69 research outputs found
On the Navier-Stokes equations with rotating effect and prescribed outflow velocity
We consider the equations of Navier-Stokes modeling viscous fluid flow past a
moving or rotating obstacle in subject to a prescribed velocity
condition at infinity. In contrast to previously known results, where the
prescribed velocity vector is assumed to be parallel to the axis of rotation,
in this paper we are interested in a general outflow velocity. In order to use
-techniques we introduce a new coordinate system, in which we obtain a
non-autonomous partial differential equation with an unbounded drift term. We
prove that the linearized problem in is solved by an evolution
system on for . For this we use
results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove,
for and initial data , the
existence of a unique mild solution to the full Navier-Stokes system.Comment: 18 pages, to appear in J. Math. Fluid Mech. (published online first
Saturated Hydraulic Conductivity and Land Use Change, New Insights to the Payments for Ecosystem Services Programs: a Case Study from a Tropical Montane Cloud Forest Watershed in Eastern Central Mexico
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
We present an abstract framework for analyzing the weak error of fully
discrete approximation schemes for linear evolution equations driven by
additive Gaussian noise. First, an abstract representation formula is derived
for sufficiently smooth test functions. The formula is then applied to the wave
equation, where the spatial approximation is done via the standard continuous
finite element method and the time discretization via an I-stable rational
approximation to the exponential function. It is found that the rate of weak
convergence is twice that of strong convergence. Furthermore, in contrast to
the parabolic case, higher order schemes in time, such as the Crank-Nicolson
scheme, are worthwhile to use if the solution is not very regular. Finally we
apply the theory to parabolic equations and detail a weak error estimate for
the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic
heat equation
Adjoint bi-continuous semigroups and semigroups on the space of measures
For a given bi-continuous semigroup T on a Banach space X we define its
adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some
abstract conditions this adjoint semigroup is again bi-continuous with respect
to the weak topology (X^o,X). An application is the following: For K a Polish
space we consider operator semigroups on the space C(K) of bounded, continuous
functions (endowed with the compact-open topology) and on the space M(K) of
bounded Baire measures (endowed with the weak*-topology). We show that
bi-continuous semigroups on M(K) are precisely those that are adjoints of a
bi-continuous semigroups on C(K). We also prove that the class of bi-continuous
semigroups on C(K) with respect to the compact-open topology coincides with the
class of equicontinuous semigroups with respect to the strict topology. In
general, if K is not Polish space this is not the case
Error estimates for the discretization of the velocity tracking problem
In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281â2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary NavierâStokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 â 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary NavierâStokes equations
Asymptotic behavior and hypercontractivity in non-autonomous Ornstein-Uhlenbeck equations
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