Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.

THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER

The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.

Stress and Displacement Intensity Factors of Cracks in Anisotropic Media

A relation connecting stress intensity factors (SIF) with displacement intensity factors (DIF) at the crack front is derived by solving a pseudodifferential equation connecting stress and displacement discontinuity fields for a plane crack in an elastic anisotropic medium with arbitrary anisotropy. It is found that at a particular point on the crack front, the vector valued SIF is uniquely determined by the corresponding DIF evaluated at the same point.

ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

Stokes waves with constant vorticity: folds, gaps and fluid bubbles

The Stokes wave problem in a constant vorticity flow is formulated, by virtue of conformal mapping techniques, as a nonlinear pseudodifferential equation, involving the periodic Hilbert transform, which becomes the Babenko equation in the irrotational flow setting. The associated linearized operator is self-adjoint, whereby the modified Babenko equation is efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a ‘fold’ appears in the wave speed versus amplitude plane, and a ‘gap’ as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself at the trough line, enclosing a bubble of air. More folds and gaps follow for stronger vorticity. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely or, equivalently, gravitational acceleration vanishes, while the profile encloses a circular bubble of fluid in rigid body rotation at the ends of the gaps. Touching waves at the beginnings of the second gaps tend to the circular vortex wave on top of the limiting Crapper wave in the infinite vorticity limit, or the zero gravity limit, and the circular vortex wave on top of itself at the ends of the gaps. Touching waves for higher gaps accommodate more circular bubbles of fluid.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.