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     

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.     

On one method of approximation of initial boundary value problems for the Navier-Stokes equations

Solutions of the initial boundary value problems for Navier-Stokes equations are approximated by solutions of the initial boundary value problem $$\partial _t u(t) + u_k (t)\partial _k u(t) - v\Delta u(t) - \frac{1}{\varepsilon }\triangledown div u(t) + \frac{1}{2}u(t)div u(t) = f(t),u(0) = u_0 in \Omega , u(t) = 0 on \partial \Omega $$ We study the nearness of solutions of these problems in suitable norms and also the nearness of their minimal global B-attractors. Bibliography:11 titles.     

Remark on compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier-Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier-Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.     

On exact boundary zero-controlability of two-dimensional Navier-Stokes equations

For two-dimensional Navier-Stokes equations defined in a bounded domain Ω and for an arbitrary initial vector field, we construct the boundary Dirichlet condition that is tangent to the boundary ?Ω of Ω and satisfies the property: the solutionυ(t, x) of the mentioned boundary-value problem equals zero at a certain finite time momentT. Moreover, $$\parallel x(t, \cdot )\parallel _{L_2 (\Omega )} \leqslant c\exp \left( {\tfrac{{ - k}}{{(T - t)^2 }}} \right)ast \to T,$$ wherec > 0,k > 0 constants.     

On an initial boundary value problem for the equations of ideal magneto-hydrodynamics

We study an initial boundary value problem for the symmetric hyperbolic quasilinear system of the equations of ideal magneto-hydrodynamics with a perfectly conducting wall boundary condition. Existence of a unique classical solution is proved inside a suitable class of functions of Sobolev type. Moreover, solutions inside the above class are shown to depend continuously in strong norm on the data.     

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.     

Non-linear functional boundary value problem for discontinuous functional differential equations

This paper is devoted to investigating the existence of solutionsfor discontinuous functional differential equations with non-linearfunctional boundary conditions. Some existence results of solutionsare obtained by employing a generalized method of upper andlower solutions, which may be discontinuous.     

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.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

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.     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

A problem with free boundary for the Navier-Stokes equations. II

Institute of Mathematics and Cybernetics, Lithuanian Academy of Sciences. Vilnius University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 2, pp. 337–350, April–June, 1991.     

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 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.     

On fourth-order boundary value problem with singular data

We consider a simply supported beam with restoring and external forces given as a sum of a continuous function and a Dirac delta distribution. We present sufficient conditions on these data in order to guarantee a unique positive or negative solution, respectively.