岡本 久

We consider the equation f(xxt) + ff(xxx) - af(x)f(xx) = nuf(xxxx), x is an element of (0, 1), t > 0, where a e R is a constant, with the periodic boundary condition. We show that any solution exists globally in time if -3 less than or equal to a less than or equal to 1.

We consider the system (x) over dot = ayz + bz + cy, (y) over dot = dzx + ex + fz, (z) over dot = gxy + hy + kx for real functions x(t), y(t) and z(t), where the overdot denotes differentiation with respect to a time-like independent variable t, and the coefficients a to k are real constants. Such equations arise in mechanical and fluid-dynamical contexts. Depending on parameter values, solutions may exhibit blowup in finite time; or they may be bounded oscillatory, or unbounded, as time t --> infinity. The local shape of the latter unbounded solutions is typically helical, sometimes with and sometimes without a 90degrees bend in the axis of the helix. Complete solutions are obtained in cases where certain coefficients are zero. Other cases are investigated numerically and asymptotically. The numerical solutions reveal an interesting "four-leaf' structure connected to the helical trajectories: this structure largely determines whether these trajectories bend through 90degrees or not, A fluid-dynamical application is discussed in Appendix A. (C) 2002 Elsevier Science B.V. All rights reserved.

We consider a special class of similarity solutions of the stationary Navier-Stokes equations and prove the existence of the solution for all the Reynolds numbers. We further prove that the solution exhibits interior and boundary layers as the Reynolds number tends to +∞ and -∞, respectively.

Oseen's spiral flows for viscous incompressible fluid are considered. Their limiting behavior as the Reynolds number tends to infinity is rigorously analyzed and the width of the interior layer is proved to be of O(R-1/2), where R is the Reynolds number.

The equations da/dt = sigmaa + mua* + lambdaa*b, db/dt = -sigma(1)b + nua(2) describe second-harmonic resonance of two small-amplitude standing water waves with Faraday excitation. Identical equations govern small oscillations of forced, resonant, coupled pendula. Real positive sigma and sigma(1) correspond to linear damping with perfect tuning of both forcing and internal resonance. With detuning from resonance, sigma and sigma(1) are complex. The degenerate case sigma(1) = 0 has a continuum of fixed points, and the solutions then take an unusual form: a number of decreasing "bounces" are separated by pauses at unstable fixed points, while small disturbances grow. When sigma(1) is small but non-zero, a similar qualitative structure remains; but all solutions terminate at one of a pair of stable fixed points of the now non-degenerate system. When sigma and sigma(1) are complex, there is double detuning, and the solution structure is far more complex. We mainly investigate cases where sigma(1) is imaginary and sigma is complex, exploiting a continuum of simple periodic orbits which then exists. A rich dynamical structure is revealed by sample computer solutions of the full equations; and stability analyses of fixed points and of periodic orbits explain many of the observed features. The latter also provide useful insights into expected behaviour at all other parameter values.

We consider a semilinear equation arising from the Navier-Stokes equations - the governing equations of viscous fluid motion and related model equations. The solutions of the semilinear equation represent a certain class of exact solutions of the Navier-Stokes equations. Both the equation and our models have nonlocal terms. We will show that the nonlocality will play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term, which is often neglected in the study of the blow-up problems, plays a very decisive role. In addition to our new contributions, open problems and known facts are surveyed.

The fast summation method of Draghicescu is applied to computation of 2D vortex sheet motion. This paper reports our numerical experiments which show the effectiveness as well as difficulties of Draghicescu's fast summation method in 2D vortex-sheet computations. For instance, the fast summation method is nearly thirty times faster than the direct summation method when we use 65536 = 2(16) vortex blobs. As a test problem, we re-examine Krasny's problem of computing a vortex sheet for a fairly long time.

