应兰州大学数学与统计学院副院长马闪教授邀请,俄罗斯科学院Keldysh应用数学研究所Alexei Ilyin教授将于2024年9月1日至9月6日访问我校并作系列学术报告。

报告题目一：Inequalities for orthonormal families and optimal bounds for the dimension of attractors of dissipative dynamical systems：a few classical and new results

时间：2024年9月2日（星期一）下午14: 30-15: 30

地点：大学生活动中心506报告厅

报告题目二：Inequalities for orthonormal families and optimal bounds for the dimension of attractors of dissipative dynamical systems：applications on weakly damped nonlinear hyperbolic system

时间：2024年9月3日（星期二）下午14: 30-15: 30

地点：大学生活动中心506报告厅

报告摘要：

Estimates for the fractal dimension of the global attractors of dissipative evolution PDEs are traditionally related with the number of the degrees of freedom involved in the description of the long-time behaviour of the solutions. The dimension estimates, in turn, are based on the bounds for the N-traces of the linearized evolution overator. Therefore inequalities for systems that are orthonormal with respect to the underlying Hilbert phase space naturally come into play.

In the case of the 2D Navier-Stokes equations inequalities for the L^2-orthonormal systems of vector functions (the celebrated Lieb–Thirring inequalities) play the esential role in finding good or even optimal estimates for the dimension of the global attractors. We reviewa few classical and new results for certain models in mathematical fluid mechanics both in 2D and 3D.

Another popular example of an equation served by the attractor theory is a weakly damped nonlinear hyperbolic system. Here the key role is played by the inequalities for systems with orthonormal gradients. Based on them, we prove an explicit estimate for the fractal dimension of the attractor. Remarkably, the case of the spatial dimension d≥3 is simpler and in the case of a system with non-gradient perturbation the upper bound for the fractal dimension is supplemented with the lower bound of the same order in the limit of a small damping coefficient. The lower dimensional case is surprisingly more difficult, less complete, and requires a rather different technique.

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报告人简介

Alexei Ilyin教授，现任俄罗斯科学院Keldysh应用数学研究所首席科学研究员，是国际无穷维动力系统领域的专家之一。1975至1980年在莫斯科国立大学攻读学士学位；博士期间师从Andrey Tikhonov教授，并于1990年获得博士学位；2006年在Steklov数学研究所获得科学博士学位。Alexei Ilyin教授特别擅长偏微分方程中积分不等式以及谱不等式的最佳常数估计，在一些Sobolev不等式，吸引子分形维数上下界的估计等方面做出了突出的成果。对无穷维动力系统吸引子的存在性、正则性等相关问题也有深入的研究，尤其关于经典Navier-Stokes方程以及Euler方程的吸引子问题上取得了一系列深刻结果。1994年主持国际科学基金项目（美国），先后主持多项俄罗斯基础研究基金项目，并在Comm. Pure Appl. Math., J. London Math. Soc. Int. Math. Research Notices等国际重要学术期刊上发表论文六十余篇。

甘肃应用数学中心

甘肃省高校应用数学与复杂系统省级重点实验室

数学与统计学院

萃英学院

2024年8月27日