## Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture SeriesThis text contains a series of self-contained reviews on the state of the art in different areas of partial differential equations, presented by French mathematicians. Topics include qualitative properties of reaction-diffusion equations, multiscale methods coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation laws. |

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### Contents

1 | |

Adaptive semiLagrangian schemes for Vlasov equations | 69 |

Coupling of a scalar conservation law with a parabolic problem | 115 |

Standing waves in nonlinear Schrödinger equations | 151 |

some examples of mathematical analysis | 193 |

Maximal regularity and applications to PDEs | 247 |

289 | |

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adaptive algorithms Anal analysis analytic semigroup assume assumption atomistic atomistic model atoms Banach space boundary conditions bounded Cauchy problem characteristic classical solution Comput conservation laws consider constant continuous continuum mechanics convergence convex coupled deﬁned deﬁnition deformation denote discontinuity curve domain energy entropy solution equation 4.2 estimate exists ﬁnd finite ﬁrst flux function fracture function f G H1(RN graph Hence Hilbert space Hopf equation hyperbolic inequality initial datum integral identity interpolation Kruzhkov Lemma linear Lipschitz Lipschitz continuous macroscopic Math mathematical maximal LP-regularity property Mech mesh multiscale nonlinear Schrödinger equations nonnegative numerical obtain operator partial differential equations particles Phys piecewise smooth Proposition prove quasilinear quasilinear equations Remark result Riemann problem satisﬁes Section semi-Lagrangian semigroup sequence smooth function smooth solution solution of 3.1 solution of equation stability standing waves theory tion unique vector W-tree wavelet