We study the linearized stability of planer flows of incompressible, viscous fluid in two-dimensional annular domains. A certain family of steady, explicit solutions which have spiral streamlines are considered. The Navier-Stokes equations are linearized at these solutions and we show analytically or numerically that these solutions are stable to perturbation of steady states, whatever the Reynolds number or the aspect ratio of the annuli may be. Hopf bifurcations from them are also examined numerically. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

Interfacial water waves of permanent profile between two fluids of different densities are considered. We will show that interfacial waves are generalizations of surface waves, which have been studied extensively in both mathematical and physical papers. The purpose of the present paper is to give a mathematical explanation for numerical results on bifurcations of surface waves by Shoji and ourselves. A hypothesis of degeneracy plays a key role in the present analysis. In fact, we showed in an early paper that a certain degenerate bifurcation point, if it is assumed to be present, can elucidate the complicated bifurcation structure of the surface waves by Shoji. However, in previous papers, we proved unexpectedly that any degenerate bifurcation point does not appear if we vary the depth of the flow. So, the idea of degeneracy has not been physically substantiated in the category of surface waves. In this paper we prove that such a degenerate bifurcation point actually exists when we vary the ratio of the propagation speeds between the upper and lower fluids. Consequently the complicated structure of the surface waves can be explained by regarding the surface waves as special cases of the interfacial waves.

J. Leray considered a backward self-similar solution of the Navier-Stokes equations in the hope that it gives us an example of the finite-time blow-up of the three dimensional nonstationary Navier-Stokes equations. However, he showed no example of solutions. We list here some particular solutions and discuss their fluid mechanical properties. We also consider a forward self-similar solution, which describes solutions decaying as time tends to infinity.

The motion of a planar vortex sheet in the presence of a uniform shear flow is considered. We use the same numerical method as Krasny in his study of vortex sheets, namely, Chorin's vortex blob method. A linear analysis shows that the background shear influences the motion of the vortex sheet in such a way that with respect to low frequency modes it is stabilized in some cases. We examine this phenomenon not only by the linear analysis but also by a numerical simulation of the nonlinear integral equation. In particular, we note carefully how the number and the location of the roll-ups of a vortex sheet change as the strength of shear changes.

We study the convergence of two-dimensional stationary Kolmogorov flows as the Reynolds number increases to infinity. Since the flows considered are stationary solutions of Navier-Stokes equations, they are smooth whatever the Reynolds number may be. However, in the limit of an infinite Reynolds number, they can, at least theoretically, converge to a nonsmooth function. Through numerical experiments, we show that, under a certain condition, some smooth solutions of the Navier-Stokes equations converge to a nonsmooth solution of the Euler equations and develop internal layers. Therefore the Navier-Stokes flows are "nearly singular" for large Reynolds numbers. In view of this nearly singular solution, we propose a possible scenario of turbulence, which is of an intermediate nature between Leray's and Ruelle-Taken's scenarios. © 1996 Plenum Publishing Corporation.

We report here our numerical computation of two-dimensional capillary-gravity waves on surface of irrotational flow of incompressible inviscid fluid. In previous papers [1-4], we reported rather detailed numerical experiment. These computations were, however, limited to solutions near double bifurcation points of mode (1,2) and (1,3). Here mode (n, m) means the linear and/or nonlinear interaction of waves of wave number m and those of wave number n. In the present paper we report what we found numerically near bifurcation points of mode (1,4) and (2,3). As in the case of modes (1,2) and (1,3) our numerical simulation shows much more solutions than were mathematically predicted in other works. Copyright © 1996 Elsevier Science B.V. All rights reserved.

We propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed Dirichlet data. Our algorithm is a variant of what is called a ''Fundamental Solution Method'' [1-3]. This method requires us to select 2N points in the two-dimensional plane, N of which are called collocation points and the remaining N are called charge points representing the position of the singularities of the fundamental solution. It is known that there exists a set of 2N points by which the error is exponentially small [1,3-6]. However, these papers are concerned mainly with existence, and, as far as the authors know, few fast and reliable algorithms are known for good position of the points. In this paper, we propose a new rule for the position of the points and examine its efficiency by numerical experiments. The new rule uses FFT effectively.

A family of exact solutions of the stationary Navier-Stokes equations are given. They satisfy the Navier-Stokes equations outside a cylinder and tend to a certain inviscid explicit Euler flow at infinity. Our solutions are generalizations of those obtained by Wang, Stuart, Tamada, and Dorrepaal.

