Hisashi Okamoto

Abstract In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Abstract Stationary waves of constant shape and constant propagation speed on rotational flows of two layers are computed numerically. Two layers are assumed to be of distinct constant vorticity distributions. Three different kinds of waves of finite depth are considered: pure capillary, capillary-gravity, and gravity waves. The problem is formulated as a bifurcation problem, which involves many parameters and produces a complicated structure of solutions. We adopted a numerical method by which waves with stagnation points can be computed, and obtained variety of new solutions. It is also reported that the locations of the stagnation points vary curiously with the prescribed parameters and that they offer an interesting problem.

We consider 1D equations with nonlocal velocity of the form w(t) + uw(x) delta u(x)w = -v Lambda(gamma)w where the nonlocal velocity u is given by (1) u = (1-partial derivative(xx))-(beta)w, beta > 0 or (2) u = Hw H is the Hilbert transform). In this paper, we address several local well-posedness results with blow-up criteria for smooth initial data. We then establish the global well-posedness by using the blow-up criteria. (C) 2017 Elsevier Ltd. All rights reserved.

We consider the Navier-Stokes equations in 2D flat tori. With various external forces and aspect ratios of the tori, we compute steady-states and time-periodic solutions at large Reynolds numbers. Our external forces are more general than those considered previously. We demonstrate, by numerical experiments, that certain solutions having streamline patterns of simple topology appear at large Reynolds numbers. (C) 2017 Elsevier Masson SAS. All rights reserved.

Solutions of a nonlinear heat equation are numerically computed in the time variable t lying in the complex plane, and possible singularities are sought. It turns out that in the complex half plane , where denotes the real part of a complex number, there is no singularity other than that which exists on the real line. However, if we compute further in the Riemann surface, new singularities are found. A certain nonlinear Schrodinger equation which is associated with our problem is also computed numerically and we propose a conjecture that it is well-posed globally in time.

<p>At very large Reynolds numbers the Navier-Stokes equations can possess a pair of large stationary vortices. This seemingly counter-intuitive phenomenon seems to be true in two-dimensions, and is demonstrated by numerically computing many Kolmogorov flows.</p>

We study stability and bifurcation of stationary and time-periodic solutions of Kolmogorov's problem for the Navier-Stokes equations in two-dimensional (2D) flat tori. Specifically we look for a unimodal solution, which is characterized by having a large, topologically simple pattern of streamlines. We present a conjecture that such simple patterns emerge in steady-states or time-periodic solutions at large Reynolds numbers, no matter what the external force may be. We confirm this conjecture by some numerical experiments. Thus the well-noted fact that a large structure appears in 2D large Reynolds number flows is reinforced in another form.

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

We consider a three-dimensional dynamical system proposed in Physica D, 164, (2002), 168-186. It is a conservative system and is unusual in that most of the solutions are unbounded. The paper presented a conjecture that an unstable periodic orbit determines directions of unbounded orbits of helical form. In the present paper we prove existence and local uniqueness of the conjectured periodic orbit by a method of numerical verification.

We consider the generalized Proudman-Johnson equation with an external force. By varying the Reynolds number R and another nondimensional parameter a, branching stationary solutions are computed numerically for the global picture of bifurcations of the equation. Asymptotic behavior of solutions as the Reynolds number tends to zero or infinity is also studied by a combination of heuristic analysis and the asymptotic expansion. In doing so, singular perturbation problems of new type are derived and analyzed. As a consequence, through the asymptotic analysis argument, the peculiarity of two dimensional Navier-Stokes flows related to the unimodality is re-confirmed.

Steady-states and traveling-waves of the generalized Constantin Lax-Majda equation are computed and their asymptotic behavior is described. Their relation with possible blow-up and the Benjamin-Ono equation is discussed.

Two partial differential equations are studied from the viewpoint of critical exponents. They are equations for a scalar unknown of one spatial variable, and produce self-similar solutions of the Navier-Stokes equations. Global existence and blow-up are examined for them, and the critical exponent separating them is determined.

We consider the generalized Proudman-Johnson equation with an external force. Solving its steady states numerically, we demonstrate that a unimodal pattern appears at large Reynolds numbers if and only if the parameter endowed in the equation is equal to unity. This may be interpreted as a strong connection of unimodal patterns to the 2D Navier-Stokes equations.

