00780nas a2200109 4500008004100000245008400041210006900125520032000194100002500514700002300539856010800562 2013 eng d00aA note on non-homogeneous hyperbolic operators with low-regularity coefficients0 anote on nonhomogeneous hyperbolic operators with lowregularity c3 a

In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.

1 aColombini, Ferruccio1 aFanelli, Francesco uhttps://math.sissa.it/publication/note-non-homogeneous-hyperbolic-operators-low-regularity-coefficients01031nas a2200133 4500008004100000245009200041210006900133260002100202300001400223490000700237520058100244100002300825856004900848 2012 eng d00aConservation of Geometric Structures for Non-Homogeneous Inviscid Incompressible Fluids0 aConservation of Geometric Structures for NonHomogeneous Inviscid bTaylor & Francis a1553-15950 v373 a

In this article we get a result on propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension N ≥ 2. In particular, we investigate conservation of striated and conormal regularity, which generalize the 2-D structure of vortex patches. The results we get are only local in time, even for N = 2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N = 2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.

1 aFanelli, Francesco uhttps://doi.org/10.1080/03605302.2012.69834302515nas a2200205 4500008004100000022001400041245009600055210006900151300001400220490000700234520182300241653002202064653002402086653003502110653001302145653003502158100002202193700002302215856007102238 2011 eng d a0021-782400aThe well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces0 awellposedness issue for the densitydependent Euler equations in  a253 - 2780 v963 a

This work is the continuation of the recent paper (Danchin, 2010) [9] devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type B∞,rs embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces C1,α and of the endpoint Besov space B∞,11. For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda (1984) [2]. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. Résumé Ce travail complète lʼarticle récent (Danchin, 2010) [9] consacré au système dʼEuler incompressible à densité variable. Lorsque lʼétat initial ne comporte pas de vide, on montre ici que le système est bien posé dans tous les espaces de Besov B∞,rs inclus dans lʼensemble des fonctions lipschitziennes. Ce cadre fonctionnel contient en particulier les espaces de Hölder C1,α et lʼespace de Besov limite B∞,11. On établit également un critère de prolongement dans lʼesprit de celui de Beale, Kato et Majda (1984) [2] pour le cas homogène. Dans la dernière partie de lʼarticle, on donne des minorations pour le temps de vie des solutions du système. En dimension deux, on montre que ce temps de vie tend vers lʼinfini lorsque la densité tend à être homogène. À notre connaissance, il sʼagit du premier résultat de ce type pour le système dʼEuler incompressible à densité variable.

10aBlow-up criterion10aCritical regularity10aIncompressible Euler equations10aLifespan10aNonhomogeneous inviscid fluids1 aDanchin, Raphaël1 aFanelli, Francesco uhttp://www.sciencedirect.com/science/article/pii/S0021782411000511