We consider the problem to determine the two dimensional water waves of permanent configuration. The problem is a bifurcation problem ([16]). In this paper, we define a mapping G between two Hilbert spaces and prove that the solutions of the above problem are in one-to-one correspondence to the zeros of the mapping G. Our most important contribution in this paper is to clarify the role of the aspect ratio, i.e., the ratio between the mean depth of the flow and the wave length. In particular, we prove that there is no degenerate bifurcation point, whatever the aspect ratio may be.

We consider the problem to determine the two dimensional water waves of permanent configuration. The problem is a bifurcation problem ([16]). In this paper, we define a mapping G between two Hilbert spaces and prove that the solutions of the above problem are in one-to-one correspondence to the zeros of the mapping G. Our most important contribution in this paper is to clarify the role of the aspect ratio, i.e., the ratio between the mean depth of the flow and the wave length. In particular, we prove that there is no degenerate bifurcation point, whatever the aspect ratio may be. © 1994, Research Institute forMathematical Sciences. All rights reserved.

We consider a family of stationary solutions of the 2-D Navier-Stokes equations parametrized by the Reynolds number. As the Reynolds number increases toward infinity, the solutions converge to a solution of the 2-D Euler equations. Our variational problem gives a necessary condition to pick up this limit solution among the continuum of the solutions to the Euler equations.

Some new results on the nonstationary Navier-Stokes equations are presented. Our results connect the well known, functional analytic theory for the Navier-Stokes equations with the blow-up solutions which were newly found by Ohkitani and others [S. Childress et al., J. Fluid Mech. 203 (1989), 1-22; K. Ohkitani, J. Phys. Soc. Japan 59 (1990), 3811-3814; J. T. Stuart, in ''Symposium to Honor C. C. Lin (D. J. Benny, F. H. Shu, and C. Yuan, Eds.), pp. 81-95, World Scientific, 1987]. Actually we consider unbounded solutions and prove a generalization of Graffi's uniqueness theorem [Ann. Mat. Pura AppL (4) 50 (1960), 379 387]. (C) 1994 Academic Press, Inc.

We consider Kolmogorov's problem of viscous incompressible fluid motions on two dimensional tori. The problem is a bifurcation problem with two parameters, the Reynolds number and the aspect ratio. Varying the aspect ratio as a splitting parameter, we compute numerically bifurcation diagrams with the Reynolds number as a bifurcation parameter. As the aspect ratio changes, we observe turning points and secondary bifurcation points appear or disappear. Furthermore, Hopf bifurcation points are also found when the aspect ratio of the torus satisfies a certain condition. This paper is an improved and enlarged version of the report [26]. Some errors in [17, 26] are corrected here. © 1993 JJIAM Publishing Committee.

The linear potential problem not only arises as it is but also appears as a part of nonlinear problem solvers. Hence a rapid solver for the potential problem is of considerable value. Although so many good computer programs are available now, it is worthwhile using programs according as their specific aims rather than solving both complicated problems and simple problems like the potential ones in a single computer code. In this paper we attempt to survey recent results on some algorithms which work only for the Laplace operator but is rapid.

A coupled pair of O(2)-equivariant bifurcation equations are developed to model steady axisymmetric Navier-Stokes flow between two concentric cylinders. The flow is driven by the rotation of the inner cylinder and axially periodic boundary conditions are applied at the two end surfaces. We examine the stabilities of steady axisymmetric flows and determine how these stabilities change upon varying the geometric parameters in the model. Our abstract bifurcation equations are of degree five and are dependent upon three physical parameters. By considering unfoldings about degenerate singularities we reproduce the qualitative behaviour generated numerically by Tavener and Cliffe [18]. New bifurcation diagrams are produced by varying a parameter held constant by these authors. © 1991 the JJIAM Publishing Committee.

We propose an algorithm for computing nondegenerate hysteresis points arising in bifurcation problems with two parameters such as G: R2 RN?RN. A combination of methods in [5] and [7] requires finding zeros of an extended system of 4N+3 variables. Shintani and Kanda [6] proposes another algorithm, in which computation of extended system of 3N+3 variables is sufficient. On the contrary, in our algorithm, we should find zeros of another extended system of 3N+2 variables. © 1991, Research Institute forMathematical Sciences. All rights reserved.