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web page: https://sites.google.com/view/atusitani/

**Skills and Expertise**

## Publications

Publications (91)

In this paper we study a nonlinear partial differential equation describing the evolution of a forced foam drainage in one dimensional case. A foam drainage equation was proposed by Goldfarb et al. [5] in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles...

This expository article deals with contact problems with friction for a linearized (visco)elasticity in two dimension, which are arising from a wide variety of phenomena in mechanical engineering and concerning with some inverse problems and control problems. Contact conditions for cracks are so-called non-penetration conditions defined as unilater...

In order to study the mechanism of earthquake through fracture mechanics, as a first step we discuss a dynamic unilateral contact problem with friction for a cracked viscoelastic body. Here we adopt the viscoelastic model proposed by Landau and Lifshitz [24] and a non-local friction law. The existence of a solution to the problem is obtained by usi...

The main purpose of this paper is to investigate mathematically gas discharge. Townsend discovered α- and γ-mechanisms which are essential for ionization of gas, and then derived a threshold of voltage at which gas discharge can happen. In this derivation, he used some simplification such as discretization of time. Therefore, it is an interesting p...

Nonlinear parabolic equations of “divergence form,” u t =(φ(u)ψ(u x )) x , are considered under the assumption that the “material flux,” φ(u)ψ(v), is bounded for all values of arguments, u and v. In literature such equations have been referred to as “strongly degenerate” equations. This is due to the fact that the coefficient, φ(u)ψ ′ (u x ), of th...

The purpose of this paper is to investigate mathematically the fundamental mechanism of gas discharge. Townsend discovered that α- and γ -mechanisms are essential for a gas ionization process. Morrow, Degond, and Lucquin-Desreux derived a mathematical model taking these two mechanisms into account. This model consists of nonlinear hyperbolic, parab...

A new basic equation in nonlinear acoustics similar to Kuznetsov equation [7] and Blackstock equation [1] is derived from the compressible Navier–Stokes equations of sound mode via weakly nonlinear approximations. Then global-in-time solutions of the initial value problems for these equations are proved, and the error estimates between them are est...

In this chapter, we discuss the mathematical analysis of synchronization with focusing on that of the Kuramoto–Sakaguchi equation. We also introduce related topics from the perspective of network science. The solvability and existence of vanishing diffusion coefficient are investigated.

We discuss two-phase radial viscous fingering problem in a Hele-Shaw cell, which is a nonlinear problem with a free boundary for elliptic equations. Unlike the Stefan problem for heat equations Hele-Shaw problem is of hydrodynamic type. In this paper the classical solvability of two-phase Hele-Shaw problem with radial geometry is established by app...

In this paper we are concerned with the drift wave turbulence in a strong magnetic field. The existence and the uniqueness of a strong Stepanov- Almost-periodic solution to the initial boundary value problems are established both for the model equations of the resistive drift wave turbulence and for the three-dimensional Hasegawa-Wakatani equations...

The Kuramoto-Sakaguchi (or simply Kuramoto) equation is considered when the "frequency distribution", the frequency being an independent variable in the model equation, has unbounded support. This equation is a nonlinear, Fokker-Planck-type, parabolic integro- differential equation, and arises from the statistical description of the dynamical behav...

The pattern formation of a radially growing interface in a Hele-Shaw cell is studied. In contrast to the previous studies based on the Young-Laplace equation, a boundary condition is employed that included the effect of the wetting layer of the displacing fluid in the cell. Under this boundary condition, a weakly nonlinear analysis is carried out a...

In [5, 6] it was proved that there exist the unique strong solutions, local in time, to the free boundary problems for the primitive equations of the atmosphere and the ocean in three-dimensional strip. Moreover, on the small time interval the strict positivity of the temperatures of both the atmosphere and the ocean, the moisture of the atmosphere...

In this paper we consider a system of equations describing the one-dimensional motion of a viscous and heat-conductive gas bounded by the free-surface. The motion is driven by the self-gravitation of the gas. This system of equations, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian-mass tr...

We consider an asymptotic behaviour of a solution near a tip of a rigid line inclusion in two dimensional homogeneous isotropic linearized elasticity. By means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution around there. Furthermore, we give expressions of the invariant integral and the Irwin's fo...

Primitive equations derived originally by Richardson in 1920's have been considered as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean.
In this paper, we discuss the free boundary problem of the primitive equations for the ocean in three-dimensional strip with surface tension.
Using the so-called $p$-...

In this paper we study two-dimensional flows of incompressible viscoelastic Maxwell media with Jaumann corotational derivative in the rheological constitutive law. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. Group properties of this system are studied. On this basi...

We prove that a free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness. Moreover, it is shown that the solution of this problem converges to the solution of the problem without surface tension as the coefficient of the surface tension tends...

