For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that 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. IntroductionIn this paper we consider the 3D incompressible Navier-Stokes equationposed on T 3 × R, with periodic boundary conditions in x ∈ T 3 = R 3 /2πZ 3 . We consider solutions normalized to have zero spatial mean, i.e.,´T 3 v(x, t)dx = 0. The constant ν ∈ (0, 1] is the kinematic viscosity. We define weak solutions to the Navier-Stokes equations [49, Definition 1], [19, pp. 226]:Definition 1.1. We say v ∈ C 0 (R; L 2 (T 3 )) is a weak solution of (1.1) if for any t ∈ R the vector field v(·, t) is weakly divergence free, has zero mean, and (1.1a) is satisfied in D ′ (T 3 × R), i.e.,holds for any test function ϕ ∈ C ∞ 0 (T 3 × R) such that ϕ(·, t) is divergence-free for all t.As a direct result of the work of Fabes-Jones-Riviere [19], since the weak solutions defined above lie in C 0 (R; L 2 (T 3 )), they are in fact solutions of the integral form of the Navier-Stokes equations v(·, t) = e νt∆ v(·, 0) +ˆt 0 e ν(t−s)∆ Pdiv (v(·, s) ⊗ v(·, s))ds ,(1.2) and are sometimes called mild or Oseen solutions (cf.[19] and [39, Definition 6.5]). Here P is the Leray projector and e t∆ denotes convolution with the heat kernel.

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We prove that given any β < 1/3, a time interval [0, T ], and given any smooth energy profile e : [0, T ] → (0, ∞), there exists a weak solution v of the three-dimensional Euler equations such that v ∈ C β ([0, T ] × T 3 ), with e(t) = ´T3 |v(x, t)| 2 dx for all t ∈ [0, T ]. Moreover, we show that a suitable h-principle holds in the regularity class C β t,x , for any β < 1/3. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.Date: January 31, 2017. 1 The smallest constant C satisfying (1.2) will be denoted by [v] β , cf. Appendix A. We will write v ∈ C β (T 3 ×[0, T ]) when v is Hölder continuous in the whole space-time.

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. In view of the results of Marchand in 2008, we establish that for the dissipative SQG equation, weak solutions may be constructed in the same function space both via classical weak compactness arguments and via convex integration. © 2019 Wiley Periodicals, Inc.

In recent works by Isett [16], and later by Buckmaster, De Lellis, Isett and Székelyhidi Jr. [2], iterative schemes were presented for constructing solutions belonging to the Hölder class C 1/5−ε of the 3D incompressible Euler equations which do not conserve the total kinetic energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of C 1/3−ε solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solutions which for almost every time belong to the critical Onsager Hölder regularity C 1/3−ε and have compact temporal support.