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The incompressible Navier-Stokes problem is discretized in time by
the two-step backward differentiation formula.
Error estimates are proved under feasible assumptions on the
regularity of the exact solution avoiding hardly fulfillable
compatibility conditions. Whereas the time-weighted velocity error is
of optimal second order, the time-weighted error in the pressure is
of first order. Suboptimal estimates are shown for a
linearisation. The results cover both the two- and
three-dimensional...
We prove the essential m-dissipativity of the Kolmogorov operator associated with the stochastic Navier-Stokes flow with periodic boundary conditions in a space where is an invariant measure
In this paper, a estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on , or on a regular bounded open set of . The proof is based partially on the Strauss inequality [Strauss,Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc. 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions...
In this paper, a estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on , or on a regular bounded open set of . The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math.23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc.9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions...
Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved -estimates of second order derivatives uniformly in the angular and translational velocities, ω and...
We formulate sufficient conditions for regularity up to the boundary of a weak solution v in a subdomain Ω × (t₁,t₂) of the time-space cylinder Ω × (0,T) by means of requirements on one of the eigenvalues of the rate of deformation tensor. We assume that Ω is a cube.
We prove the exact boundary controllability of the 3-D Euler equation
of incompressible inviscid fluids on a regular connected bounded open set when the
control operates on an open part of the boundary that
meets any of the connected components of the boundary.
We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in and a corresponding example for a Stokes-like system.
In the paper [13], we give the full system of equations modelling the motion of a fluid/viscoelastic solid system, and obtain a differential model similar to the so-called Oldroyd model for a viscoelastic fluid. Moreover, existence results in bounded domains are obtained. In this paper we extend the results in [13] to unbounded domains. The unique solvability of the system of equations is established locally in time and globally in time with so-called smallness restrictions. Moreover, existence...
We study the well-posedness of an unsteady fluid-structure interaction problem.
We consider a viscous incompressible flow, which is modelled by the
Navier-Stokes equations. The structure is a collection of rigid moving bodies. The fluid
domain depends on time and is defined by the position of the structure, itself resulting
from a stress distribution coming from the fluid. The problem is then
nonlinear and the equations we deal with are coupled. We prove its local
solvability in time through two...
We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded...
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