
Author(s) 
Antontsev, S.N. Khludnev, A.M. Hoffmann, K.H. 

Title  FREE BOUNDARY PROBLEMS IN CONTINUUM MECHANICS 
Publisher  Birkhäuser Verlag AG 
Year of publication  1992 
Reviewed by  S.V. Rogozin 
In the last decade the method of free boundary became one of the powerful and highly applicable methods using in many mathematical and mechanical problems. There are some mathematical centres in different countries where this method is worked out and applied. One of the famous of them is the Lavrentyev Institute of Hydrodynamics (Novosibirsk, Siberian Branch of Russian Academy of Science). At this centre in July of 1991 was held an International Conference on Free Boundary Problems in Continuum Mechanics. The reviewed book is a collection of articles presented on this conference. It includes for about 40 papers of the authors from many countries of Europe, America and Asia. The problems which are interested them are from several branches of modern mechanics and mathematics. The contents of these articles show the progress in there directions and modern state of theory and applications in them. More precisely the investigated problems will be described further in the short reviews of articles.
The book begins with the article of G.V. Alekseyev, A. Yu. Chebotarev ("SOME EXTREMUM AND UNILATERAL BOUNDARY VALUE PROBLEMS IN VISCOUS HYDRODYNAMICS"). It is considered the special unilateral boundary value prohlem for Stokes system <formula>, where ½ bounded domain in the space R^m, m = 2, 3, with a smooth boundary <formula>, under the following boundary conditions <formula> (the latters are called unilateral boundary condition).
Is proved the unique solvability of such a problem and studied some properties of the solution. Besides an inverse unilateral boundary problem is on the discussion.
The article "ON AXISYMMETRIC MOTION OF THE FLUID A FREE SURFACE" of V.K. Andreev is devoted to the problem with a free boundary for the equations of the ideal incompressible fluid under condition of the axial symmetry in Lagrangian coordinates <formula>. It is solved the problem of new groups classification with respect to an arbitrary element <formula>, where V is a square of initial angular momentum of a liquid particle impulse about the axis Z .
D. Andreucci , A. Fasano, M. Primicerio have discussed in their papers ("ON THE OCCURRENCE OF SINGULARITIES IN AXISYMMETRICAL PROBLEMS OF HELESHAW TYPE") some problems for so called HeleShaw cell (it is formed by two parallel plates very close to each other, the space between them being partly filled by a liquid, which can be injected or extracted through a horiface). In the article is investigated that case of such a problem which leads to Stefan problem and one of its generalization. It deals with the question of finite time extinction and of essential blowup, while regularization is treated briefly, mainly by reffering to previous papers of these authors. It is also shown that case of nonessential blowup could occur.
The new asymptotic method is applied by I.V. Andrianov and A.O. Ivanov ("NEW ASYMPTOTIC METHOD FOR SOLVING OF MIXED BOUNDARY VALUE PROBLEMS") to the stability analysis of rectangular plate (0,5a <= x <= 0,5a; 0,5b <= y <= 0,5b) uniformly compressed in xdirection under some additional conditions. The idea of this method is in introduction the parameter c into the boundary conditions in such a way that c = 0  case corresponds to the simple boundary value problems and c = 1  case corresponds to that problem under cosideration. It is appeared that in this approach the Pade approximation become very useful for analysis of cexpansion of the solution.
S.N. Antontsev and M.Chipot ("SOME RESULTS ON THE THERMISTOR PROBLEM") have discussed the problem of heating a conductor by an electric current (Thermistor Problems). The corresponding mathematical statement is as follows: <formula> where ½ is a smooth bounded open set in R^n with a boundary Gamma. It is studied the question of existence and uniqueness of the weak solution as well as global existence or blowup.
"NEW APPLICATIONS OF ENERGY METHODS TO PARABOLIC AND ELLIPTIC FREE BOUNDARY PROBLEMS" are considered by S.N. Antontsev, J.I. Diaz and S.I. Shmarev. The precise mechanical problems are: free boundary problems in stationary gas dynamics, the flow of immiscible fluids through a porous medium, the boundary layer for dilatant fluids, the problem of formation of "dead cores" in reactiondiffusion equations under strong absorption. In all these cases the classical maximum principle does not work.
In the article "A LOCALIZED FINITE ELEMENT METHOD FOR NONLINEAR WATER WAVE PROBLEMS" by Kwang June Bat and Jang Whan Kim is applied the method which is based on a variational principle equivalent to Luke's variational principle. It is stated the problem, described the method and presented two computed results in which are treated the nonlinear problems.