This paper is devoted to study the global existence of a solution of bounded variation to the initial value problem for a system of conservation laws with artificial viscosity. The method of finite difference schemes of implicit type is used. We prove that the difference approximations converge to a weak solution of the problem. Moreover, this weak...

We study an initial-boundary value problem in an infinite viscoelastic strip with a semi-infinite fixed crack. For this problem we prove the existence and uniqueness of a weak solution which is prescribed on each side of the extended crack in Sobolev-type spaces.

In this article, we study the inverse problem on the identification of the leading coefficient of the multi-dimensional pseudoparabolic equation. The existence and the uniqueness of its strong solution are proved. The regularity of the solution is also investigated. These results cover the identification of the coefficient of piezo-conductivity for...

The inverse problems concerning the identification of the coefficient in the second order terms of linear pseudoparabolic equations of filtration in a fissured rock are investigated. The physical and mathematical justification of possible statements of the inverse problem for pseudoparabolic equations is given. New boundary conditions of overdeterm...

In this paper, we are concerned with the drift wave turbulence in a strong magnetic field. We prove the existence and uniqueness of a strong global in time solution to the initial boundary value problem for the model equations of drift wave turbulence similar to Hasegawa–Mima equation. Then we prove that the solution of Hasegawa–Wakatani equations...

In this paper we are concerned with a flow of inhomogeneous incompressible fluid-like bodies (IIFB). The concept of IIFB is
arised from the analysis of a certain type of granular flows. It is esssentially important to assign the so-called ‘slip’
boundary condition due to its behaviour at the surface, thus we take into account the Navier’s slip cond...

We consider the drift wave turbulence in a strong magnetic field. Here we first establish the existence and uniqueness of a strong global solution to the initial boundary value problem for the model equations of the resistive drift wave turbulence. Second, by the standard method of passing to the limit, we establish the existence and uniqueness of...

In this article, the properties of the solution to the inverse problem on the identification of the leading coefficient of the multi-dimensional pseudoparabolic equation are studied. The stabilization of its strong solution to the solution of the inverse problem for an elliptic equation is established.

This paper is concerned with the drift wave turbulence in a strong magnetic field. We prove two results: one is the existence and uniqueness of a strong solution on some time interval to the initial boundary value problem for the model equations of drift wave turbulence with zero resistivity; another is the convergence of the solution of the Hasega...

A topological derivative is defined, which is caused by kinking of a crack, thus, representing the topological change. Using variational methods, the anti-plane model of a solid subject to a non-penetration condition imposed at the kinked crack is considered. The objective function of the potential energy is expanded with respect to the diminishing...

We consider the Eguchi–Oki–Matsumura equation describing phase transition in binary alloys. We show that the corresponding semigroup possesses a maximal attractor and prove the existence of an inertial set.

We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the...

As a model problem of the nonstationary free boundary problem for the Navier-Stokes equations in a vessel whose wall has a
contact with a free surface, we are concerned in this paper with the boundary value problem for the stationary Stokes equations
with a parameter in an infinite sector with the slip and the stress boundary conditions. Existence...

The steady solution and the asymptotic behavior of the corresponding nonsteady solution are studied for Navier–Stokes equations
under the general Navier slip boundary condition. The existence of a unique stationary solution is established. It is also
proved that this solution is asymptotically stable under some restrictions on the data. Bibliograph...

Some \textit{a priori} estimates of the pressure for the Stokes equations in an infinite sector with the slip and the stress conditions on the boundary are established in weighted Sobolev spaces. The estimates for its higher order derivatives are obtained by the general scheme of Kondrat’ev. Instead the estimates for the lower order ones are derive...

In this paper, a novel adaptive gradient smoothing method (GSM) based on irregular cells and strong form of governing equations for fluid dynamics problems with arbitrary geometrical boundaries is presented. The spatial derivatives at a location of interest are consistently approximated by integrally averaging of gradients over a smoothing domain c...

In this paper we consider a system of equations describing the one-dimensional motion of a self-gravitating, radiative and chemically reactive gas having the free-boundary. For arbitrary large, smooth initial data we prove the unique existence, global in time, of a classical solution of the corresponding problem with fixed domain, obtained by the L...

The paper is concerned with the analysis of a new class of overlapping domain problems for elastic bodies having cracks. Inequality type boundary conditions are imposed on the crack faces. We prove an existence of invariant integrals and analyze the asymptotic behavior of the solution. It is shown that the limit problem describes an equilibrium sta...

The paper concerns an analysis of unilateral contact problems between two inclined elastic plates and between a plate and a beam. Considered problems are characterized by a contact set having a dimension less than one of that of a domain. This property leads to a new class of free boundary problems with inequality type boundary conditions. The main...

