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.

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 obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations.

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times t Re 1/3 , and in particular exhibits inviscid damping (e.g. the vorticity weakly approaches a shear flow). For times t Re 1/3 , which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterward, the remaining shear flow decays on very long time scales t Re back to the Couette flow. When properly defined, the dissipative length-scale in this setting is D ∼ Re −1/3 , larger than the scale D ∼ Re −1/2 predicted in classical Batchelor-Kraichnan 2D turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re −1 ) L 2 function.