M.Ja. Barnyak has investigated small normal oscillations of viscous incompressible liquid partially filling a vessel ("APPROXIMATE METHOD OF NORMAL OSCILLATIONS OF VISCOUS INCOMPRESSIBLE LIQUID IN CONTAINER"). This problem deals with construction of solution for a spectral boundary value problem of the form <formula> where ½ is a domain filled by liquid, S is a solid wall of the container, · is a nonperturbed free surface of the liquid. Are studied the generalized solutions of this problem and realized on this basis the projection method for construction of its approximate solutions.
In the next article (B.V. Bazaliy, S.P. Degtyarev "THE CLASSICAL STEFAN PROBLEM AS THE LIMIT CASE OF THE STEFAN PROBLEM WITH A KINETIC CONDITION AT THE FREE BOUNDARY") is studied the question of the limit transition at the set of classical solutions of the modified Stefan problem when the kinetic condition at the interface is reduced to the classical one.
A new mathematical model of energy dissipation of capillary liquid of small viscosity partly filling the tank is suggested in "A MATHEMATICAL MODEL OF OSCILLATIONS ENERGY DISSIPATION OF VISCOUS LIQUID IN A TANK" by I.B. Bogoryad. This model is introduced for the flow with Re >> 1. Besides it is taken into account that dissipation of the energy of liquid oscillations occurs due to its viscosity not only at the level of the undisturbed free surface ·0, but above ·0 in a thin liquid film which remains on the surface of the tank after the preceding cycle of oscillations. At last it is supposed that at some Re > Re* the turbulization of the laminar boundary layer is possible.
M.A. Borodin ("EXISTENCE OF THE CLASSICAL SOLUTION OF A TWO PHASE MULTIDIMENSIONAL STEFAN PROBLEM ON ANY FINITE TIME INTERVAL") has introduced the method which makes possible to prove the existence of the classical solution of the two phase multidimensional Stefan problem on any finite time interval and to establish the smoothness of free (unknown) boundary. The idea of the method is: firstly, a sequence of elliptic differentialdifference approximate problems should be constructed, secondly, uniform estimates should be established, and thirdly, the pass to the limit should be performed.
In the article "ASYMPTOTIC THEORY OF PROPAGATION OF NONSTATIONARY SURFACE AND INTERNAL WAVES OVER UNEVEN BOTTOM" by S.Yu. Dobrokhotov, P.N. Zhevandrov, V.V. Korobkin, I.V. Sturova are presented some theoretical and numerical results in asymptotic theory of wave propagation in dispersive weakly inhomogeneous media. On the discussion are wave diffraction by bottom obstacle, evolution of an initial disturbance for one and twolayer on the elastic base. The final result is obtained with the help of Maslov's canonical operator.
The conformal mapping methods are used for a mathematical modelling of twodimensional steady seepage flows with initially unknown boundary sections by V.N. Emikh ("MULTIPARAMETRIC PROBLEMS OF TWODIMENSIONAL FREEBOUNDARY SEEPAGE"). The main problem is the calculation of the unknown parameters of the corresponding conformal mapping. They are precomputed on the basis of the system of equations resulting after assignment of set of govering physical parameters of flow.
R.E. Ewing and V.N. Monachov ("NONISOTHERMAL TWOPHASE FILTRATION IN POROUS MEDIA") have discussed so called MLT problem. It is determined the dependence of smoothness of the solutions of the initial boundary value problems for MLT model on the coefficients of equations and boundary conditions. This result is applied then to the proof of the convergence of the iterative method for solving of the MLT problem and to the finding of its velocity.
The problem of solving the Laplace equation on a moving boundary is considered by Y.E. Hohlov ("EXPLICIT SOLUTION OF TIMEDEPENDENT FREE BOUNDARY PROBLEMS"). The author is looking for explicit analytic solutions of such a problem in the case when the local velocity of this moving is determined by a normal component of potennial fieldgradient. He has systematically used here the methods and results of geometric complex analysis.
In the article of I.A. Kaliev ("NONEQUILIBRIUM PHASE TRANSITIONS IN FROZEN GROUNDS") it is considered a mathematical model of water freezing and melting with phase relaxation. This model is described by the following initialboundary value problem for real concentration of water w and for temperature u <formula>. Are studied the properties of this problem's solutions as well as their connection with the solutions of some other problems.
N. Kenmochi, M.Niezgodka have studied a class of nonlinear parabolic systems including variational inequalities that in particular arise from the modelling of solidliquid phase transitions ("SYSTEM OF VARIATIONAL INEQUALITIES ARISING IN NONLINEAR DIFFUSION WITH PHASE CHANGE"). They have proposed an extension of the known PenroseFife model which leads to the system of equations <formula> and some variational inequalities.
