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We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\sfrac{1}{3})^-}$ regularity, for an arbitrary initial datum in $\dot H^1 (\T^3)$. This is the first rigorous derivation of zeroth law of scalar turbulence, where the scalar is advected by solution to an equation of hydrodynamics (unforced and deterministic). As a byproduct of our method, we provide a typicality statement for the drift, and recover certain desired properties of turbulence, including a lower bound on scalar variance commensurate with the Richardson pair dispersion hypothesis., Comment: Updated version 66 pages. The new version includes an extended introduction discussing implications for and the relationship to hydrodynamic turbulence, and a more general statement of the main result. Some minor typos were also corrected

An evidence of temporal dis-continuity of the solution in $F^s_{1, \infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed., Comment: 12 pages, 1 figure

Local stay of the solutions to the Euler equations for an ideal incompressible fluid in the end-point Triebel-Lizorkin spaces $F^s_{1, \infty}(\mathbb{R}^d)$ with $s \geq d + 1$ is clarified., Comment: 15 pages

We study the scenario of discretely self-similar blow-up for Navier-Stokes equations. We prove that at the possible blow-up time such solutions only one point singularity. In case of the scaling parameter $ \lambda $ near $ 1$ we remove the singularity., Comment: 14 pages

We prove the existence of a forward discretely self-similar solutions to the Navier-Stokes equations in $ \Bbb R^{3}\times (0,+\infty)$ for a discretely self-similar initial velocity belonging to $ L^2_{ loc}(\Bbb R^{3})$., Comment: 25 pages

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in $\Bbb R^3$. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the velocity has certain decay property at infinity, then the solution becomes trivial. This decay condition of of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formula for the tangential and the normal components of the weak solutions to the stationary Euler equations., Comment: 10 pages

In this paper we consider the locally backward self-similar solutions for the Euler system, and focus on the case that the possible nontrivial velocity profiles have non-decaying asymptotics. We derive the meaningful representation formula of the pressure profile in terms of velocity profiles in this case, and by using it and the local energy inequality of profiles, we prove some nonexistence results and show the energy behavior concerning the possible velocity profiles., Comment: 18 pages

We study scenarios of self-similar type blow-up for the incompressible Navier-Stokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to $C^1(\Bbb R ; L^3(\Bbb R^3)\cap C^2 (\Bbb R^3))$. For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions., Comment: 11 pages(to appear in J. Nonlinear Analysis)

This notes gives several criteria which exclude the existence of discretely self-similar solutions of the three dimensional incompressible Euler equations., Comment: 10 pages