Stability of reflection and refraction of shocks on interface

In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection-refraction problem in this case is conditionally stable.To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection-refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic-hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems.     

Existence of undercompressive weak solutions to the Euler equations

We prove the local existence of an undercompressive hydrodynamical shock to the isothermal Euler equations with a non-monotone pressure function. This nonlinear problem will be formulated as an abstract hyperbolic initial boundary value problem. The existence of a weak solution to a linearized version of the problem is shown with the use of Riesz theorem. Using the results of the linear system yields by an iteration scheme (local in time) well-posedness of the nonlinear problem. The system of equations is obtained by modeling the motion of sharp liquid-vapor interfaces including configurational forces as well as surface tension. The considered non-viscous Van der Waals fluid is compressible and allows phase transitions. The propagating phase boundary is controlled by a modified version of the Rankine-Hugoniot jump condition obtained by the Young-Laplace law. Entropy dissipation at the interface is precisely described by a kinetic relation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuation of periodic motions of a reversible system in non-structurally stable cases. Application to the N-planet problem

The problem of continuing symmetric periodic solutions of an autonomous or periodic system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not guarantee that the solution can be continued, are considered. Three approaches are proposed to solving the problem: (a) particular consideration of terms that depend on the small parameter and the selection of generating solutions; (b) the selection of a generating system depending on the small parameter; (c) reduction to a quasi-linear system which is then analysed using the first approach. Within the framework of the third approach the existence of a periodic motion is also established that differs from the generating one by a quantity whose order is a fractional power of the small parameter. The theoretical results are used to prove the existence of two families of periodic three-dimensional orbits in the N-planet problem. The orbit of each planet is nearly elliptical and situated in the neighbourhood of its fixed plane; the angle between the planes is arbitrary. The average motions of the planets in these orbits relate to one another as natural numbers (the resonance property), and at instants of time that are multiples of the half-period the planes are either aligned in a straight line—the line of nodes (the first family), or cross the same fixed plane (the second family). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbit are independent.     

Equilibrium analysis for a mass-conserving model in presence of cavitation

We study the existence of equilibrium positions for the load problem in Lubrication Theory. The problem consists of two surfaces in relative motion separated by a small distance filled by a lubricant. The system is described by the modified Reynolds equation (Elrod–Adams model) which describes the behavior of the lubricant and an extra integral equation given the balance of forces. The balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom.     

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.     

On the Heat Convection Equations with Dissipation Term in Regions with Moving Boundaries

We consider an initial value problem for a system of equations describing the motion and the heat convection in a viscous and incompressible fluid which occupies a smooth region Ω_t⊂ℝ³ depending on time. In the equation for the distribution of temperature in the fluid we take into account not only the convective term but also the term responsible for the dissipation of energy. We prove local in time existence and uniqueness of solutions of the considered problem, and global in time existence for sufficiently small data. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids

Local (in time) unique solvability of a problem on the motion of two compressible fluids, one of which has finite volume, is obtained in Hölder spaces of functions with a power-like decay at infinity. After passage to Lagrangian coordinates, we arrive at a nonlinear initial boundary value problem with a given closed interface between the liquids. We establish an existence theorem for this problem on the basis of the solvability of a linearized problem by means of the fixed-point theorem. To obtain estimates and to prove the solvability for the linearized problem, we use the Schauder method and an explicit solution of a model linear problem with a plane interface between the liquids. The results are obtained under some restrictions on the fluid density and viscosities, which mean that the fluids are not much different from each other. Bibliography: 8 titles.To Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 57–89.     

Problem of the motion of two viscous incompressible fluids separated by a closed free interface

A local existence theorem for the problem of unsteady motion of a drop in a viscous incompressible capillary fluid is proved in Sobolev spaces. A linearized problem with known closed interface is also studied in Holder spaces of functions.     

