Blow up in finite time and dynamics of blow up solutions for the -critical generalized KdV equation

In this paper, we describe the dynamics of blow up solutions for the critical generalized KdV equation such that the initial data is close to the soliton in and has decay in at the right. In particular, we prove that blow up occurs in finite time, and we obtain an upper bound on the blow up rate.

     

Capture and Controlled Release of Fragrances by CD Finished Textiles

Cotton, wool and polyester fabrics finished with cyclodextrins (by the intermediateof polycarboxylic acids) were impregnated with six different fragrant molecules(-citronellol, camphor, menthol, cis-jasmone, benzyl acetate, geraniol) and citronella oil. The smell intensity of the samples has been investigated during almostone year. In most cases, the odour retention of untreated textiles towards the substratesdid not persist over one or two weeks. By contrast, most of the treated sampleskept smelling one year after the beginning of the experiment. We observed that theuse of polyacrylic acid as crosslinking agent was less effective than citric andbutanetetracarboxylic acids. The efficiency of the different CDs varied in the order-CD > -CD > -CD, and the durability of the fragranteffect was directly dependent on the amount of CD grafted onto the fabrics.     

Current saturation and electrical breakdown in multiwalled carbon nanotubes

We investigate the limits of high energy transport in multiwalled carbon nanotubes (MWNTs). In contrast to metal wires, MWNTs do not fail in the continuous, accelerating manner typical of electromigration. Instead, they fail via a series of sharp, equally sized current steps. We assign these steps to the sequential destruction of individual nanotube shells, consistent with the MWNT's concentric-shell geometry. Furthermore, the initiation of this failure is very sensitive to air exposure. In air failure is initiated by oxidation at a particular power, whereas in vacuum MWNTs can withstand much higher power densities and reach their full current carrying capacities.     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

On the Gibbs states of the noncritical Potts model on \mathbb Z ^2

We prove that all Gibbs states of the $q$ -state nearest neighbor Potts model on $\mathbb Z ^2$ below the critical temperature are convex combinations of the $q$ pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature $\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)$ .     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

Surfactant Foam Selection for Enhanced Light Non-Aqueous Phase Liquids (LNAPL) Recovery in Contaminated Aquifers

Transport in Porous Media - Foam is promising for the remediation of non-aqueous phase liquids (NAPL) source zones; however, the production of foam and its behavior in porous media are poorly...     

A C-glycoside route to leukotrienes

The stereospecific syntheses of three isomers of 7-hydroxy-5,6-epoxy heptanoic acid methyl ester, $\text{7}$, $\text{12b}$ and $\text{19}$, have been achieved via a $\text{C}$-glycoside route.