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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.

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 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 using a penalty method. Several estimates are obtained on the solution to the penalized problem, which enable us to pass to the limit by using compactness results [30].

In this paper we are concerned with the drift wave turbulence in a strong magnetic field. The existence and the uniqueness of a strong Stepanovalmost- 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 when the initial data are Stepanov-almost-periodic in the magnetic field direction.

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 established. Moreover, we discuss the problem of passing to the limit as the Stokes number tends to zero.

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.

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 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 applying the same method as for the Stefan problem and justifying the vanishing the coefficients of the derivative with respect to time in parabolic equations.

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.

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 attention is paid to a suitable description of boundary conditions along the contact zone. Asymptotic properties of solutions are established provided that elasticity parameters of the contacting bodies are changing, in particular, in the case when a stiffness of the elastic body goes to infinity.

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.

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.

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 threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary-value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.

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.

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

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 1980 BEALE [2] studied the problem of describing the motion of a layer of heavy, viscous, incompressible fluid lying above an infinite rigid bottom and having a non-compact free surface with no surface tension. He proved its local solvability in an anisotropic Sobolev space for any initial data. Its global solvability was discussed by SYLV~SXER [20]. But a crucial point of her proof does not appear clear to me: the regularity of the free surface. With surface tension taken into account, BEALE [31 and BEALE & NISHIDA [4] respectively studied the large-time existence and regularity and the large-time behavior of the solution to this problem with initial data near equilibrium. TERAMOTO [24, 25] considered these problems for fluids lying above an inclined plane. For any initial data the same problem was solved locally in time by ALLAIN [11 in the two-dimensional case. The aim of this paper is to establish the analogous result in the three-dimensional case. A similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by SOLONNIKOV. He proved local solvability in a H61der space in [12] and global solvability in the space W 2' 1, p > n (where n is the dimension of the domain) in [16] without surface tension and in [-13 15, 17 19,21] with surface tension. Let us formulate our problem. Given an initial domain f~ _ R 3 with x3 being the vertical component and an initial velocity vector field Vo in f~, we want to know the domain f~(t), t > 0, occupied by the fluid, which is bounded by the fixed bottom SB and the free surface Se(t), the velocity vector field v = v(x, t) = (vl, v2, v3) and the pressure p = p(x, t) so that

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.

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.

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 of the unique solution is proved in weighted Sobolev spaces.

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.

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 and the time evolution of the interface is numerically calculated. These results, compared to those obtained in the previous studies, suggest that our model reflects the nonlinear features of the viscous fingering phenomena more precisely.