In this paper, we consider a system of classical model equations describing a motion of gaseous star, spherically symmetric, with a free-surface and a rigid core in the center under the influence of gaseous self-gravitation and potential force of the core. In addition to it, we investigate the above model equations in the physically more realistic...

In this paper we consider a system of equations describing the one-dimensional motion of a viscous, heat-conductive, self-gravitating and reactive gas with the free-boundary. This problem, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian mass transformation. Temporally global solution to th...

We establish the wellposedness of the time-independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb-type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed th...

In this paper we consider a system of equations describing a motion of a self-gravitating one-dimensional gaseous medium in the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the problem in a fixed domain with an explicit gravitational term. Based on the fundamental lo...

The nonlinear evolution problem for a crack with a kink in elastic body is considered. This nonlinear formulation accounts the condition of mutual non-penetration between the crack faces. The kinking crack is presented with the help of two unknown shape parameters of the kink angle and of the crack length, which minimize an energy due to the Griffi...

We are concerned with the boundary value problem for the steady Navier–Stokes equations in a 2D bounded domain with piecewise smooth boundary. Existence and uniqueness of the solution to the above problem is proved in weighted Sobolev spaces by means of the Mellin transform and the regularizer method.

In this paper we study linear elasticity equations in an infinite elastic strip with a semi-infinite crack. We find the derivative of the energy functional as the crack shifts with an angle. Then we obtain the formula given by surface force and the angle.

We are concerned with two-dimensional flow of an incompressible ideal fluid, which is formulated as the free boundary problem for the Euler equation. We show the unique solvability of this problem when the initial vorticity is large.

We consider a boundary-value problem for the stationary flow of an incompressible second-grade fluid in a bounded domain. The boundary condition allows for no-slip, Navier type slip, and free slip on different parts of the boundary. We first establish the well-posedness of a linear auxiliary problem by means of a fixed-point argument in which the p...

In this paper we consider a boundary value problem for an infinite elastic strip with a semi-infinite crack. The mass forces are supposed to be zero; on the crack the free traction boundary conditions are posed. The usage of the plane elastic single and double layer potentials reduces the problem to a system of singular integral equations. It is sh...

In this paper, we are concerned with free boundary problem for
compressible viscous isotropic Newtonian fluid.
Our problem is to find the three-dimensional domain occupied by
the fluid which is bounded below by the fixed bottom and above
by the free surface together with the density, the velocity
vector field and the absolute temperature of the flu...

The initial-boundary value problem for the Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established in Hlder spaces locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

We are concerned with a free boundary problem for two-dimensional Euler equation which describes the motion of incompressible perfect fluid between a free surface and a rigid bottom. We show that this problem is uniquely solvable, locally in time, in a class of finite smoothness.

In this paper we consider the Eguchi–Oki–Matsumura equation which consists of the fourth- and second-order coupled equations of parabolic type. It is shown that this system admits the unique global solution.

The initial-boundary value problem for the non-homogeneous Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

In this paper we study the two-phase Stefan problem for a viscous incompressible fluid which is a model of melting a solid to a liquid or of soldificating a liquid to a solid with a liquid flowing. The unique solvability is established in Hölder spaces locally in time. The method of the proof is rather standard, however the result obtained is compl...

The equations x t =x s ×x ss +ax sss + 3 2 x ss × (x s ×x ss ) which describe the motion of a vertex filament with or without an axial flow inside its core are considered. The initial and the initial-boundary value problems are proved to have unique and smooth solutions globally in time. These results are obtained by adding vanishing parabolic term...

The initial and the initial-boundary value problems for the localized induction equation which describes the motion of a vortex filament are considered. We prove the existence of solutions of both problems globally in time in the sense of distribution by the method of regularization.

We construct a family of solutions of the stationary free boundary problem for the Navier-Stokes equations corresponding to the slow rotation of a viscous compressible liquid as a rigid body about a certain axis. We suppose that the liquid is barotropic and we take into account capillary forces at the free boundary.

It is proved that this problem has a unique solution, belonging to some Sobolev spaces, on a finite time interval, whose length depends on the data of the problem.

We consider an initial value problem for a flow of a compressible viscous fluid with some slip boundary condition in a domain Ω ⊂ R3 We assume that Ω is a bounded or unbounded domain whose boundary Γ belongs to the class C2+α, α ε(0,1). Our aim is to prove the unique existence (locally in time) of a classical solution of the problem in Holder space...

The outstanding feature of many famous hydrodynamical problems is the somewhat paradoxical fact that the boundary of the flow, on which certain conditions have to be satisfied, is itself not given. There is a great variety of problems with free boundaries, some of which were already investigated in Newton's time. All these problems are essentially...

This communication is devoted to studying the periodic solution and the long-time dynamics to Burgers’ original model system of turbulence with two components in the secondary motion.