50 research outputs found
On measuring unboundedness of the -calculus for generators of analytic semigroups
We investigate the boundedness of the -calculus by estimating the
bound of the mapping :
for near zero. Here, generates
the analytic semigroup and is the space of bounded analytic
functions on a domain strictly containing the spectrum of . We show that
in general, whereas
for bounded calculi. This generalizes a result
by Vitse and complements work by Haase and Rozendaal for non-analytic
semigroups. We discuss the sharpness of our bounds and show that single square
function estimates yield
.Comment: Preprint of the final, published version. In comparison with previous
version, Prop. 2.2 was added and Thm. 3.5 has been slightly adapted in order
to point out the major assertio
On the smallest scale for the incompressible Navier-Stokes equations
It is proven that for solutions to the two- and three-dimensional incompressible Navier-Stokes equations the minimum scale is inversely proportional to the square root of the Reynolds number based on the kinematic viscosity and the maximum of the velocity gradients. The bounds on the velocity gradients can be obtained for two-dimensional flows, but have to be assumed to be three-dimensional. Numerical results in two dimensions are given which illustrate and substantiate the features of the proof. Implications of the minimum scale result to the decay rate of the energy spectrum are discussed
On the stability of totally upwind schemes for the hyperbolic initial boundary value problem
In this paper, we present a numerical strategy to check the strong stability
(or GKS-stability) of one-step explicit totally upwind scheme in 1D with
numerical boundary conditions. The underlying approximated continuous problem
is a hyperbolic partial differential equation. Our approach is based on the
Uniform Kreiss-Lopatinskii Condition, using linear algebra and complex analysis
to count the number of zeros of the associated determinant. The study is
illustrated with the Beam-Warming scheme together with the simplified inverse
Lax-Wendroff procedure at the boundary
Higher order finite difference schemes for the magnetic induction equations
We describe high order accurate and stable finite difference schemes for the
initial-boundary value problem associated with the magnetic induction
equations. These equations model the evolution of a magnetic field due to a
given velocity field. The finite difference schemes are based on Summation by
Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various
numerical experiments that demonstrate both the stability as well as high order
of accuracy of the schemes.Comment: 20 page
Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
We consider three problems for the Helmholtz equation in interior and
exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and
Neumann-to-Dirichlet problems for outgoing solutions, and the interior
impedance problem. We derive sharp estimates for solutions to these problems
that, in combination, give bounds on the inverses of the combined-field
boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee
comments and added several reference
Strongly Kreiss Bounded Operators in UMD Banach Spaces
In this paper we give growth estimates for for in the
case is a strongly Kreiss bounded operator on a UMD Banach space . In
several special cases we provide explicit growth rates. This includes known
cases such as Hilbert and -spaces, but also intermediate UMD spaces such
as non-commutative -spaces and variable Lebesgue spaces.Comment: 27 page
Resolvent conditions and growth of powers of operators
Following Berm\'udez et al. (ArXiv: 1706.03638v1), we study the rate of
growth of the norms of the powers of a linear operator, under various resolvent
conditions or Ces\`aro boundedness assumptions. We show that is
power-bounded if (and only if) both and are absolutely Ces\`aro
bounded. In Hilbert spaces, we prove that if satisfies the Kreiss
condition, ; if is absolutely Ces\`aro
bounded, for some (which
depends on ); if is strongly Kreiss bounded, then for some . We show that a Kreiss bounded operator on a
reflexive space is Abel ergodic, and its Ces\`aro means of order
converge strongly when .Comment: Added references [35] and [38] and updated some remark