The problem of a contact between a beam and a rigid punch is investigated by A.M. Khludnev ("CONTACT VISCOELASTOPLASTIC PROBLEM FOR A BEAM"). The problem contains two restriction of inequality type. The first of them has a geometrical character and describes a nonpenetration condition, while the second one is of mechanical nature and corresponds to yield condition. It is proved a solvability of the exact formulation of the problem (in particular all the boundary conditions are fulfilled).
The article of S.N. Korobeinikov and V.V. Allyokhin ("APPLICATION OF A FINITEELEMENT METHOD TO TWODIMENSIONAL CONTACT PROBLEM") deals with the solution of the contact problems using the Lagrange multiplier method and penalty function simultaneously. Both of these methods have been introduced into PIONER computer program and on the base of this program are obtained the numerical solutions of some nonlinear static problems as well as dynamics ones (including 2/D contact problems).
V.A. Korobitsyn has obtained a computation algorithm of potential flows of incompressible ideal liquid with free surface in terms of homogeneous difference scheme ("COMPUTATION OF A GAS BUBBLE MOTION IN LIQUID"). For this situation are valid the different laws of conservation of mass, impulse and energy. It shows the effecteveness of algorithm for a number of problems.
The lowgravity problem on wave motion of the liquidgas free surface · placed in a limited volume Q is studied by I.A. Lukowsky and A.N. Timoha ("WAVES ON THE LIQUIDGAS FREE SURFACE IN THE PRESENCE OF THE ACOUSTIC FIELD IN GAS"). It is supposed that the highfrequency vibrator is contained in gas on S0 < dQ. The problem is investigated in nonlinear hydrodynamics formulation under hypothesis about idealized and potential motions of mediums.
N.I. Makarenko ("SMOOTH BORE IN A TWOLAYER FLUID") has considered the steady plane irrotational flow of a two layer fluid with different densities. The fluid is bounded by an even bottom from below and by a rigid cover from above. The state before the waves to which the flow tends asymptotically as <formula> is assumed to be piecewise constant with the given velocities and heights. The basic dimensionless parameters are the Fronde numbers. For the investigation of this problem is used the bifurcation theory.
In the article of L.A. Merzhievsky and A.D. Resnyansky ("NUMERICAL CALCULATION OF MOVABLE FREE AND CONTACT BOUNDARIES IN PROBLEMS OF DYNAMICS DEFORMATION OF VISCOELASTIC BODIES") is studied the problem of loading and fracture of metal specimens in viscoelastic Maxwelllike problem. To simulate fracture is assumed that a crack is simulated by introducing two free surfaces on the boundaries of two adjacent cells. It is specified for numerical solution a concept of calculation region structure.
A planar vortex flow of incompressible fluid is considered in O.I. Mokhov "ON THE CANONICAL VARIABLES FOR TWODIMENSIONAL VORTEX HYDRODYNAMICS OF INCOMPRESSIBLE FLUID". The corresponding equations are represented in the canonical Hamiltonian forms. For this structure it is proposed a general method of group analysis. Among the equations for which this method can be applied are so called the equations for Rossby waves in atmospere of rotating planet.
R.V. Namm has given ("ABOUT THE METHOD WITH REGULARIZATION FOR SOLVING THE CONTACT PROBLEM IN ELASTICITY") the iterative algorithm with regularization on each step for solving the problem of minimization the quadric functional on a close convex set in Hilbert space. This algorithm is applied then to the investigation of the contact problem in elasticity.
In the article (V.V. Nikulin "SPACE EVOLUTION OF TORNADOLIKE VORTEX CORE") is proposed the theoretical model for study the flow in a core of tornadolike vortex with taking into account inconstancy of vertical velocity in horizontal core section. Due to this is established a possibility of extension of continuous solution to finite or infinite height. Besides are classified the nonextended solutions.
S.P. Ohezin ("OPTIMAL SHAPE DESING FOR PARABOLIC SYSTEM AND TWOPHASE STEFAN PROBLEM") has considered the classical mathematical model of onedimensional twophase Stefan problem. This problem is reduced to minimization problem for special functinal. The containing in Stefan problem differential equations are approximated by means of parabolic equations with penalty terms which are defined in a priori known large cylindrical domain.
Some new examples of the variational methods' application to the investigation of the flow with free surface are given by A.G. Petrov ("INCOMPRESSIBLE FLUID FLOWS WITH FREE BOUNDARY AND THE METHODS FOR THEIR RESEARCH"). Besides on the discussion are some other methods of study this kind of problem (of integral limiting equations for harmonic functions, of duplex estimators of fluid energy dissipation).
