Perfect Incompressible FluidsAn accessible and self-contained introduction to recent advances in fluid dynamics, this book provides an authoritative account of the Euler equations for a perfect incompressible fluid. The book begins with a derivation of the Euler equations from a variational principle. It then recalls the relations on vorticity and pressure and proposes various weak formulations. The book develops the key tools for analysis: the Littlewood-Paley theory, action of Fourier multipliers on L spaces, and partial differential calculus. These techniques are used to prove various recent results concerning vortex patches or sheets; the main results include the persistence of the smoothness of the boundary of a vortex patch, even if that smoothness allows singular points, and the existence of weak solutions of the vorticity sheet type. The text also presents properties of microlocal (analytic or Gevrey) regularity of the solutions of Euler equations and links such properties to the smoothness in time of the flow of the solution vector field. |
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bilinear operator bounded functions Cauchy sequence chapter Chemin class C1+ closed set compactly supported completes the proof conic set constant C satisfying converges Corollary deduce definition denote diffeomorphism divergence-free vector field embedded ensures estimate Euler equations Euler system exists a constant flow following properties Fourier transform function f Furthermore gradient Hence Hölder spaces implies infer infinitely differentiable function initial data integer interval let us consider let us define Let us suppose Lipschitzian log h m=1 mfl norm open set operator paraproduct positive real number proof of Theorem Proposition regularity satisfying the following sequence singular smooth Sobolev spaces strictly larger strictly positive strictly positive real t₁ tempered distribution upper bound vanishes vector field belonging vn)nen vortex patches vorticity wave-front set Xx(x zero