On well-posedness for the Benjamin-Ono equation

We prove existence of solutions for the Benjamin-Ono equation with data in $H^s(\R)$, $s>0$. Thanks to conservation laws, this yields global solutions for $H^\frac 1 2(\R)$ data, which is the natural ``finite energy'' class. Moreover, inconditional uniqueness is obtained in $L^\infty_t(H^\frac 1 2(\R))$, which includes weak solutions, while for $s>\frac 3 {20}$, uniqueness holds in a natural space which includes the obtained solutions.Comment: Important changes. We improved both existence and uniqueness results. In particular, uniqueness holds in the natural $L^\infty_t; H^{1/2}_x$ energy spac

Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schr\"odinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.Comment: Presentation improve

Modified energies for the periodic generalized KdV equation and applications

We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high order Gaussian measures along with $L^p$ regularity on the corresponding Radon-Nykodim density, as well as new bounds on the growth of the Sobolev norms of the solutions

Almost sure existence of global weak solutions for super-critical Navier-Stokes equations

In this paper we show that after suitable data randomization there exists a large set of super-critical periodic initial data, in $H^{-\alpha}({\mathbb T}^d)$ for some $\alpha(d) > 0$, for both 2d and 3d Navier-Stokes equations for which global energy bounds are proved. As a consequence, we obtain almost sure super-critical global weak solutions. We also show that in 2d these global weak solutions are unique.Comment: 22 pages, a revised argument in Section 5, the $d=3$ cas

Fine properties of self-similar solutions of the Navier-Stokes equations

We study the solutions of the nonstationary incompressible Navier--Stokes equations in $\R^d$, $d\ge2$, of self-similar form $u(x,t)=\frac{1}{\sqrt t}U\bigl(\frac{x}{\sqrt t}\bigr)$, obtained from small and homogeneous initial data $a(x)$. We construct an explicit asymptotic formula relating the self-similar profile $U(x)$ of the velocity field to its corresponding initial datum $a(x)$

The relative importance of phytoplankton aggregates and zooplankton fecal pellets to carbon export: insights from free-drifting sediment trap deployments in naturally iron-fertilised waters near the Kerguelen Plateau

The first KErguelen Ocean and Plateau compared Study (KEOPS1), conducted in the naturally iron-fertilised Kerguelen bloom, demonstrated that fecal material was the main pathway for exporting carbon to the deep ocean during summer (January–February 2005), suggesting a limited role of direct export via phytodetrital aggregates. The KEOPS2 project reinvestigated this issue during the spring bloom initiation (October–November 2011), when zooplankton communities may exert limited grazing pressure, and further explored the link between carbon flux, export efficiency and dominant sinking particles depending upon surface plankton community structure. Sinking particles were collected in polyacrylamide gel-filled and standard free-drifting sediment traps (PPS3/3), deployed at six stations between 100 and 400 m, to examine flux composition, particle origin and their size distributions. Results revealed an important contribution of phytodetrital aggregates (49 ± 10 and 45 ± 22% of the total number and volume of particles respectively, all stations and depths averaged). This high contribution dropped when converted to carbon content (30 ± 16% of total carbon, all stations and depths averaged), with cylindrical fecal pellets then representing the dominant fraction (56 ± 19%).At 100 and 200 m depth, iron- and biomass-enriched sites exhibited the highest carbon fluxes (maxima of 180 and 84 ± 27 mg C m-2 d-1, based on gel and PPS3/3 trap collection respectively), especially where large fecal pellets dominated over phytodetrital aggregates. Below these depths, carbon fluxes decreased (48 ± 21% decrease on average between 200 and 400 m), and mixed aggregates composed of phytodetritus and fecal matter dominated, suggesting an important role played by physical aggregation in deep carbon export.Export efficiencies determined from gels, PPS3/3 traps and 234Th disequilibria (200 m carbon flux/net primary productivity) were negatively correlated to net primary productivity with observed decreases from ~ 0.2 at low-iron sites to ~ 0.02 at high-iron sites. Varying phytoplankton communities and grazing pressure appear to explain this negative relationship. Our work emphasises the need to consider detailed plankton communities to accurately identify the controls on carbon export efficiency, which appear to include small spatio-temporal variations in ecosystem structure

A lower bound on blowup rates for the 3D incompressible Euler equation and a single exponential Beale-Kato-Majda type estimate

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.Comment: AMS Latex, 15 page

The relative importance of phytoplankton aggregates and zooplankton fecal pellets to carbon export: insights from free-drifting sediment trap deployments in naturally iron-fertilised waters near the Kerguelen Plateau

The first KErguelen Ocean and Plateau compared Study (KEOPS1), conducted in the naturally iron-fertilised Kerguelen bloom, demonstrated that fecal material was the main pathway for exporting carbon to the deep ocean during summer (January–February 2005), suggesting a limited role of direct export via phytodetrital aggregates. The KEOPS2 project reinvestigated this issue during the spring bloom initiation (October–November 2011), when zooplankton communities may exert limited grazing pressure, and further explored the link between carbon flux, export efficiency and dominant sinking particles depending upon surface plankton community structure. Sinking particles were collected in polyacrylamide gel-filled and standard free-drifting sediment traps (PPS3/3), deployed at six stations between 100 and 400 m, to examine flux composition, particle origin and their size distributions. Results revealed an important contribution of phytodetrital aggregates (49+/-10 and 45+/-22% of the total number and volume of particles respectively, all stations and depths averaged). This high contribution dropped when converted to carbon content (30+/-16% of total carbon, all stations and depths averaged), with cylindrical fecal pellets then representing the dominant fraction (56+/-19 %). At 100 and 200m depth, iron- and biomass-enriched sites exhibited the highest carbon fluxes (maxima of 180 and 84+/- 27 mgCm-2 d-1, based on gel and PPS3/3 trap collection respectively), especially where large fecal pellets dominated over phytodetrital aggregates. Below these depths, carbon fluxes decreased (48+/-21%decrease on average between 200 and 400 m), and mixed aggregates composed of phytodetritus and fecal matter dominated, suggesting an important role played by physical aggregation in deep carbon export. Export efficiencies determined from gels, PPS3/3 traps and 234Th disequilibria (200m carbon flux/net primary productivity) were negatively correlated to net primary productivity with observed decreases from ~0.2 at low-iron sites to ~0.02 at high-iron sites. Varying phytoplankton communities and grazing pressure appear to explain this negative relationship. Our work emphasises the need to consider detailed plankton communities to accurately identify the controls on carbon export efficiency, which appear to include small spatio-temporal variations in ecosystem structure