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Nonuniqueness of weak solutions to the Navier-Stokes equation. (English) Zbl 1412.35215
In this very interesting paper are analyzed the weak solutions of the 3D Navier-Stokes equations. The main result is the non uniqueness of weak solutions with bounded kinetic energy. An important part of the introduction is concerning a brief history of the results obtained for the Cauchy-problem for Navier-Stokes equations with initial data of finite kinetic energy, containing also recent results of the authors. The new element of the paper is a convex integration scheme in Sobolev spaces, based on the convex integration framework in Hölder spaces for the Euler equations, introduced in [C. De Lellis and L. Székelyhidi jun., Invent. Math. 193, No. 2, 377–407 (2013; Zbl 1280.35103)]. The main idea is to solve by iterations the Navier-Stokes-Reynolds system where the Reynolds stress is a trace-free symmetric matrix; the velocity is considered as a sum of terms of particular Beltrami waves. A 3D Beltrami vector \(F\) is parallel to its own curl: \( F \times (\nabla \times F)=0\), therefore \(\exists \lambda \in R \) s.t. \( (\nabla \times F) = \lambda F\). If \( \operatorname{div} F=0\), then \( \nabla \times (\nabla \times F)= - \Delta F\). If in addition \(\lambda\) is constant we get \(- \Delta F = \lambda^2 F\). Beltrami flows are solutions of the steady Euler equations describing a 3D steady inviscid and incompressible flows. It is worth noting that Beltrami 3D fields are also linked to two dimensional steady Euler equations. In section 3 is described in detail the construction of a special class of Beltrami wave, called intermittent Beltrami waves. These waves form the building blocks used for the above mentioned convex integration scheme. The second important result of the paper is following: the Hölder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

MSC:
35Q30 Navier-Stokes equations
35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
76F02 Fundamentals of turbulence
35D30 Weak solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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