Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

We consider concentrated vorticities for the Euler equation on a smooth domain $\mathrm{\Omega }?{\mathbf{R}}^2$ in the form of

$\omega =\sum _{j=1}^N{\omega }_j{\chi }_{{\mathrm{\Omega }}_j},|{\mathrm{\Omega }}_j|=\pi r_j^2,\underset{{\mathrm{\Omega }}_j}{\int }{\omega }_{j}d\mu ={\mu }_j\ne 0,$

supported on well-separated vortical domains ${\mathrm{\Omega }}_j$, $j=1,\dots ,N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with ${\mathrm{\Omega }}_j$ being part of unknowns. For any given vorticities ${\mu }_1,\dots ,{\mu }_N$ and small $r_1,\dots ,r_N\in {\mathbf{R}}^{+}$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff–Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $?{\mathrm{\Omega }}_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the 2N-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2N-codim directions corresponding to the vortical domain shapes.     

Triacetic acid lactone as a bioprivileged molecule in organic synthesis

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Stability of O3 + N2, O3 + O2 and O3 + CO2 double hydrates: DFT study

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Condensation of all-cis-tetraphenylcyclotetrasiloxanetetraol in ammonia: new method for preparation of ladder-like polyphenylsilsesquioxanes

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Synthesis and antimycobacterial activity of purine conjugates with (S)-lysine and (S)-ornithine

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Spectral theory approach for a class of radial indefinite variational problems

Considering the radial nonlinear Schrödinger equation

($P_r$)$?\mathrm{\Delta }u+V(x)u=g(x,u)\text{in}{\mathbb{R}}^N,N\ge 3$

we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem ($P_r$) is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem ($P_r$) to an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator $A:=\mathrm{\Delta }+V(x)$ restricted to $H_{rad}^1({\mathbb{R}}^N)$, the space of radially symmetric functions in $H^1({\mathbb{R}}^N)$, for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Identification and quantification of main flavonoids in the leaves of Bambusa multiplex cv. Fernleaf

Abstract

The chemical profile of Bambusa multiplex cv. Fernleaf (B. multiplex) leaves was analysed by UPLC-DAD-Q-TOF-MS. Twelve compounds were identified and C-glycosyl flavonoids, including vitexin, isovitexin, isoorientin and its derivatives, are the main constitutes of the plant. Besides, a HPLC method for isoorientin quantification was developed. The RSD of retention time and peak area were 0.05% and 2.04% for six times analysis of isoorientin with concentration of 20?μg/mL. The recovery of isoorientin in real sample was 99.2%. The general trend of isoorientin content in B. multiplex leaves was that it steady increased from Jan. to May, and then quickly decreased. The maximum was found on May with value of 4.7?mg/g. The lowest level of isoorientin was found during Aug. to Nov. with value of about 1.66?mg/g. In different seasons, isoorientin is always the most dominant flavonoid which was accounted for about 50% of total flavonoids in the sample.     

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi-Hamiltonian structures

For the orthosymplectic Lie superalgebra $\text{◂⋅▸}OSP(2,2)$, we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi-Hamiltonian structures.