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.

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.

We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from L ∞ to C α , without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to L ∞ ∩ H 1 . Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in H 1 , obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in H 1 along solution trajectories, and use this property to bound the dimension of the global attractor.

We establish the local existence of pathwise solutions for the stochastic Euler equations in a threedimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Lastly, we show that, in the three dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. CONTENTS2010 Mathematics Subject Classification. 35Q35, 60H15, 76B03.

We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with L 2 initial data and divergence free drift velocity that lies in L ∞ t BM O −1 x . We apply this result to prove global regularity for a family of active scalar equations which includes the advection-diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core.2000 Mathematics Subject Classification. 76D03, 35Q35, 76W05.