Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

Synthesizing SMOS Zero-Baselines with Aquarius Brightness Temperature Simulator

SMOS [1] and Aquarius [2] are ESA and NASA missions, respectively, to make L-band measurements from the Low Earth Orbit. SMOS makes passive measurements whereas Aquarius measures both passive and active. SMOS was launched in November 2009 and Aquarius in June 2011.The scientific objectives of the missions are overlapping: both missions aim at mapping the global Sea Surface Salinity (SSS). Additionally, SMOS mission produces soil moisture product (however, Aquarius data will eventually be used for retrieving soil moisture too). The consistency of the brightness temperature observations made by the two instruments is essential for long-term studies of SSS and soil moisture. For resolving the consistency, the calibration of the instruments is the key. The basis of the SMOS brightness temperature level is the measurements performed with the so-called zero-baselines [3]; SMOS employs an interferometric measurement technique which forms a brightness temperature image from several baselines constructed by combination of multiple receivers in an array; zero-length baseline defines the overall brightness temperature level. The basis of the Aquarius brightness temperature level is resolved from the brightness temperature simulator combined with ancillary data such as antenna patterns and environmental models [4]. Consistency between the SMOS zero-baseline measurements and the simulator output would provide a robust basis for establishing the overall comparability of the missions

Monitoring freeze-thaw state by means of GNSS reflectometry. An analysis of TechDemoSat-1 data

The article of the freeze/thaw dynamic of high-latitude Earth surfaces is extremely important and informative for monitoring the carbon cycle, the climate change, and the security of infrastructures. Current methodologies mainly rely on the use of active and passive microwave sensors, while very few efforts have been devoted to the assessment of the potential of observations based on signals of opportunity. This article aims at assessing the performance of spaceborne Global Navigation Satellite System Reflectometry (GNSS-R) for high-spatial and highoral resolution monitoring of the Earth-surface freeze/thaw state. To this aim, reflectivity values derived from the TechDemoSat-1 (TDS-1) data have been collected and elaborated, and thus compared against the soil moisture active passive (SMAP) freeze/thaw information. Shallow subsurface soil temperature values recorded by a network of in situ stations have been considered as well. Even if an extensive and timeliness cross availability of both types of experimental data is limited by the spatial coverage and density of TDS-1 observations, the proposed analysis clearly indicates a significant seasonal cycle in the calibrated reflectivity. This opens new perspectives for the bistatic L-band high-resolution satellite monitoring of the freeze/thaw state, as well as to support the development of next-generation of GNSS-R satellite missions designed to provide enhanced performance and improved temporal and spatial coverage over high latitude areas

Well-posedness and stability results for the Gardner equation

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also cover the energy space H^1(R) where global well-posedness follows from the conservation laws of the system. Moreover, we construct solitons of the Gardner equation explicitly and prove that, under certain conditions, this family is orbitally stable in the energy space.Comment: 1 figure. Accepted for publication in Nonlin.Diff Eq.and App

Global well-posedness for a coupled modified kdv system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modi ed Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura's Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura's Transform that takes a KdV system to the system we are considering. To overcome this di culty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura's Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.Fundação para a Ciência e a Tecnologia (FCT

The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the same sign (KdVI) or two boundary conditions if $q_{t}$ and $q_{xxx}$ have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if $\{q(x,0),q(0,t) \}$ and $\{q(x,0),q(0,t),q_{x}(0,t) \}$ are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values $\{q_{x}(0,t),q_{xx}(0,t) \}$ and $\{q_{xx}(0,t) \}$, respectively. We show that this can be achieved without solving for $q(x,t)$ by analysing a certain ``global relation'' which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function $\Phi^{(t)}(t,k)$, where $\Phi^{(t)}$ satisifies the $t$-part of the associated Lax pair evaluated at $x=0$. Indeed, by employing a Gelfand--Levitan--Marchenko triangular representation for $\Phi^{(t)}$, the global relation can be solved \emph{explicitly} for the unknown boundary values in terms of the given initial and boundary conditions and the function $\Phi^{(t)}$. This yields the unknown boundary values in terms of a nonlinear Volterra integral equation.Comment: 21 pages, 3 figure

Development of SMAP Mission Cal/Val Activities

The Soil Moisture Active Passive (SMAP) mission is a NASA directed mission to map global land surface soil moisture and freeze-thaw state. Instrument and mission details are shown. The key SMAP soil moisture product is provided at 10 km resolution with 0.04cubic cm/cubic cm accuracy. The freeze/thaw product is provided at 3 km resolution and 80% frozen-thawed classification accuracy. The full list of SMAP data products is shown

Satellite-observed changes in vegetation sensitivities to surface soil moisture and total water storage variations since the 2011 Texas drought

We combine soil moisture (SM) data from AMSR-E and AMSR-2, and changes in terrestrial water storage (TWS) from time-variable gravity data from GRACE to delineate and characterize the evolution of drought and its impact on vegetation growth. GRACE-derived TWS provides spatially continuous observations of changes in overall water supply and regional drought extent, persistence and severity, while satellite-derived SM provides enhanced delineation of shallow-depth soil water supply. Together these data provide complementary metrics quantifying available plant water supply. We use these data to investigate the supply changes from water components at different depths in relation to satellite-based enhanced vegetation index (EVI) and gross primary productivity (GPP) from MODIS and solar-induced fluorescence (SIF) from GOME-2, during and following major drought events observed in the state of Texas, USA and its surrounding semiarid area for the past decade. We find that in normal years the spatial pattern of the vegetation–moisture relationship follows the gradient in mean annual precipitation. However since the 2011 hydrological drought, vegetation growth shows enhanced sensitivity to surface SM variations in the grassland area located in central Texas, implying that the grassland, although susceptible to drought, has the capacity for a speedy recovery. Vegetation dependency on TWS weakens in the shrub-dominated west and strengthens in the grassland and forest area spanning from central to eastern Texas, consistent with changes in water supply pattern. We find that in normal years GRACE TWS shows strong coupling and similar characteristic time scale to surface SM, while in drier years GRACE TWS manifests stronger persistence, implying longer recovery time and prolonged water supply constraint on vegetation growth. The synergistic combination of GRACE TWS and surface SM, along with remote-sensing vegetation observations provides new insights into drought impact on vegetation–moisture relationship, and unique information regarding vegetation resilience and the recovery of hydrological drought