@article {1305.1133, title = {A note on non-homogeneous hyperbolic operators with low-regularity coefficients}, year = {2013}, abstract = {

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 $t$.

}, author = {Ferruccio Colombini and Francesco Fanelli} } @article {doi:10.1080/03605302.2012.698343, title = {Conservation of Geometric Structures for Non-Homogeneous Inviscid Incompressible Fluids}, journal = {Communications in Partial Differential Equations}, volume = {37}, number = {9}, year = {2012}, pages = {1553-1595}, publisher = {Taylor \& Francis}, abstract = {

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 H{\"o}lder regularity in the interior of a bounded domain.

}, doi = {10.1080/03605302.2012.698343}, url = {https://doi.org/10.1080/03605302.2012.698343}, author = {Francesco Fanelli} } @article {DANCHIN2011253, title = {The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {96}, number = {3}, year = {2011}, pages = {253 - 278}, abstract = {

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$\infty$,rs embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of H{\"o}lder spaces C1,α and of the endpoint Besov space B$\infty$,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. R{\'e}sum{\'e} Ce travail compl{\`e}te l'article r{\'e}cent (Danchin, 2010) [9] consacr{\'e} au syst{\`e}me d'Euler incompressible {\`a} densit{\'e} variable. Lorsque l'{\'e}tat initial ne comporte pas de vide, on montre ici que le syst{\`e}me est bien pos{\'e} dans tous les espaces de Besov B$\infty$,rs inclus dans l'ensemble des fonctions lipschitziennes. Ce cadre fonctionnel contient en particulier les espaces de H{\"o}lder C1,α et l'espace de Besov limite B$\infty$,11. On {\'e}tablit {\'e}galement un crit{\`e}re de prolongement dans l'esprit de celui de Beale, Kato et Majda (1984) [2] pour le cas homog{\`e}ne. Dans la derni{\`e}re partie de l'article, on donne des minorations pour le temps de vie des solutions du syst{\`e}me. En dimension deux, on montre que ce temps de vie tend vers l'infini lorsque la densit{\'e} tend {\`a} {\^e}tre homog{\`e}ne. {\`A} notre connaissance, il s'agit du premier r{\'e}sultat de ce type pour le syst{\`e}me d'Euler incompressible {\`a} densit{\'e} variable.

}, keywords = {Blow-up criterion, Critical regularity, Incompressible Euler equations, Lifespan, Nonhomogeneous inviscid fluids}, issn = {0021-7824}, doi = {https://doi.org/10.1016/j.matpur.2011.04.005}, url = {http://www.sciencedirect.com/science/article/pii/S0021782411000511}, author = {Rapha{\"e}l Danchin and Francesco Fanelli} }