Search

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 problem to analyze the threshold by using the Degond–Lucquin-Desreux–Morrow model and also to compare the results of analysis with Townsend's theory. Note that gas discharge never happens in Townsend's theory if γ-mechanism is not taken into account. In this paper, we study an initial–boundary value problem to the model with α-mechanism but no γ-mechanism. This problem has a trivial stationary solution of which the electron and ion densities are zero. It is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable. Then we conclude that gas discharge can happen for a voltage greater than this threshold even if γ-mechanism is not taken into account. It is also of interest to know the asymptotic behavior of solutions to this initial–boundary value problem for the case that the trivial solution is unstable. To this end, we establish bifurcation of non-trivial stationary solutions by applying Crandall and Rabinowitz's Theorem, and show the linear stability and instability of those non-trivial solutions.

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 the second derivative, u x x , can be arbitrarily small for large value of the gradient, u x . The “hyperbolic phenomena” (unbounded growth of space derivatives within a finite time) have been established in literature for solutions to Cauchy problem for the above-mentioned equations. Accordingly one can expect a correct statement of the initial-boundary value problem for such equations only under additional assumptions on the problem data. In this paper we describe several restrictions, under which the initial-boundary value problems for strongly degenerate parabolic equations are well-posed.

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 solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.

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. Bibliography: 16 titles.

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 state for the elastic body with a thin inclusion.

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 a boundary value problem for an infinite elastic strip with a semi-infinite crack. By using the single and double layer potentials this problem is reduced to a singular integral equation, which is uniquely solved in the Holder spaces by the Fredholm alternative.

In [9, 10] we established the existence of classical solutions to two-phase and one-phase radial viscous fingering problems, respectively, in a Hele-Shaw cell by the parabolic regularization and by vanishing the coefficient of the derivative with respect to time in a parabolic equation. In this paper we show the uniqueness of such solutions to the respective problems

The equations which describe the motion of a vortex 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 terms which conserve the length of the filament.

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 $p$-coordinates and a coordinate transformation similar to that in [2] in order to fix the time-dependent domain, we prove temporally local existence of the unique strong solution to the transformed problem in Sobolev-Slobodetskiĭ spaces.

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.

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 established in [5] converges to that of Hasegawa–Mima like equation established at the first stage as the resistivity tends to zero on some time interval.

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 derived by making use of the Mellin transformation and the explicit representation formula of the pressure.

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 Griffith hypothesis. Based on the obtained results of the shape sensitivity analysis, solvability of the evolutionary minimization problem is proved, and the necessary conditions for the optimal crack are derived.

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.

E. Hopf discussed in details on the Cauchy problem of Burgers' equation in his famous paper [5]. Since then, many papers on the equation and its related topics have been published. However, they have not treated the initial-boundary value problem of it. The author previously discussed on the first initial-boundary value problem of this equation in [17]. Recently, N. Itaya has shown the existence and the uniqueness, in a certain sense, of the temporally global solution of the Cauchy problem of the following generalized Burgers' equation

Up io the present day many kinds of mathematical discussions on incompressible viscous fluid motion have fully developed (cf. [32, 36]). As for compressible viscous one, however, there have been only a few works on it. In 1959 Serrin [50] proved the uniqueness theorem in a bounded domain, making use of the classical energy method. In 1962 Nash [44] tried to show the existence theorem in R, but it seems to the author that he has failed. Independently of them Itaya succeeded to prove the existence and the uniqueness theorems on the Cauchy problem for it in [24-28], using Tikhonov's fixed point theorem. Now in the present paper, we shall show that the first initial-boundary value problem for it can uniquely be solved under suitable assumptions for the initial-boundary data and for the boundary of the domain, from the classical point of view. In § 1 an exact statement and the main theorem (Theorem 1) will be found. In §2 we perform the characteristic transformation and mention the theorem of the transformed problem (Theorem 2). Firstly we prove Theorem 2 and then show that Theorem 2 implies Theorem 1 in the last section § 8. In §§ 3-5 linear equations connected with the transformed equations are treated. In more detail, in § 3. 1 we briefly state some basic results for a fundamental solution in the whole space R due to Eidel'man [9, 18] and Pogorzelski [46-48] (cf. [25]). In §3.2 we check the basic condition of uniform solvability due to Solomjak [52, 54], which is essential for the study of the boundary value problem in applied mathematics, corresponding to the Lopatinsky condition for the

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.

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 branch of the incipient crack. The respective sensitivity analysis is provided by a Saint-Venant principle and a local decomposition of the solution of the variational problem in the Fourier series.

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 local existence result and a priori estimates, we can construct a classical unique global solution.

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.