Mathematical analysis of models of non-homogeneous fluids and of hyperbolic equations with low regularity coefficients
type: Dataset
description: The present thesis is devoted both to the study of strictly hyperbolic operators with low regularity coefficients and of the density-dependent incompressible Euler system. On the one hand, we show a priori estimates for a second order strictly hyperbolic operator whose highest order coefficients satisfy a log-Zygmund continuity condition in time and a log-Lipschitz continuity condition with respect to space. Such an estimate involves a time increasing loss of derivatives. Nevertheless, this is enough to recover well-posedness for the associated Cauchy problem in the space H^∞ (for suitably smooth second order coefficients).In a first time, we consider acomplete operator in space dimension 1, whose first order coefficients were assumed Hölder continuous and that of order 0 only bounded. Then, we deal with the general case of any space dimension, focusing on a homogeneous second order operator: the step to higher dimension requires a really different approach. On the other hand, we consider the density-dependent incompressible Euler system. We show its well-posedness in endpoint Besov spaces embedded in the class of globally Lipschitz functions, producing also lower bounds for the lifespan of the solution in terms of initial data only. This having been done, we prove persistence of geometric structures, such as striated and conormal regularity, for solutions to this system. In contrast with the classical case of constant density, even in dimension 2 the vorticity is not transported by the velocity field. Hence, a priori one can expect to get only local in time results. For the same reason, we also have to dismiss the vortex patch structure. Littlewood-Paley theory and paradifferential calculus allow us to handle these two different problems. A new version of paradifferential calculus, depending on a parameter g≥1, is also needed in dealing with hyperbolic operators with nonregular coefficients. The general framework is that of Besov spaces, which includes in particula…
landing page: https://theses.hal.science/tel-00794508
identifier: NNT: 2012PEST1077
keyword: besov-spaces
keyword: calcul-paradifferentiel
keyword: decomposition-de-littlewood-paley
keyword: densite-variable
keyword: equations-deuler
keyword: espaces-de-besov
keyword: euler-equation
keyword: fluide-incompressible
keyword: incompressible-fluid
keyword: infoeu-reposemanticsdoctoralthesis
keyword: littlewood-paley-decomposition
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: paradifferential-calculus
keyword: theses
keyword: variable-density
issued: 2026-05-14T03:57:43.478197
modified: 2026-05-14T03:57:43.478202
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
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Mathematical analysis of models of non-homogeneous fluids and of hyperbolic equations with low regularity coefficients
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format: HTML
access URL: https://theses.hal.science/tel-00794508
issued: 2026-05-14T03:57:43.555591
modified: 2026-05-14T03:57:43.427459