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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 shown that this system is uniquely solvable in the appropriate Holder spaces by the Fredholm alternative. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Publisher Summary 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 nonlinear. This chapter explores the free boundary problems for the system of differential equations of motion of compressible viscous isotropic Newtonian fluids (say, general fluids).

We consider an initial value problem for a flow of a compressible viscous fluid with some slip boundary condition in a domain Ω ⊂ R 3 We assume that Ω is a bounded or unbounded domain whose boundary Γ belongs to the class C 2+α , α e(0,1). Our aim is to prove the unique existence (locally in time) of a classical solution of the problem in Holder spaces.