Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userKolmogorovs theory of turbulence and its rigorous 1d model
167
0
0.0
(
0
)
Authors
Sergei Kuksin
Ask ChatGPT about the research
No Arabic abstract
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 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.
rate research
Read More
In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlovi{c} for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE.
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:1907.04531]. 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 prove rigorously that the exact N-electron Hohenberg-Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained-search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N=2 [CFK11]. The correct limit problem has been derived in the physics literature by Seidl [Se99] and Seidl, Gori-Giorgi and Savin [SGS07]; in these papers the lack of a rigorous proof was pointed out. We also give a mathematical counterexample to this type of result, by replacing the constraint of given one-body density -- an infinite-dimensional quadratic expression in the wavefunction -- by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated.
We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the range of validity of the Brunet--Derrida formula for a general class of reaction terms.
We consider quantum walks with position dependent coin on 1D lattice $mathbb{Z}$. The dispersive estimate $|U^tP_c u_0|_{l^infty}lesssim (1+|t|)^{-1/3} |u_0|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ is the projection to the continuous spectrum. This is an analogous result for Schrodinger operators and discrete Schrodinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
