2020/04/14 by Stefano Bianchini, Stefano, Bianchini, Sara Daneri +1
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Trauma, Hemostasis, Coagulopathy, Resuscitation
paper · pdf · doi:10.48550/arxiv.2004.06557
openalex publication_date 2020/04/14 · openalex created_date 2023/03/18 · openalex updated_date 2026/07/28
In this paper we consider the multi-dimensional pressureless Euler system and\nwe tackle the problem of existence and uniqueness of sticky particle solutions\nfor general measure-type initial data. Although explicit counterexamples to\nboth existence and uniqueness are known since citeBressan-Nguyen, the\nproblem of whether one can still find sticky particle solutions for a large set\nof data and of how one can select them was up to our knowledge still completely\nopen.\n In this paper we prove that for a comeager set of initial data in the weak\ntopology the pressureless Euler system admits a unique sticky particle solution\ngiven by a free flow where trajectories are disjoint straight lines.\n Indeed, such an existence and uniqueness result holds for a broader class of\nsolutions decreasing their kinetic energy, which we call dissipative solutions,\nand which turns out to be the compact weak closure of the classical sticky\nparticle solutions. Therefore any scheme for which the energy is l.s.c. and is\ndissipated will converge, for a comeager set of data, to our solution, i.e. the\nfree flow.\n