We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $lambda$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $lambdato
0$ and then $Lto infty$. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit $Ltoinfty$ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to
be $C_0^infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.
This paper is a synopsis of the recent book A. Boritchev, S. Kuksin, textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}, AMS Publications, 2021 (to appear). The book is dedicated to the stochastic Burgers equation as a model for 1
d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.
We consider a damped/driven nonlinear Schrodinger equation in an $n$-cube $K^{n}subsetmathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions [ u_t- uDelta u+i|u|^2u=sqrt{ u}eta(t,x),quad xin K^{n},quad u|_{partial K^{n}}=0, quad u>0, ]
where $eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ | u(t)|_m^2 le C u^{-m}, $ uniformly in $tge0$ and $ u>0$. In this work we prove that for small $ u>0$ and any initial data, with large probability the Sobolev norms $|u(t,cdot)|_m$ of the solutions with $m>2$ become large at least to the order of $ u^{-kappa_{n,m}}$ with $kappa_{n,m}>0$, on time intervals of order $mathcal{O}(frac{1}{ u})$.
We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a typical steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a v
icinity of the original steady one. More precisely, we establish that there exist stationary solutions $u_0$ of the Euler equation on $mathbb S^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose (non-steady) evolution by the Euler flow cannot converge in the $C^k$ Holder norm ($k>10$ non-integer) to any stationary state in a small (but fixed a priori) $C^k$-neighbourhood of $u_0$. The set of such initial conditions $v_0$ is open and dense in the vicinity of $u_0$. A similar (but weaker) statement also holds for the Euler flow on $mathbb T^3$. Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.
In this note we present the main results of the papers cite{DK, DK2}, dedicated to rigorous study of the limiting properties of the stochastic model for wave turbulence due to Zakharov-Lvov. Proofs of the assertions, stated below without reference, may be found in those works.
We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove
that when the amplitude goes to zero and the torus size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.
We continue the study of small amplitude solutions of the damped/driven cubic NLS equation, written as formal series in the amplitude, initiated in our previous work [Formal expansions in stochastic model for wave turbulence 1: kinetic limit, arXiv:1
907.04531]. We are interested in behaviour of the formal series under the wave turbulence limit the amplitude goes to zero, while the space-period goes to infinity
We present the modified approach to the classical Bogolyubov-Krylov averaging, developed recently for the purpose of PDEs. It allows to treat Lipschitz perturbations of linear systems with pure imaginary spectrum and may be generalized to treat PDEs with small nonlinearities.
We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that
the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.
