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

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

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

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

Structural Characterization of Phase Separation in Fe-Cr: A Current Comparison of Experimental Methods

Self-assembly due to phase separation within a miscibility gap is important in numerous material systems and applications. A system of particular interest is the binary alloy system Fe-Cr, since it is both a suitable model material and the base system for the stainless steel alloy category, suffering from low-temperature embrittlement due to phase separation. Structural characterization of the minute nano-scale concentration fluctuations during early phase separation has for a long time been considered a major challenge within material characterization. However, recent developments present new opportunities in this field. Here, we present an overview of the current capabilities and limitations of different techniques. A set of Fe-Cr alloys were investigated using small-angle neutron scattering (SANS), atom probe tomography, and analytical transmission electron microscopy. The complementarity of the characterization techniques is clear, and combinatorial studies can provide complete quantitative structure information during phase separation in Fe-Cr alloys. Furthermore, we argue that SANS provides a unique in-situ access to the nanostructure, and that direct comparisons between SANS and phase-field modeling, solving the non-linear Cahn Hilliard equation with proper physical input, should be pursued

Soil Moisture Retrieval with Airborne PALS Instrument over Agricultural Areas in SMAPVEX16

NASA's SMAP (Soil Moisture Active Passive) calibration and validation program revealed that the soil moisture products are experiencing difficulties in meeting the mission requirements in certain agricultural areas. Therefore, the mission organized airborne field experiments at two core validation sites to investigate these anomalies. The SMAP Validation Experiment 2016 included airborne observations with the PALS (Passive Active L-band Sensor) instrument and intensive ground sampling. The goal of the PALS measurements are to investigate the soil moisture retrieval algorithm formulation and parameterization under the varying (spatially and temporally) conditions of the agricultural domains and to obtain high resolution soil moisture maps within the SMAP pixels. In this paper the soil moisture retrieval using the PALS brightness temperature observations in SMAPVEX16 is presented

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html