\(L^2\)-critical nonuniqueness for the 2D Navier-Stokes equations. (English) Zbl 1531.35213
As it is well known [G. Furioli et al., Rev. Mat. Iberoam. 16, No. 3, 605–667 (2000; Zbl 0970.35101); J.-q. Mo, Appl. Math. Mech., Engl. Ed. 31, No. 12, 1577–1584 (2010; Zbl 1207.35090)], the existence and uniqueness for the Cauchy problem for the two-dimensional Navier-Stokes equations hold for \(L^2\) initial data, even in a bigger class than the Leray solutions. The authors discuss the case of \(L^p\), \(p<1\), data (on the two-dimensional torus \(\mathbb{T}^2\)) and shown nonuniqueness of solutions in general case. In particular, solutions with “almost” prescribed smooth energy function are shown to exist thanks to convex integration arguments.
Reviewer: Piotr Biler (Wrocław)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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