In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.
The stability and evolution of very thin, single component, metallic-melt films is studied by analysis of coupled strongly nonlinear equations for gas-melt (GM) and crystal-melt (CM) interfaces, derived using the lubrication approximation. The crystal-melt interface is deformable by freezing and melting, and there is a thermal gradient applied across the film. Linear analysis reveals that there is a maximum applied far-field temperature in the gas, beyond which there is no film instability. Instabilities...
The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.
We prove the existence of a weak solution and of a strong solution (locally in time) of the equations which govern the motion of viscous incompressible non-homogeneous fluids. Then we discuss the decay problem.
We investigate the inviscid limit for the stationary Navier-Stokes equations in a two dimensional bounded domain with slip boundary conditions admitting nontrivial inflow across the boundary. We analyze admissible regularity of the boundary necessary to obtain convergence to a solution of the Euler system. The main result says that the boundary of the domain must be at least C²-piecewise smooth with possible interior angles between regular components less than π.
This paper is devoted to the study of smooth flows of density-dependent fluids in or in the torus . We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity when goes to . A blow-up criterion involving the norm of vorticity in is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time...
We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.
We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain , which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves and (lower and upper parts of ), the Dirichlet boundary conditions on (the inflow) and (boundary of the profile) and an artificial “do nothing”-type boundary condition...
We consider a free boundary value problem for a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel, with gravity and surface tension, governed by the Navier-Stokes equations. We obtain existence results for the linear and nonlinear time-dependent problem. We analyse the qualitative behavior of the flow using tools of bifurcation theory. The main result is a Hopf bifurcation theorem with -symmetry.
We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.