arXiv subjects
Publications and source records attributed to Sergei Kuksin.
In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.
We prove the persistence of analyticity for classical solution of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy-Kowalewski and Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of space-variables, then a classical solution also is analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space-analyticity (and the partial space-analyticity) of their classical solutions.
We consider the stochastic CGL equation $$ \dot u- νΔu+(i+a) |u|^2u =η(t,x),\;\;\; \text {dim} \,x=n, $$ where $ν>0$ and $a\ge 0$, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force $η$ is white in time, regular in $x$ and non-degenerate. We study this equation in the space of continuous complex functions $u(x)$, and prove that for any $n$ it defines there a unique mixing Markov process. So for a large class of functionals $f(u(\cdot))$ and for any solution $u(t,x)$, the averaged observable $\E f(u(t,\cdot))$ converges to a quantity, independent from the initial data $u(0,x)$, and equal to the integral of $f(u)$ against the unique stationary measure of the equation.
We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the $\ell^2$ space and derive for it lower and upper bounds in terms of some functions of the $\ell^2$-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum and its analysis using the conformal mapping theory.
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $κ$-smoothing map ($κ\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $κ$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $κ=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier--Stokes equations perturbed by various random forces of low dimension.