Randomness and Nonlinear Evolution Equations

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.     

Sharp global well-posedness for KdV and modified KdV on

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all -based Sobolev spaces where local well-posedness is presently known, apart from the endpoint for mKdV and the endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

     

Bilinear estimates and applications to 2d NLS

The three bilinearities for functions are sharply estimated in function spaces associated to the Schrödinger operator . These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.

     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

Coordination properties of sulfonyl-N-aminoacids: Crystal and molecular structure of the [Zn(II) (N-(p-toluenesulfonyl)-L-glutaminate)2(H2O)2] complex

The coordination behavior of N-tosyl-L-glutamine (hereafter abbreviated as tsglnH, tosyl =toluenesulfonyl group) toward the Zn(II) ion is investigated by means of structural and spectroscopic techniques. From aqueous solution, at acidic pH the Zn(II) ion separates a compound of empirical formula Zn(tsgln)₂·2H₂O whose crystal structure is determined. The crystal is monoclinic, space group C2, with cell dimensions:a=19.775(5),b=5.3146(3),c=16.543(8) Å, =119.93(3)°,Z=2. The Zn(II) is tetrahedrally coordinated by two symmetry-related tsgln monoanions acting as monodentate carboxylate ligands and two symmetry related water molecules. The Zn–O bond distances are similar while the angles at Zn(II) ion differ significantly from those of regular tetrahedron. The side chain amide group does not participate in metal coordination and forms intermolecular hydrogen bonds.     

Thermodynamic aspects of the adsorption of hexametaphosphate on kaolinite

The adsorption of hexametaphosphate ion, an important deflocculant used in the ceramic industry, from aqueous solutions onto kaolinite has been studied at different temperatures. The adsorption isotherm follows the Langmuir model: the thermodynamic parameters DeltaG(ads)(0), DeltaH(ads)(0), and DeltaS(ads)(0) were calculated and found to be consistent with an interaction model involving the formation of an inner-sphere complex between HMP and aluminol groups. Also, the dependence of the adsorption behavior on the kaolinite volume fraction has been studied and discussed in term of association processes between the clay particles.