We compute trajectories of fluid particles in a water wave that propagates with a constant shape at a constant speed. The Stokes drift, which asserts that fluid particles are pushed forward by a wave, is proved using a new method. Numerical examples with various gravity and surface tension coefficients are presented.

We consider Kolmogorov&apos;s problem for the two-dimensional (2D) Navier-Stokes equations. Stability of and bifurcation from the trivial solution are studied numerically. More specifically, we compute solutions with large Reynolds numbers with a family of prescribed external forces of increasing degree of oscillation. We find that, whatever the external force may be, a stable steady-state of simple geometric character exits for sufficiently large Reynolds numbers. We thus observe a kind of universal outlook of the solutions, which is independent of the external force. This observation is reinforced further by an asymptotic analysis of a simple equation called the Proudman-Johnson equation.

The movement of fluid particles around a running cylinder or a sphere is considered. Particle trajectories viewed from a fixed object are contours of the stream function and well known in many cases. Here, we are concerned with trajectories viewed from the absolute coordinates where the object is moving. In 1870, Maxwell considered the problem in irrotational flow of inviscid fluid, and found that the trajectory of a particle is a curve of elastica having a self-intersection point. We consider here a similar problem in three-dimensional (3D) irrotational flow, 3D Stokes flow around a sphere and Brinkman's porous-media flow. In the 3D Stokes case, we found that the trajectories are unbounded and have no self-intersection. In the Brinkman case, we treated both flow around a cylinder and flow around a sphere: our numerical examinations revealed both self-intersecting and non-self-intersecting trajectories.

The generalized Proudman-Johnson equation, which was derived from the Navier-Stokes equations by Jinghui Zhu and the author, are considered in the case where the viscosity is neglected and the periodic boundary condition is imposed. The equation possesses two nonlinear terms: the convection and stretching terms. We prove that the solution exists globally in time if the stretching term is weak in the sense to be specified below. We also discuss on blow-up solutions when the stretching term is strong.

We present evidence on the global existence of solutions of De Gregorio&apos;s equation, based on numerical computation and a mathematical criterion analogous to the Beale-Kato-Majda theorem. Its meaning in the context of a generalized Constantin-Lax-Majda equation will be discussed. We then argue that a convection term, if set in a proper form and in a proper magnitude, can deplete solutions of blow-up.

A parameterized family of continuous functions which was considered by the first author is re-visited in the case when they are monotonically increasing. We prove that the functions are not only continuous and strictly increasing but also singular, i.e., their derivatives are zero almost everywhere.

We consider an overdetermined system of elliptic partial differential equations arising in the Navier-Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier-Stokes and Euler equations.

We consider the equations of motion of incompressible fluid. Several examples of blow-up for model equations are studied in order to suggest that the existence of the convection term plays a crucial role in the global existence of smooth solutions of some equations including the two-dimensional Euler equations. If we may say so, the convection term suppresses the blow-up of solutions.

We consider a parameterized family of continuous functions, which contains as its members Bourbaki's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

We consider a model equation for 3D vorticity dynamics of incompressible viscous fluid proposed by K. Ohkitani and the second author of the present paper. We prove that a solution blows up in finite time if the L 1-norm of the initial vorticity is greater than the viscosity.

We consider a parameterized, nonlocally constrained boundary - value problem, whose solutions are known to yield exact solutions, called Oseen's spiral flows, of the Navier-Stokes equations. We represent all solutions explicitly in terms of elliptic functions, and clarify completely the structure of the set of all the global branches of the solutions.

We consider viscous incompressible fluid motions on two-dimensional tori. Unlike many papers which treat the cases where the basic flow is a parallel flow, we consider here the case where the flow pattern is rhombic. We numerically compute the bifurcating solutions and find that (1) the bifurcating branch is a transcritical one, (2) on one side of the branch, the solution displays a sharp interior layer in the inviscid limit, and (3) on the other side of the branch, the solutions in the inviscid limit do not possess an interior layer.

We propose a numerical method for solving integral equations whose solutions possess singularities at the end points. Our method is based on the double exponential transform and effective use of the Fast Fourier Transform. Its accuracy is tested by Nekrasov's integral equation for water-waves and Yamada's equation for solitary waves. (C) 2002 Elsevier Science B.V. All rights reserved.