Regularity of solutions and interfaces of a generalized porous medium equation inR N

Summary We consider the Cauchy problem for the generalized porous medium equation u_t=(u) where u=u(x, t), xRⁿ and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity (u) is a C¹ function defined for uO which grows like a power of u. Our assumptions generalize the porous medium case, (u)=u^m, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t 0 and t. Our main result deals with the regularity of the solutions. We show that after a certain time t₀ the pressure, defined by v=(u), with (u)=(u)/u and (0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in R^N+1; the solution u is Hölder continuous for all times t> 0.Both authors partially supported by CAICYT, Project 2805-83. The second author also supported by USA-Spain Joint Research Grant CCB-8402023.     

Synthesis of chiral β-3-amino-2-hydroxyalkanoates and 3-alkyl-3-hydroxy-β-lactams by double asymmetric induction

Reactions of chiral (2S)-enolates of dioxolan-4-ones, derived from lactic, mandelic, and phenyllactic acids, with aliphatic (S_S)- and (S_R)-tert-butylsulfinyl aldimines afforded conformationally restrained C2-disubstituted N,O-orthogonally protected 3-amino-2-hydroxyalkanoates in the form of N-sulfinyl protected 1′-aminodioxolan-4-ones. The product distribution showed that there is significant kinetic selectivity, due to the presence of ‘matched’ and ‘mismatched’ components, between the (S)- or (R)-tert-butylsulfinyl aldimines and the (2S)-enolates of the 1,3-dioxolan-4-ones. Selective methoxide-induced removal of the acetal group of the N-sulfinyl-1′-aminodioxolanones yielded the corresponding N-sulfinyl protected methyl alkanoates. In addition, the selective acid-induced removal of the sulfinyl group of the N-sulfinyl-1′-aminodioxolanones provided the corresponding N-unprotected 1′-aminodioxolanones, whose base-induced cyclization afforded the corresponding β-lactams.     

Playing with the weakest supramolecular interactions in a 3D crystalline hexakis[60]fullerene induces control over hydrogenation selectivity

Weak forces can play an essential role in chemical reactions. Controlling such subtle forces in reorganization processes by applying thermal or chemical stimuli represents a novel synthetic strategy and one of the main targets in supramolecular chemistry. Actually, to separate the different supramolecular contributions to the stability of the 3D assemblies is still a major challenge. Therefore, a clear differentiation of these contributions would help in understanding the intrinsic nature as well as the chemical reactivity of supramolecular ensembles. In the present work, a controlled reorganization of an hexakis[60]fullerene-based molecular compound purely governed by the weakest van der Waals interactions known, i.e. the dihydrogen interaction – usually called sticky fingers – is illustrated. This pre-reorganization of the hexakis[60]fullerene under mild conditions allows a further selective hydrogenation of the crystalline material via hydrazine vapors exposure. This unique two-step transformation process is monitored by single-crystal to single-crystal diffraction (SCSC) which allows the direct observation of the molecular movements in the lattice and the subsequent solid–gas hydrogenation reaction.

Weak forces play an essential role in chemistry. Controlling these supramolecular interactions will contribute to the creation dynamic absorbent materials with a variety of technological applications as chemosensors and environmental remediation.     

A Practical Enantioselective Synthesis of (-)-NS-49 #

Asymmetric synthesis of (-)-NS-49 (1) was achieved by a convenient and short route.

     

On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.     

Synthesis and characterization of new tetraalkylaminophosphonium chlorides

New tetraalkylaminophosphonium chlorides were readily prepared by four-fold condensation of commercially available [P(CH₂OH)₄]Cl with a range of fifteen aryl based primary amines, in EtOH, at ambient temperature. All compounds have been characterized by FT–IR spectroscopy and elemental analysis. Solution ³¹P{¹H} NMR studies of these chloride salts reveal their instability with respect to various P^III/P^V species. The structures of three examples have been determined by single crystal X-ray diffraction and confirm the pseudotetrahedral arrangement around the P^V center. Extensive N–H···Cl hydrogen bonding is observed.