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Non Uniqueness of Power-Law Flows

2021; Springer Science+Business Media; Volume: 388; Issue: 1 Linguagem: Inglês

10.1007/s00220-021-04231-7

1432-0916

Jan Burczak, Stefano Modena, László Székelyhidi,

Stability and Controllability of Differential Equations

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension d≥3 . For the power index q below the compactness threshold, i.e. q∈(1,2dd+2) , we show ill-posedness of Leray-Hopf solutions. For a wider class of indices q∈(1,3d+2d+2) we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in L2 .