Roman Shvydkoy

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood?Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

CONVEX INTEGRATION FOR A CLASS OF ACTIVE SCALAR EQUATIONS

We show that a general class of active scalar equa- tions, including porous media and certain magnetostrophic tur- bulence models, admit non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in (2, 3).

Hyperbolicity of semigroups and Fourier multipliers

We present a characterization of hyperbolicity for strongly continuous semigroups on Banach spaces in terms of Fourier multiplier properties of the resolvent of the generator. Hyperbolicity with respect to classical solutions is also considered. Our approach unifies and simplifies the M. Kaashoek-S. Verduyn Lunel theory and multiplier-type results previously obtained by S. Clark, M. Hieber, S. Montgomery-Smith, F. Rabiger, T. Randolph, and L. Weis.

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the Lp-condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables

Nonlinear Instability for the Navier-Stokes Equations

{u \in L^p}

Euler Equations and Turbulence: Analytical Approach to Intermittency

and

2D Homogeneous Solutions to the Euler Equation

{u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}