Lagrangian approximations and weak solutions of the Navier-Stokes equations

Abstract

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations. AMS-MSC (2000): 35B65, 35D05, 76D05

Justification of the fast multipole method for the stokes system. Part II. Exterior domain problems

“…As in [29], we choose for the solution of (16) the potential ansatz and obtain for the velocity part of the solution the representation…”

“…By the methods presented in [1,29], we obtain a representation of the solution in form of hydrodynamical potentials, whose unknown densities are solutions of uniquely solvable boundary integral equations. For the numerical solution of these integral equations with m points in the discretization of the boundary, a dense, non-symmetrical linear system have to be solved.…”

“…In the following two lemmata, we outline the main results for the solvability of the Stokes system in cases (2) and (3) without detailed investigation of the corresponding problems (one can find it, e.g., in [14], [21] and [29]). …”

Boundary integral equations for a three‐dimensional Brinkman flow problem

“…[9,10,14,19,30,31,45]). On the other hand, Costabel [3] showed for strongly elliptic second order systems that the corresponding layer potential operators on a Lipschitz boundary still provide coerciveness on appropriate Sobolev-Slobodetski spaces on the boundary and regularity properties.…”

The planar Dirichlet problem for the Stokes equations

“…We look for a solution in a modified form. We were inspirited by [6], where the Dirichlet problem for the Stokes equations was studied on an exterior planar domain with connected boundary, and by [16], where the Dirichlet problem for the Laplace equation was studied by the integral equation method on planar domains and a solution of the corresponding integral equation was given in the form of a Neumann series.…”