Analyticité locale pour les solutions de l'équation d'Euler. (Local analyticity for the solutions of Euler's equation)
The author considers the Euler equations for non viscous incompressible fluids with non constant density. The flow is contained in an open bounded subset \(\Omega\) of \({\mathbb{R}}^ n\) with analytical boundary. The local analyticity of the solution up to the boundary is studied. More precisely, given a point \(x_ 0\in \partial \Omega\), set \(\gamma_ T(x_ 0)=\{(t,\phi_{t,0}(x_ 0))\), \(t\in [0,T]\}\), where \(\phi_{t,0}(x_ 0)\) is the position at time t of the particle which is in \(x_ 0\) at time \(t=0\) (the solution is known to exist up to time T). Since the velocity field is tangential on the boundary, \(\phi_{t,0}(x_ 0)\in \partial \Omega\). If the solution at time \(t=0\) is analytical up to the boundary near \(x_ 0\), and the external force field is analytical up to the boundary near \(\gamma_ T(x_ 0)\), then the solution is analytical up to the boundary near \(\gamma_ T(x_ 0)\).
scientific article; zbMATH DE number 4034195