The Singular Limit Problem to the Extended Cahn–Hillard Equation

A mathematical model with a small parameter, which describes the hardening process of the binary tin–lead alloy, is investigated on the basis of nonlinear asymptotic analysis. A singular limit problem, namely an extended Stefan problem in the case of short relaxation time in the phase transformation zone, is derived. We prove the existence of an asymptotic solution with any accuracy on the time interval where the solution to the singular limit problem exists. The phase-separation interface is determined uniquely by three leading approximations. We also show that the stability of the separation interface depends on the so-called dissipation condition obtained for the solutions of the interface problem. Nonsymmetry of the surface tension tensor leads to a situation where the limit values of concentration distributions are in dependence on the geometry of the interface. This provokes the dispersion of the interface problem solutions on the part of the interface that not is tangent to the main crystallographic axis.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

Surface and internal waves: The two-dimensional problem on forward motion of a body intersecting the interface between two fluids

A linear two-dimensional boundary value problem, that describes steady-state surface and internal waves due to the forward motion of a body in a fluid consisting of two superposed layers with different densities, is considered. The body is fully submerged and intersects the interface between the two layers. Two well-posed formulations of the problem are proposed in which, along with the Laplace equation, boundary conditions, coupling conditions on the interface, and conditions at infinity, a pair of supplementary conditions are imposed at the points where the body contour intersects the interface. In one of the well-posed formulations (where the differences between the horizontal momentum components are given at the intersection points), the existence of the unique solution is proved for all values of the parameters except for a certain (possibly empty) nowhere dense set of values.     

Well-posedness in Sobolev spaces of the full water wave problem in 2-D

. We consider the motion of the interface of 2-D irrotational, incompressible, inviscid water wave, with air above water and surface tension zero. We show that the interface is always not accelerating into the water region, normal to itself, as rapidly as the normal acceleration of gravity, as long as the interface is nonself-intersect. We therefore obtain the existence and uniqueness of solutions of the full water wave problem, locally in time, for any initial interface which is nonself-intersect, including the case that the interface is of multiple heights. Oblatum 1b-II-1996 & XI-1996     

Oberbeck–Boussinesq approximation for the motion of two incompressible fluids

The Oberbeck–Boussinesq approximation for unsteady motion of a drop in another fluid is considered. On the unknown interface between the liquids, the surface tension is taken into account. This problem is studied in H?lder classes of functions, where the local existence theorem for the problem is proved. The proof is based on the fact that the solvability of the problem with a temperature independent right-hand side was obtained earlier. For a given velocity vector field of the fluids, a diffraction problem is obtained for the heat equation the solvability of which is established by well-known methods. The existence of a solution to the complete problem is proved by successive approximations. Bibliography: 10 titles. Dedicated to V. A. Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 92–ll9.     

The fractional Landau�CLifshitz�CGilbert equation and the heat flow of harmonic maps

We consider a quasi-static droplet motion based on contact angle dynamics on a planar surface. We derive a natural time-discretization and prove the existence of a weak global-in-time solution in the continuum limit. The time discrete interface motion is described in comparison with barrier functions, which are classical sub- and super-solutions in a local neighborhood. This barrier property is different from standard viscosity solutions since there is no comparison principle for our problem. In the continuum limit the barrier properties still hold in a modified sense.     

Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates

We discuss the generation and motion of interfaces for Lotka-Volterra competition-diffusion system with large interaction. An asymptotic analysis of solutions shows that the two competing species are segregated and an interface appears on the common boundary of their habitats. The motion of the interface is governed by a free boundary problem. In this paper we establish a mathematical theory for the formation of interfaces (at the initial stage) by using an upper and lower solutions method. In addition, combining our results and a known result for the motion of interfaces (after the initial stage), we obtain some information on the generation and motion of interfaces for given almost any smooth initial data.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.     

Operator methods in the problem of perturbed motion of a rotating body partially filled with liquid

We apply operator methods to the investigation of an initial boundary-value problem which describes the perturbed motion of a body with cavity partially filled with an ideal liquid relative to the uniform rotation of this system about a fixed axis. We prove the existence and uniqueness of generalized solutions with finite energy and establish a sufficient condition for the stability of motion and some properties of the spectrum of the problem under consideration.     

Initial‐to‐Interface Maps for the Heat Equation on Composite Domains

A map from the initial conditions to the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with n interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed‐form solution to the interface problem.     

Goursat problem of the continuous conjugation of radial rectilinear motions of a gas

Gas dynamics equations have an isentropic solution describing the radial rectilinear motion of particles to the center and from the center with constant velocities. Two such solutions can be continuously conjugated if the Goursat problem is solved in a spatially similar domain with matched data on the characteristics. We prove the existence and uniqueness of a smooth solution of the Goursat problem in a small ball for a polytropic gas with exponent 5/3.