Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page

L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method

We consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L^2 perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L^2 norm of a perturbed solution relative to the shock wave is bounded above by the L^2 norm of the initial perturbation.Comment: 17 page

Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit

Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary assumption removed, plus some minor change

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Observation of a New Charmed Strange Meson

Using the CLEO-II detector, we have obtained evidence for a new meson decaying to $D^0 K^+$. Its mass is $2573.2^{+1.7}_{-1.6}\pm 0.8\pm 0.5$ {}~MeV/$c^2$ and its width is $16^{+5}_{-4}\pm 3$~MeV/$c^2$. Although we do not establish its spin and parity, the new meson is consistent with predictions for an $L=1$, $S=1$, $J_P=2^+$ charmed strange state.Comment: 9 pages uuencoded compressed postscript (process with uudecode then gunzip). hardcopies with figures can be obtained by sending mail to: [email protected]

Precision Measurement of the Ds∗+−Ds+D_s^{*+}- D_s^+ Mass Difference

We have measured the vector-pseudoscalar mass splitting $M(D_s^{*+})-M(D_s^+) = 144.22\pm 0.47\pm 0.37 MeV$, significantly more precise than the previous world average. We minimize the systematic errors by also measuring the vector-pseudoscalar mass difference $M(D^{*0})-M(D^0)$ using the radiative decay $D^{*0}\rightarrow D^0\gamma$, obtaining $[M(D_s^{*+})-M(D_s^+)]-[M(D^{*0})-M(D^0)] = 2.09\pm 0.47\pm 0.37 MeV$. This is then combined with our previous high-precision measurement of $M(D^{*0})-M(D^0)$, which used the decay $D^{*0}\rightarrow D^0\pi^0$. We also measure the mass difference $M(D_s^+)-M(D^+)=99.5\pm 0.6\pm 0.3$ MeV, using the $\phi\pi^+$ decay modes of the $D_s^+$ and $D^+$ mesons.Comment: 18 pages uuencoded compressed postscript (process with uudecode then gunzip). hardcopies with figures can be obtained by sending mail to: [email protected]