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Masakazu Yamamoto

Publications and source records attributed to Masakazu Yamamoto.

12 recordsLinked to original sources
arXiv2026

Characteristics of drift effects arising from nonlinear symmetry of the quasi-geostrophic equation

This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of differentiation is the same. Naturally, their scales also coincide. However, the direction in which the nonlinear effects act differs: one acts along the isothermal surface, while the other acts along the temperature gradient in a specified direction. The main assertion quantifies this difference through the large-time behavior of their solutions. In particular, the nonlinear distortions in the asymptotic profiles of both equations are compared. In this context, the spatial symmetry of the first approximation plays a crucial role, but the solutions require no symmetry. As an appendix, the mixed problem of those models are studied.

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arXiv2026

Nonlinear profiles on solutions to the surface quasi-geostrophic equation

The quasi-geostrophic equation is well known as a model for predicting potential temperature around low-pressure systems in high-latitude regions. When diffusion effects are added to this equation, it serves as a model for potential temperature at the sea surface. In either case, the nonlinear term represents the effect of the Coriolis force. This paper yields nonlinear profile in solutions to the surface quasi-geostrophic equation. This profile is determined uniquely from the perspective of large-time behavior of solutions. Components of solutions are arranged sequentially from slow to fast decay based on the parabolic scale. Then such an expansion is determined uniquely. Since the equation is subcritical in the context of large-time effects, the main components exhibit linear features. Nonlinear characteristics appear in components that have a smaller effect on solutions. From the perspective of the correlation between spacetime variables, nonlinear profiles exhibit characteristics that are distinctly different from those of linear profiles. Moreover, the nonlinear effects of this equation are expected to be weaker than those of other equations with the same scale due to their spatial structure. The main theorem publishes minute, but unique rotational flow arising from the Coriolis force.

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arXiv2026

Distortion of charge distribution due to internal electric fields described by the drift-diffusion semiconductor model

In this paper, the initial value problem for the Debye--Hueckel drift-diffusion equation is studied. This equation was introduced as a model describing plasma behavior and is also known as a simulation model of MOSFET, and so its solution describes charge density. It is well-known that, if the initial density is localized, then the density is adjusted to be radially symmetric due to the linear diffusion. Consequently, the electric field is also governed by a radially symmetric potential, and its effects are expected to act radially symmetrically. The main result express the electric field and its effect on the charge density as concrete functions. It also denotes the distortion of symmetry and the shift of scale on the density due to the internal electric field. Unlike the historical paper via Escobedo and Zuazua and the followers, the main result captures stronger nonlinearity than the logarithmic shift.

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arXiv2025

Nonlinear distortion of symmetry in solutions to the convection-diffusion equation of Burgers type

In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. Regarding the solutions to this model, the Spanish school in the 1990s performed asymptotic expansions based on the linear diffusion. Those profiles exhibit symmetries characteristic of linear phenomena. In this paper, the distortion of symmetry arising from the nonlinear effects is described explicitly. Furthermore, it is demonstrated that the extent of this distortion differs significantly depending on the parity of the spatial dimension. This contradicts the conventional expectation that the manifestation of nonlinearity depends on the scale of the equation. This interpretation is supported by comparison with similar Navier--Stokes equations. The Burgers type is applicable as an indicator for considering several bilinear problems.

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arXiv2025

Logarithmic evolutions on the incompressible Navier--Stokes flow

Through the asymptotic expansion, the large-time behavior of the incompressible Navier-Stokes flow in $n$-dimensional whole space is drawn. In particular, the logarithmic evolution included in the flow velocity is the focus of attention. When the components of velocity are ordered from major to minor according to the parabolic scales, the logarithmically evolving components appear in a certain pattern. This work asserts that this pattern varies depending on the even-oddness of the space dimension. This means that the parity of the nonlinear drift is different in even and odd dimensions. The logarithmic term represents the nonlinear component of the phenomenon. No symmetry of the initial condition is required to prove this fact. In the preceding works, the expansion with the $2n$th order was already derived. The assertion is derived by reexamining these works in detail.

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arXiv2025

Parabolic-scalings on large-time behavior of the incompressible Navier--Stokes flow

Through asymptotic expansion, the large-time behavior of incompressible Navier--Stokes flow in $n$-dimensional whole space is depicted. Especially, from their parabolic scalings, large-time behaviors of any terms on the expansion are clarified. The parabolic scalings also guarantee the uniqueness of the expansion. In the preceding work, the expansion with the $n$th order has already been derived. They also predicted that the flow has some logarithmic evolutions in higher-order decay. In this paper, an asymptotic expansion with $2n$th order is presented. Furthermore, logarithmic evolutions are discovered.

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arXiv2023

Time evolution of the Navier-Stokes flow in far-field

Asymptotic expansion in far-field for the incompressive Navier-Stokes flow are established. Under moment conditions on the initial vorticity, technique of renormalization together with Biot-Savard law derives an asymptotic expansion for the velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given.

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arXiv2018

Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation

The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.

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arXiv2018

Large-time behavior and far field asymptotics of solutions to the Navier-Stokes equations

Asymptotic expansions of global solutions to the incompressible Navier-Stokes equation as $t$ tends to infinity with high-order is studied and large-time behavior of the expansion is clarified. Furthermore, far field asymptotics also is derived. Those expansions are provided without moment conditions on the initial velocity. The Biot-Savard law together with the renormalization for the vorticity equations yields those expansions.

math.AP↗
arXiv2018

Large-time asymptotics of a fractional drift-diffusion-Poisson system via the entropy method

The self-similar asymptotics for solutions to the drift-diffusion equation with fractional dissipation, coupled to the Poisson equation, is analyzed in the whole space. It is shown that in the subcritical and supercritical cases, the solutions converge to the fractional heat kernel with algebraic rate. The proof is based on the entropy method and leads to a decay rate in the $L^1(\mathbb{R}^d)$ norm. The technique is applied to other semilinear equations with fractional dissipation.

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arXiv2015

Asymptotic expansion of solutions to the drift-diffusion equation with fractional dissipation

The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian. When the exponent of the fractional Laplacian is large, large-time behavior of solutions is known. However, when the exponent is small, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with small exponent is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.

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