Yakov Sinai

We study complex-valued blowups of solutions for several hydrodynamic models. For complex-valued initial conditions, smooth local solutions can have finite-time singularities since the energy inequality does not hold. By using some version of the renormalization group method, we derive the equations for corresponding fixed points and analyze the spectrum of the linearized operator. We describe the open set of initial conditions for which blowups at finite time can occur.

We study the behavior of a class of convolution-type nonlinear transformations. Under some smallness conditions we prove the existence of fixed points and analyze the spectrum of the associated linearized operator.

We consider the recurrent equation
$\Lambda_p = \frac{1}{p-1} \sum\limits_{p_1=1}^{p-1} f (\frac{p1}{p}) \Lambda_{p_1} · \Lambda_{p-p_1}$
which depends on the initial condition $\Lambda_1=x$. Under some conditions on $f$ we show that there exists the value of x for which $\Lambda_p$ tends to a constant as p tends to infinity.

The article presents a review of geometrical proof of the results earlier obtained by Foias, Temam with respect to Gevre class regularity for the solution of the Navier–Stokes equations. The articleis based on the lectures, delivered at the Moscow Independent University in june 1999.