edoc: No conditions. Results ordered -Date Deposited. 2024-11-07T02:34:11ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2021-04-08T19:56:44Z2022-05-02T11:47:04Zhttps://edoc.unibas.ch/id/eprint/82583This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/825832021-04-08T19:56:44ZStrong convergence of the vorticity for the 2D Euler Equations in the inviscid limitIn this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $\omega^\nu$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $\omega$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $\omega^\nu$ to $\omega$ in $L^p$. Finally, we show that solutions of the Euler equations with $L^p$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach. Gennaro CiampaGianluca CrippaStefano Spirito2021-03-24T07:12:49Z2021-06-10T01:30:04Zhttps://edoc.unibas.ch/id/eprint/81399This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/813992021-03-24T07:12:49ZWeak Solutions Obtained by the Vortex Method for the 2D Euler Equations are Lagrangian and Conserve the EnergyGennaro CiampaGianluca CrippaStefano Spirito2021-02-24T15:11:06Z2021-02-24T15:11:06Zhttps://edoc.unibas.ch/id/eprint/81397This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/813972021-02-24T15:11:06ZSmooth approximation is not a selection principle for the transport equations with rough vector fieldIn this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation. Gennaro CiampaGianluca CrippaStefano Spirito