Smooth counterexamples to strong unique continuation for a Beltrami system in C2\mathbb{C}^2

We construct an example of a smooth map $\mathbb{C}\to\mathbb{C}^2$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $\bar z$ derivative to the norm of the $z$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.Comment: 21 page

SOME NONLINEAR DIFFERENTIAL INEQUALITIES AND AN APPLICATION TO HÖLDER CONTINUOUS ALMOST COMPLEX STRUCTURES

Abstract. We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex valued functions f(z) satisfying ∂f/∂¯z = |f | α,0<α<1, and f(0) ̸ = 0, there is also a lower bound for sup |f | on the unit disk. For each α, we construct a manifold with an α-Hölder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous. 1

Tidal Evolution Related to Changing Sea Level; Worldwide and Regional Surveys, and the Impact to Estuaries and Other Coastal Zones

Global sea level rise understanding is critical for coastal zones, and estuaries are particularly vulnerable to water level changes. Sea level is increasing worldwide due to several climactic factors, and tidal range may also change in estuaries due to sea level rise and anthropogenic harbor improvements that may modify friction and resonance, increasing risks to population centers. Tidal range changes may further complicate the risks of sea level rise, increasing the frequency of nuisance flooding, and may affect tide-sensitive ecosystems. Higher total water levels threaten to increase flood zone areas in estuarine regions, which can impact the infrastructure, industry, and public health of coastal populations, as well as disrupting sensitive biological habitats. Therefore, it is of critical interest to analyze how tidal range changes under sea level changes. This chapter describes the tidal anomaly correlation (TAC) methodology which can quantify the tidal evolution related to sea level changes. A basin-wide survey of Pacific and Atlantic Ocean tide gauges is detailed, showing that tidal changes due to sea level rise is present at most locations surveyed. A focused regional study of Hong Kong is also described as an example of how tidal evolution can impact high population density coastal zones

Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates

We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.Comment: Version 3 with new Section 4.2 based on suggestions of referee. Accepted to appear in Transactions AM

Circulations in the Pearl River Estuary: Observation and Modeling

This chapter reports a cruise survey on the Pearl River Estuary (PRE) and adjacent costal water in the period between May 3, 2014 and May 11, 2014. The circulation and salinity structure were sampled for different tidal phases. With the cruise data, a “sandwich” structure of the lateral salinity distribution and a two-layer structure of longitudinal circulation were identified, together with high variations influenced by wind and tide. Furthermore, longitudinally orientated convergence or divergence of the lateral velocity close to the channel location for certain tidal conditions was observed. The finite volume community ocean model (FVCOM) is configured and run with high spatial resolution of 100 m in the PRE. An atmospheric model, the Weather Research and Forecasting (WRF) Model, is also run to provide high spatial and temporal resolution of atmospheric forcing for the FVCOM. The FVCOM modeling skill assessment is conducted using the cruise salinity and velocity data, as well as water levels, showing that the model can well simulate the velocity and salinity structures. The numerical model reveals that there is a strong neap-spring cycle for the PRE de-tided circulation with 0.37 m s−1 during the neap tide about 42% stronger than that (0.26 m s−1) during the spring tide in the surface layer

Variations of the Absorption of Chromophoric Dissolved Organic Matter in the Pearl River Estuary

Analysis of in-situ measurements during a spring cruise survey in the Pearl River Estuary (PRE) reveals that, controlled by the two-layer gravitational circulation, chromophoric dissolved organic matter (CDOM) absorption shows a clear horizontal distribution pattern at both water surface and bottom, with higher CDOM absorption and lower spectral slope in the northwestern estuary, and a reversed pattern in the southeastern estuary and near the Hong Kong waters. The surface CDOM has higher absorption and lower spectral slope than the bottom. Horizontal transport is suggested to be the dominant hydrodynamic mechanism affecting CDOM distribution pattern in the PRE. With a regional algorithm tailored for the PRE CDOM absorption retrieval, a time series of CDOM absorption and spectral slope in the PRE and the Hong Kong waters in spring from 2012 to 2018 is produced based on satellite images obtained by four sensors with different spatial and spectral resolutions: the Visible Infrared Imaging Radiometer Suite (VIIRS), the Ocean and Land Colour Instrument (OLCI), the Hyperspectral Imager for the Coastal Ocean (HICO), and the Operational Land Imager (OLI). A correlation is revealed between the multi-temporal CDOM absorption and the monthly averaged river discharge, indicating the capability of CDOM ocean color products in identifying hydrodynamic processes in estuaries and coastal waters