Куксин Сергей Борисович
Факультет математики
Должности
- Ведущий научный сотрудник — Факультет математики
Био
- · Начал работать в НИУ ВШЭ в 2025 году.
- · Научно-педагогический стаж: 5 лет.
Образование
- 1991 · Доктор физико-математических наук: Высшая аттестационная комиссия при Совете Министров СССР
- 1981 · Кандидат физико-математических наук: Московский государственный университет им. М.В. Ломоносова
- 1977 · Специалитет: Московский государственный университет им. М.В. Ломоносова, специальность «Математика», квалификация «Математик»
Опыт работы
- · 2016: Профессор Сергей Борисович Куксин, д.ф.-м.н. (институт Математики Жюссье, Париж, Франция) — один из крупнейших математиков в области уравнений в частных производных и их приложений к математической физике, пленарный докладчик Первого Европейского математического конгресса (1992), приглашенный докладчик Международного конгресса математиков (1998), XI (1994) и XVI (2009) Международных конгрессов по математической физике. В году профессор С. Б. Куксин получил премию Ляпунова Российской академии наук за создание и развитие теории Колмогорова—Арнольда—Мозера для уравнений в частных производных. В
- · 2016: году избран Почетным профессором Эдинбургского университета, в
- · 2017: году — Национальным профессором КНР
Идентификаторы исследователя
- ORCID:
0000-0003-2322-2821 - ResearcherID:
AAC-3293-2022 - Google Scholar: https://scholar.google.com/citations?user=B3C0nl8AAAAJ&hl=ru&oi=ao
- Scopus AuthorID:
7003775978
Публикации (8)
Mixing for dynamical systems driven by stationary noise
2025 · ARTICLE · en
The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be δ-correlated in time, therefore the evolu- tion in question is not necessarily Markovian. We first prove an abstract result which implies the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate this on the ex- amples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier–Stokes system, and a complex Ginzburg–Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin’s theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with Newton’s quadratic approximation. It uses the method of Kantorovich functional, introduced earlier by the authors in the context of randomly forced PDEs, and some ideas used by them in the Markovian case to prove mixing with the help of controllability properties of an associated system.
Averaging for stochastic perturbations of integrable systems
2024 · ARTICLE · en
We are concerned with averaging theorems for ε-small stochastic perturbations of integrable equations in Rd×Tn={(I,φ)} (Formula presented.) and in R2n={v=(v1,⋯,vn),vj∈R2}, (Formula presented.) where I=(I1,⋯,In) is the vector of actions, Ij=12‖vj‖2. The vector-functions θ and W are locally Lipschitz and non-degenerate. Perturbations of these equations are assumed to be locally Lipschitz and such that some few first moments of the norms of their solutions are bounded uniformly in ε, for 0≤t≤ε-1T. For I-components of solutions for perturbations of (1) we establish their convergence in law to solutions of the corresponding averaged I-equations, when 0≤τ:=εt≤T and ε→0. Then we show that if the system of averaged I-equations is mixing, then the convergence is uniform in the slow time τ=εt≥0. Next using these results, for ε-perturbed equations (2) we construct well posed effective stochastic equations for v(τ)∈R2n (independent of ε) such that when ε→0, the actions of solutions for the perturbed equations with t:=τ/ε converge in distribution to actions of solutions for the effective equations. Again, if the effective system is mixing, this convergence is uniform in the slow time τ≥0. We provide easy sufficient conditions on the perturbed equations which ensure that our results apply to their solutions. ©
О схеме Ньютона–Канторовича для эволюционных уравнений
2024 · ARTICLE · ru
Предложена конструктивная форма метода Ньютона–Канторовича для построения решений эволюционных уравнений с малыми нелинейностями, применимая к уравнениям в линейных пространствах, не являющихся банаховыми. Описано лишь основное содержание метода без конкретизации используемых норм и необходимых ε–δ-деталей.
On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation
2021 · ARTICLE · en
We consider a damped/driven nonlinear Schrödinger equation in an n-cube Kn⊂Rn, n is arbitrary, under Dirichlet boundary conditions ut−νΔu+i|u|2u=ν−−√η(t,x),x∈Kn,u|∂Kn=0,ν>0, where η(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)∥2m≤Cν−m, uniformly in t≥0 and ν>0. In this work we prove that for small ν>0 and any initial data, with large probability the Sobolev norms ∥u(t,⋅)∥m of the solutions with m>2 become large at least to the order of ν−κn,m with κn,m>0, on time intervals of order O(1ν).
Exponential mixing for dissipative PDEs with bounded non-degenerate noise
2020 · ARTICLE · en
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
Mixing via controllability for randomly forced nonlinear dissipative PDEs
2020 · ARTICLE · en
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.
Global, local and dense non-mixing of the 3D Euler equation
2020 · ARTICLE · en
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 vicinity of the original steady one. More precisely, we establish that there exist stationary solutions u0 of the Euler equation on S3 and divergence-free vector fields v0 arbitrarily close to u0, whose (non-steady) evolution by the Euler flow cannot converge in the Ck Hölder norm (k>10 non-integer) to any stationary state in a small (but fixed a priori) Ck-neighbourhood of u0. The set of such initial conditions v0 is open and dense in the vicinity of u0. A similar (but weaker) statement also holds for the Euler flow on T3. 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.
On the Zakharov-L’vov stochastic model for wave turbulence
2020 · ARTICLE · en
In this paper we discuss a number of rigorous results in the stochastic model for wave turbulence due to Zakharov–L'vov. Namely, we consider the damped/driven (modified) cubic nonlinear Schrödinger equation on a large torus and decompose its solutions to formal series in the amplitude. We show that when the amplitude goes to zero and the torus’ size goes to infinity the energy spectrum of the quadratic truncation of this series converges to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.
Курсы (1)
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Ergodicity and Mixing for Markov Processes
2024/2025 · Дисциплина общефакультетского пула · Анг