Oseen resolvent estimates with small resolvent parameter

We consider the Oseen system with resolvent term in an exterior domain in R^3 , supplemented by homogeneous Dirichlet boundary conditions. Under the assumption that the resolvent parameter λ is close to zero and Re λ ≥ 0, λ different from 0, we estimate the L^p-norm of the velocity against the L^p-norm of the right-hand side, times a factor C |λ|^{−2} , with C > 0 independent of λ. Such an estimate cannot hold for this range of λ if |λ|^{−2} is replaced by |λ|^{−κ} with κ 0, with the constant in the resolvent estimate depending on c

Stability of stationary viscous incompressible flow around a rigid body performing a translation

Suppose a rigid body moves steadily and without rotation in a viscous incompressible fluid,and the flow around the body is steady, too. Such a flow is usually describedby the stationary Navier-Stokes system with Oseen term, in an exterior domain.An Oseen term arises because the velocity field is scaled in such a way that it vanishesat infinity. In the work at hand, such a velocity field, denoted by U, is considered as given.We study a solution of the incompressible evolutionary Navier-Stokes system withthe same right-hand side and the same Dirichlet boundary conditions as the stationaryproblem, and with U+u_0 as initial data, where u_0 is a H^1-function.Under the assumption that the H^1-norm of u_0 is small (u_0 a ``perturbation of U'')and that the eigenvalues of a certain linear operator have negative real part, we show thatthe L^2-norm of the gradient of the difference v(t)-U tends to zero when t tends to infinity (''stability of v''), where v denotes the velocity part of the solution to the initial-boundary value problem under consideration

The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space of the velocity

A representation formula without pressure term is derived for regular solutions to the 3D time-dependent Oseen system in exterior Lipschitz domains. This formula is valid even if no boundary conditions are imposed. It is used in order to exhibit how the velocity decays pointwise in space. It turns out that the rate of this decay depends on L p-integrability in time of the velocity. In addition, this work is the basis for successor papers dealing with spatial decay of L q-weak solutions and mild solutions to the time-dependent Oseen system, and with L 2-strong solutions to the stability problem related to the Navier-Stokes system with Oseen term

Point counting on reductions of CM elliptic curves

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.Comment: Minor corrections. To appear in Journal of Number Theor

A Modular Curve of Level 9 and the Class Number One Problem

In this note we give an explicit parametrization of the modular curve associated to the normalizer of a non-split Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem.Comment: 18 page

Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for (3/2)-\varep < p< 3+\epsilon. This gives an affirmative answer to a conjecture of M. Taylor.Comment: 28 page. Minor revision was made regarding the definition of the Stokes operator in Lipschitz domain

NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY

We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ D , where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate

Some genus 3 curves with many points

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves