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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.

The purpose of this paper is to investigate mathematically the fundamental mechanism of gas discharge. Townsend discovered that $\alpha$- and $\gamma$-mechanisms are essential for a gas ionization ...

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, driven by gravity and capillarity. Mathematical studies on a foam drainage equation so far are restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. This paper is aimed at proving that the initial boundary value problem for the forced foam drainage equation admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and comparison theorem. Moreover, the existence of its steady solution and its stability are shown. This is the first rigorous mathematical result for a series of researches on this equation.

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 and the salinity of the ocean was proved for the positive initial and boundary data. In this paper, we are concerned with the more exact estimates from above and below of the temperatures, moisture and salinity among the solutions.

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 transformation. For smooth initial data we first establish the temporally global solvability of the problem based on both the fundamental result for local in time and unique existence of the classical solution and a priori estimates of its solution. Second it is proved that some estimates of the global solution are independent of time under a certain restricted, but physically plausible situation. This gives the fact that the solution does not blow up even if time goes to infinity under such a situation. Simultaneously, a temporally asymptotic behavior of the solution is established.

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.

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 formula.

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 basis, two submodels of the Maxwell model are selected, which can be reduced to hyperbolic ones. More precisely, we consider plane shear flow between two parallel planes and Couette type flow caused by the inertial cylinder rotation. As a result, we obtain the closed systems of three equations of mixed type, which describe nonlinear transverse waves in an incompressible Maxwell fluid. It is demonstrated that discontinuities can develop in elastic media even from smooth initial data. Stability of shocks in the Maxwell fluid with and without retardation time is discussed.

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 overdetermination are discussed. Certain parabolic inverse problems relevant to pseudoparabolic ones are considered. The existence and uniqueness of a strong solution to one of the discussed inverse problems for the pseudoparabolic equation is proved in the case of one space variable. It is also established that the inverse problem for the pseudoparabolic equation approximates weakly some inverse problem for the parabolic one.

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.

In this paper 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 a strong solution to the initial boundary value problem for three-dimensional 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 condition. Here, the theorem on the unique solvability, local in time, is proved.

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.

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 situation: the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the one in a fixed domain. Based on the fundamental local existence result and a priori estimates, we can construct a unique global classical solution.

As a model problem of the stationary 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 in an infinite sector with the slip and the stress boundary conditions. Existence of the unique solution is proved in weighted Sobolev spaces by means of the Mellin transform.

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 solution is really a classical solution according to the Oleĭnik's argument.

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 problem is decomposed into a Stokes-type problem and two transport equations. Then we use the method of successive approximations to prove the unique solvability of the nonlinear problem with a sufficiently small body force in Holder spaces. Bibliography: 17 titles.

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.

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 Holder spaces locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

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 to zero.

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 completely new: the existence in the Hölder spaces with the same Hölder exponents as the data and the uniqueness also in the same Hölder spaces.

This paper is devoted to the study of a solidification/melting process in the case where the fluid is flowing. Such a phenomenon is described by the Stefan problem with transport terms in the equation of the temperature distribution and the initial-boundary value problem for the incompressible Navier--Stokes equations. The existence of the classical solution is proved locally in time.

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.

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 constructed around the location. Such a favorable GSM scheme corresponds to a compact stencil with positive coefficients of influence on regular cells. The error equidistribution strategy is adopted in the solution-based adaptive GSM procedure, and adaptive grids are attained with the remeshing techniques and the advancing front method. In this paper, the adaptive GSM has been tested for solutions to both Poisson and Euler equations. The sensitivity of the GSM to the irregularity of the grid is examined in the solutions to the Poisson equation. We also investigate the effects of error indicators based on the first derivatives and second derivatives of density, respectively, to the solutions to the shock flow over the NACA0012 airfoil. The adaptive GSM effectively yields much more accurate results than the non-adaptive GSM solver. The whole adaptive process is very stable and no spurious behaviors are observed in all testing cases. The cosmetic techniques for improving grid quality can effectively boost the accuracy of GSM solutions. It is also found that the adaptive GSM procedure using the second derivatives of density to estimate the error indicators can automatically and accurately resolve all key features occurring in the flow with discontinuities. Copyright © 2009 John Wiley & Sons, Ltd.

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

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.