We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.The classical Stefan problem is a nonlinear problem with free boundary for parabolic equations that describes the process of crystallization of a medium that can be in different phase states. In addition, on the free boundary, the function of distribution of the temperature takes a constant value (equal to the temperature of crystallization). Numerous works (see, e.g., survey [1]) are devoted to the problem of solvability and regularity of a solution of this problem. In the present paper, we consider a Stefan-type problem (see [2,3]) in which, first, the distribution of temperature is described by an elliptic equation and, second, the temperature of crystallization is proportional to the curvature of the sought phase boundary. In this connection, we mention [4], in which the problem considered differs from our case only by the type of boundary conditions on fixed boundaries; however, the result concerning the regularity of the free boundary is substantially weaker. We also mention [5], in which the problem of filtration with free boundary for an elliptic equation is studied, but without regard for the curvature of the free boundary.

A new stereoscopic method for collecting particle image velocimetry (PIV) measurements within ∼1 mm of a wavy air-water interface with simultaneous measurements of the morphology of the interface is described. The method, termed three-dimensional interfacial particle image velocimetry (3D-IPIV), is tested in a wind wave channel with a wind speed of 5.8 m s −1 , water depth of 10 cm and a fetch of ∼9 m. Microscale breaking waves populate the interface and their flow patterns are clearly visible in the velocimetry results. The associated capillary waves and surface divergence patterns are observed. Several statistical measurements of the flow are compared with independent measurements and good agreement is found. The method shows great promise for investigating the transfer of momentum, heat and gases across an air-water interface, both in the laboratory and in field settings. Additional methods are described for manufacturing the flow tracers needed for the 3D-IPIV method. These tracers are likely to be useful for other researchers, and have the characteristics of being fluorescent, neutrally buoyant, non-toxic, monodisperse, inexpensive and easy to manufacture.

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We prove the existence of a classical solution global in time for the first initial boundary-value problem for a nonlinear strongly degenerate parabolic equation.

ConditionKEY WORDS: second-order parabolic equation, uniformly elliptic operators, nonclassical elliptic boundary value problem, Hblder function space.Consider the following problem. Let f~ C R", n > 3, and let some coordinates w = (wl,..., w,~-l) be are uniformly elliptic operators, (b(x, t), g) > b0 > 0, b = (b~,..., bn), and ~ is the inward (with respect to f~) normal on F. Obviously, for r = qo(x)lr, system (1) is equivalent to the problem of finding a single function u(x, t) with boundary conditions containing second derivatives along the tangents to the boundary. Apparently, such a nonclassical boundary value problem was first stated in [1]. The elliptic version of this problem was treated in [2]. In [3] a similar problem was studied for the case in which the parabolic operator is given on the boundary FT, and in [4] a more complicated case of a quasilinear problem with a parabolic quasilinear operator on the boundary was treated. Finally, in [5] the fundamental solution of problem (1) was shown to exist in the half-space under some restrictions on the coefficients and the right-hand sides of the problem. Obviously, the fact that the boundary condition in (1) contains only an elliptic operator makes the linear version of the problem richer in content.The general scheme for proving the solvability of classical boundary value problems for parabolic equations (the Dirichlet problem, the Neumann problem) outlined in [6, Chap. 4], is based on the study of some model problems in the half-space, so that the properties of these model problems are then transferred to the case of an arbitrary domain and arbitrary coefficients from some classes. Following this idea, we obtain a boundary value problem in our case by freezing the arguments in the coefficients, discarding lower-order terms, and straightening a part of the boundary F, which leads to the following problem. It is required