A note on local well-posedness of generalized KdV type equations with dissipative perturbations

In this note we report local well-posedness results for the Cauchy problems associated to generalized KdV type equations with dissipative perturbation for given data in the low regularity $L^2$-based Sobolev spaces. The method of proof is based on the {\em contraction mapping principle} employed in some appropriate time weighted spaces.Comment: 14 page

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo

Direct application of compound-specific radiocarbon analysis of leaf waxes to establish lacustrine sediment chronology

Author Posting. © Springer, 2007. This is the author's version of the work. It is posted here by permission of Springer for personal use, not for redistribution. The definitive version was published in Journal of Paleolimnology 39 (2008): 43-60, doi:10.1007/s10933-007-9094-1.This study demonstrates use of compound-specific radiocarbon analysis (CSRA) for dating Holocene lacustrine sediments from carbonate-hosted Ordy Pond, Oahu, Hawaii. Long-chain odd-numbered normal alkanes (n-alkanes), biomarkers characteristic of terrestrial higher plants, were ubiquitous in Ordy Pond sediments. The δ13C of individual n-alkanes ranged from −29.9 to −25.5‰, within the expected range for n-alkanes synthesized by land plants using the C3 or C4 carbon fixation pathway. The 14C ages of n-alkanes determined by CSRA showed remarkably good agreement with 14C dates of rare plant macrofossils obtained from nearby sedimentary horizons. In general, CSRA of n-alkanes successfully refined the age-control of the sediments. The sum of n-alkanes in each sample produced 70–170 μg of carbon (C), however, greater age errors were confirmed for samples containing less than 80 μg of C. The 14C age of n-alkanes from one particular sedimentary horizon was 4,155 years older than the value expected from the refined age-control, resulting in an apparent and arguable age discrepancy. Several lines of evidence suggest that this particular sample was contaminated by introduction of 14C-free C during preparative capillary gas chromatography. This study simultaneously highlighted the promising potential of CSRA for paleo-applications and the risks of contamination associated with micro-scale 14C measurement of individual organic compounds.This project was funded by Petroleum Research Fund (PRF #40088-ACS) and in part by Sigma Xi, The Scientific Research Society (Grants in aid of research, 2003)

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture