A problem with free boundary for the Navier-Stokes equations for a compressible fluid in the presence of surface tension

It is proved that this problem has a unique solution, belonging to some Sobolev spaces, on a finite time interval, whose length depends on the data of the problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 142–148, 1990.     

Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local (in time) existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions . We assume that may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ². We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles. Dedicated to O. A. Ladyzhenskaya on the occasion of her 70^th birthday. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 159–178, 1992. Translated by E. V. Frolova.     

Nonlocal overdetermined boundary value problem for stationary Navier-Stokes equations

A stationary system of Stokes and Navier-Stokes equations describing the flow of a homogeneous incompressible fluid in a bounded domain is considered. The vector of the flow velocity and a finite number of nonlocal conditions are defined at a part of the domain boundary. It is proved that, in the linear case, the problem has at least one stable solution. In the nonlinear case, the local solvability of the problem is proved.     

A free-boundary problem for the Navier-Stokes equations. I

Institute of Mathematics and Cybernetics, Lithuanian Academy of Sciences. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 1, pp. 166–179, January-March, 1991.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

A free boundary problem for the Navier–Stokes equations with measure data

The existence of a weak solution of an initial boundary-value problem for the plane nonstationary Navier–Stokes equations with Radon measure data on the free boundary, is established. The problem may be considered as a model of the blood flow around the heart valves. An inverse problem is studied, it allows us to find the boundary forces acting on the valve from the observed values of the velocity of the fluid in a fixed subregion.     

On free boundary problems with moving contact points for the stationary two-dimensional Navier-Stokes equations

Solvability of the problem of slow drying of a plane capillary in the classical setting (i. e., with the adherence condition on a rigid wall) is established. The proof is based on a detailed study of the asymptotics of the solution near a point of contact of the free boundary with a moving wall, including estimates of the coefficients in well known asymptotic formulas. It is shown that the only value of the contact angle admitting a solution of the problem with finite energy dissipation equals π. Bibliography: 18 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 179–205. Translated by E. V. Frolova.     

A free boundary problem for quasilinear degenerate parabolic equations with general degeneracy

In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)–(1.4). The existence of generalized solutions inBV ₁, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C¹⁺ smoothness for the free boundary is obtained in the parabolic case.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Analyticity for the Navier-Stokes equations governed by surface tension on the free boundary

A free boundary problem for an incompressible viscous fluid is considerd; the boundary is to be determined by equilibrium conditions involving the fluid's stress tensor and its surface tension. It is proved that if the data of the problem are analytic then the free boundary, the velocity vector, and the pressure are all analytic.     

Regularity for the Navier-Stokes equations with slip boundary condition

For the Navier-Stokes equations with slip boundary conditions, we obtain the pressure in terms of the velocity. Based on the representation, we consider the relationship in the sense of regularity between the Navier-Stokes equations in the whole space and those in the half space with slip boundary data.

     

A problem with free boundary

One considers the Dirichlet problem for the equation u=(u), where is the Heaviside function. Under special assumptions one constructs the solution of this problem with convex and smooth level surfaces and, in particular, with a regular free surface, which coincides with the set of level zero. One proves the solvability in the small of the problem in the neighborhood of the constructed regular solution under perturbations of the boundary condition and a smooth boundary of the domain .Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 72–83, 1986.     

Solvability of some two-dimensional quasistationary free boundary problems for the Navier-Stokes equations with moving contact points

The paper is concerned with some quasistationary two-dimensional free boundary problems of viscous flow with moving contact points and with contact angle equal to π. A typical example of such a flow is filling a capillary tube in the presence of surface tension. The proof of the solvability of these problems is based on the analysis (made by the author and V. V. Pukhnachëv about 10 years ago) of the asymptotic formulas for the solutions of the Navier-Stokes equations in a neigborhood of contact points. Bibliography: 10 titles.     

Solvability of three-dimensional problems with a free boundary for a stationary system of Navier-Stokes equations

One proves the solvability of the boundary-value problem for the Navier-Stokes system, describing the stationary motion of a heavy, viscous, incompressible, capillary fluid, filling partially a certain container.     

The initial boundary value problem with a free surface condition for the ɛ-approximations of the Navier-Stokes equations and some of their regularizations

We study the unique solvability in the large on the semiaxis ℝ⁺ of the initial boundary value problems (IBVP) with the boundary slipcondition (the natural boundary condition) for the ɛ-approximations (0.6)–(0.8), (0.20); (0.13)–(0.15), (0.21), and (0.16–0.18), (0.22) of the Navier-Stokes equations (NSE), of the NSE modified in the sense of O. A. Ladyzhenskaya, and the equations of motion of the Kelvin-Voight fluids. For the classical solutions of perturbed problems we prove certain estimates which are uniform with respect to ɛ, and show that as ɛ→0 the classical solutions of the perturbed IBVP respectively converge to the classical solutions of the IBVP with the boundary slip condition for the NSE, for the NSE (0.11) modified in the sense of Ladyzhenskaya, and for the equations (0.12) of motion of the Kelvin-Voight fluids. Bibliography: 40 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 38–70. Translated by A. P. Oskolkov.     

A singular perturbation free boundary problem for elliptic equations in divergence form

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem , where A = A(x) is Holder continuous, β _ɛ converges to the Dirac delta δ₀. By studying some suitable level sets of u _ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C ^1,γ surface around a.e. point on the free boundary.     

Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition II

Summary We consider mixed finite element approximations of the stationary, incompressible Navier-Stokes equations with slip boundary condition simultaneously approximating the velocity, pressure, and normal stress component. The stability of the schemes is achieved by adding suitable, consistent penalty terms corresponding to the normal stress component and to the pressure. A new method of proving the stability of the discretizations allows, us to obtain optimal error estimates for the velocity, pressure, and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition. The schemes can easily be implemented into existing finite element codes for the Navier-Stokes equations with standard Dirichlet boundary conditions.