained. For the proof, several bilinear estimates are established. The essential tool is based on the paraproduct formula and the imbedding theorem in homogeneous Besov spaces.