A.G. Petrova has studied onedimensional onephase supercooled Stefan problems for system of two diffusion equations connected each other under the free boundary conditions ("ON THE STEFAN PROBLEMS FOR THE SYSTEM OF EQUATIONS ARISING IN THE MODELLING OF LIQUIDPHASE EPITAXY PROCESSES"). In the problems of the first type the values of functions on the free boundary are known while in those of the second type these values should be determined due to some additional conditions. The problems of these two types are considered in the cases of a restricted domain occupied by phase and semiinfinite one.
A quasistationary boundary value problem for phase field model equations is considered by P.I. Plotnikov and V.N. Starovoitov ("STEFAN PROBLEM WITH SURFACE TENSION AS A LIMIT OF THE PHASE FIELD MODEL"). It is supposed that continuum medium occupies the bounded domain in ½. The problem is to find a temperature and order parameter which are satisfied some equations depending of a small parameter c. It is proved that this problem has the solution converging as <formula> to the solution of the Stefan problem with surface tension.
In the article H. Sabar, M. Buisson, M.Berveiller "THE MODELIZATION OF TRANSFORMATION PHASE VIA THE RESOLUTION OF AN INCLUSION PROBLEM WITH MOVING BOUNDARY" is proposed a micromechanical formulation the phasetransformation problem taking simultaneously account of the growth and interactions of phases in a heterogeneous material. The method and its applications are described in the scope of the linear elastoquasistatistics associated with a time varying distribution of some inelastic strain.
V.M. Sadovskii ("TO THE PROBLEM OF CONSTRUCTING WEAK SOLUTIONS IN DYNAMIC ELASTOPLASTICITY") has given the integral formulation which is equivalent to the equations of flow theory with an arbitrary hardening curve. Besides he has obtained the system of discontinuity relationships on the basis of this formulation.
The paper of V.V.Shelukhin ("THE JUSTIFICATION OF THE CONJUGATE CONDITIONS FOR THE EULER'S AND DIRAC'S EQUATIONS") deals with a poisoning model in which the mass exchange of subsoil and surface waters is taken into account. It is shown that the conjugate condition problem for the flow under the roof can be solved by the method applied to the CauchyPoisson problem.
V.A. Solonnikov ("ON AN EVOLUTION PROBLEM OF THERMOCAPILLARY CONVECTION") has considered a free boundary problem of imcompressible viscous flow governing the motion of an isolated liquid mass. The liquid is subjected to capillary forces at the boundary, and the coefficient of the surface tension depends on the temperature satisfying the heat equation with convection and dissipation terms.
In the article Uwe Streit "FRONT TRACKING METHODS FOR ONEDIMENSIONAL MOVING BOUNDARY PROBLEMS" are proposed the front tracking methods based on such a finite differences which are connected with the following procedure: choose a fixed spacetime grid such that the discrete interface moves within one time step is not greater than the width of the space mesh. To obtain an error estimate for the field variable, the front tracking scheme is spaped into a scheme for a corresponding fixed domain formulation.
In the article of V.M. Teshukov ("ON CAUCHY PROBLEM FOR LONG WAVE EQUATIONS") is considered the Cauchy problem for system of integrodifferential equations describing a planeparallel unsteady wave motion of ideal incompressible fluid. For the initial data satisfying the hyperbolicity condition it is proved the solvability of this problem.
T. Tiihonen and J. Järvinen have proposed the trial methods for solving steady state free boundary value problems ("ON FIXED POINT (TRIAL) METHODS FOR FREE BOUNDARY PROBLEMS"). The approach is of a fixed point type. The authors have discussed the convergence of these methods for two types of test examples.
In the paper O.V. Voinov "NONLINEAR THEORY OF DYNAMICS OF A VISCOUS FLUID WITH A FREE BOUNDARY IN THE PROCESS OF A SOLID BODY WETTING" are considered the problems of the capillary phenomena paradoxial from viewpoint of the hydrodynamics appeared at the flow analysis for the case of a viscous liquid with a free boundary.
The results presented in all the papers are of a high level. Are given some deep and prominent ideas. Due to them are obtained the solutions (exact, approximate or numerical) of complicate mechanical or physical problems. The main using approach is the free boundary method and its generalizations. The contents of the book shows the modern state of the investigations in this direction. Thus it will be of a great interest for specialists in mechanics, mathematical analysis, differential equations, functional analysis, numerical analysis, computer simulations.