We present a method of self-consistent a-priori bounds, which allows to study rigorously dynamics of dissipative PDEs. As an application present a computer assisted proof of an existence of a periodic orbit for the Kuramoto-Sivashinsky equation ut = (u )x− uxx− νuxxxx, u(t, x) = u(t, x + 2π), u(t, x) = −u(t,−x),

A closure model for turbulent flows is developed based on a dynamical system theory. An appropriately discretized formulation of the governing equations considered this process. The key ingredient an approximation system’s attractor, where all trajectories in phase space are confined. This approximate inertial manifold approach provides path to track lower-dimensional subspace. Unlike conventio...

We study the spatial power spectra of Nikolaevskii turbulence in one-dimensional space. First, we show that the energy distribution in wave-number space is extensive in nature. Then, we demonstrate that, when varying a particular parameter, the spectrum becomes qualitatively indistinguishable from that of Kuramoto-Sivashinsky turbulence. Next, we derive the critical exponents of turbulent fluct...

Initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves (that may serve as mathematical models wildfires) were considered. The existence and uniqueness of global strong solutions well their exponential decay have been established.