This talk is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompress-
ible magneto-hydrodynamics near the transition point between the linearized stability and instability. In
order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [1]
have introduced an asymptotic quadratically nonlinear integro-di erential equation for the amplitude
of small perturbations of the planar discontinuity. We study such amplitude equation and prove its
nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the
fluid along the discontinuity. We first present the problem and discuss some known results about the stability
of current-vortex sheets; then we give some new results on the well-posedness of the Cauchy problem
associated to the amplitude equation.
This is a joint work with P. Secchi and P. Trebeschi.
[1]: Hunter, J. K. and Thoo, J. B., On the weakly nonlinear Kelvin-Helmholtz instability of tangential
discontinuities in MHD, J. Hyperbolic Di er. Equ., 8 (4), 2011, 691-726.
Andrea Parlangeli è fisico (PhD) e caporedattore del mensile Focus (Mondadori Scienza), dove lavora dal 2000. Ha un’ampia esperienza in campo giornalistico ed editoriale. È stato anche caporedattore centrale di Focus Storia, Focus Storia Collection, Focus Storia Biografie e Geo. Ha scritto e curato diversi libri, tra i quali “Benvenuti nell’Antropocene” (Mondadori) con il premio Nobel Paul Crutzen, “La nascita imperfetta delle cose” (Rizzoli) di Guido Tonelli, "La musica nascosta dell'universo" (Einaudi) di Adalberto Giazotto e "Uno spirito puro; Ennio De Giorgi, genio
della matematica" (Milella), recentemente pubblicato in inglese da Springer. Si è laureato nel 1995 alla Scuola Normale Superiore di Pisa, dove ha conosciuto Ennio De Giorgi.
ellittico-paraboliche, paraboliche sia in avanti che all'indietro.
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Abstract: Un'importante congettura afferma che il numero di modelli minimali di una varieta' complessa proiettiva sia finito modulo isomorfismi. Daro' una introduzione sulla congettura e sui risultati noti e parlero' di un work in progress sulla possibilita' di limitare il numero di modelli minimali di una threefold di tipo generale usando invarianti topologici.
" Lymph nodes are organs scattered throughout the lymphatic system which play a vital role in our immune response by breaking down bacteria, viruses, and waste; the interstitial fluid, called lymph once inside the lymphatic system, is of fundamental importance in this process as it transports these substances inside the lymph node. The main mechanical features of the lymph node include the presence of a porous bulk region (called lymphoid compartment), surrounded by a thin layer (called subcapsular sinus) where the fluid can flow freely. Lymph nodes are critical sites for the filtration and processing of lymph fluid, which contains a variety of immune cells, antigens, and other molecules. Understanding the fluid dynamics within lymph nodes is crucial for elucidating the mechanisms of immune response and for the development of therapies for lymphatic disorders.
During last ten years the author has been developing a discrete-velocity-type numerical method to solve model kinetic equation with the E.M. Shakhov collision integral for three-dimensional flows. The method combines a high-order TVD advection scheme for arbitrary spatial meshes, unstructured mesh in velocity space with adaptation for high-speed flows and one-step LU-SGS implicit time evolution method. These features allow applications to industrial-type problems with complex geometries. This presentation is a review of all recent developments of this numerical method for high-speed flows, which have been carried out by the author alone or with the collaborators from the Russian academy of Sciences. These developments primarily concern various studies of the stationary high-speed flows up to free-stream Mach number M=25 as well as comparison of kinetic and direct simulation Monte-Carlo (DSMC) solutions. Possible extension to diatomic gases will also be discussed.
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Sunto.
Nel seminario illustrerò come in tutti gli spazi metrici di misura sia possibile sviluppare un calcolo differenziale del prim’ordine.
Descriverò poi come nel spazi con curvatura di Ricci limitata dal basso sia presente anche una struttura differenziale del secondo ordine: in particolare oggetti come l’Hessiano di una funzione, derivata covariante e curvatura di Ricci sono tutti ben definiti.