Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order $s\geq0$ on the group \Diff_c(M) of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on \Diff_c(S^1) vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on \Diff_c(M) vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on \Diff_c(M) is positive for $\dim(M)=1, s>\frac12$ and $\dim(M)\geq2, s\geq1$. For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$ we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$, the modified Constantin-Lax-Majda equation for $s=\frac 12$ and the Camassa-Holm equation for $s=1$.Comment: 16 pages. Final versio

Generation and annealing of defects in virgin fused silica (type III) upon ArF laser irradiation: Transmission measurements and kinetic model

Abstract The transmission of ArF laser pulses in virgin fused silica (type III) samples changes during N = 10 6 pulses at an incoming fluence H in = 5 mJ cm À2 pulse À1 . The related absorption is determined by the pulse energy absorption coefficient a(N, H in ) using a modified Beer&apos;s law, yielding initial values a ini around 0.005 cm À1 , a maximum a max 6 0.02 cm À1 at N = 10 3 -10 4 and stationary values 0.0045 cm À1 6 a end 6 0.0094 cm À1 after N % 6 · 10 5 pulses. The development a(N, H in = const.) is simulated by a rate equation model assuming a pulse number dependent E 0 center density E 0 (N). E 0 (N) is determined by a dynamic equilibrium between E 0 center generation and annealing. Generation occurs photolytically from the precursors ODC II and unstable SiH structures upon single photon absorption and from strained SiO bonds via two-photon excitation. Annealing in the dark periods between the laser pulses is dominated by the reaction of E 0 with H 2 present in the SiO 2 network. The development a(N, H in = const.) is observed for the very first sample irradiation only (virgin state). The values a end are not accessible by simple spectrophotometer measurements

Polyharmonic approximation on the sphere

The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$ approximation order for this kind of approximation is $\sigma$ for functions having $L_p$ smoothness $\sigma$ (for $\sigma$ up to the order of the underlying differential operator, just as in univariate spline theory). This is an improvement over previous error estimates, which penalized the approximation order when measuring error in $L_p$, p>2 and held only in a restrictive setting when measuring error in $L_p$, p<2.Comment: 16 pages; revised version; to appear in Constr. Appro

Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off

Accepted versio

Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already been publishe

Well-posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.Comment: The introduction is rewritte

On the Usefulness of Modulation Spaces in Deformation Quantization

We discuss the relevance to deformation quantization of Feichtinger's modulation spaces, especially of the weighted Sjoestrand classes. These function spaces are good classes of symbols of pseudo-differential operators (observables). They have a widespread use in time-frequency analysis and related topics, but are not very well-known in physics. It turns out that they are particularly well adapted to the study of the Moyal star-product and of the star-exponential.Comment: Submitte

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain $\Omega$ of R N , N=2,3, surrounded by a thin layer $\Sigma \epsilon$, along a part $\Gamma$2 of its boundary $\partial \Omega$, we consider a Navier-Stokes flow in $\Omega \cup \partial \Omega \cup \Sigma \epsilon$ with Reynolds' number of order 1/$\epsilon$ in $\Sigma \epsilon$. Using $\Gamma$-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface $\Gamma$2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context