In the framework of spatially extended dynamical systems, we present three examples in which the presence of walls lead to dynamic behavior qualitatively different from the one obtained in an infinite domain or under periodic boundary conditions. For a nonlinear reaction-diffusion model we obtain boundary-induced spatially chaotic configurations. Nontrivial average patterns arising from boundar...

We introduce a method of estimating the space analyticity radius of solutions for the Navier Stokes and related equations in terms of L p and L norms of the initial data. The method enables us to express the space analyticity radius for 3D Navier Stokes equations in terms of the Reynolds number of the flow. Also, for the Kuramoto Sivashinsky equation, we give a partial answer to a conjecture th...

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes Hopf bifurcation. Furthermore, we obtain an essentially constructive family time-periodic near To this end, point rewrite time periodic problem for desingularized formulation. then apply parametrized Newton–Kantorovich approach...

We are interested in solution techniques for backward–in–time evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in the recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto–Sivashinsky equation in a 1D periodic domain. Such backward problems are ty...

We describe a wavelet-based approach to the investigation of spatiotemporally complex dynamics, and show through extensive numerical studies that the dynamics of the Kuramoto-Sivashinsky equation in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale (wave number). A projection onto a spline wavelet basis enables good separation of scales...

We consider the family of destabilized Kuramoto-Sivashinsky equations in one spatial dimension ut + νuxxxx + βuxx + γuux = αu for α,ν ≥ 0 and β ,γ ∈ R. For certain parameter values, shock-like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self-consistent bounds and validated numerics.