應(yīng)yl7703永利官網(wǎng)副院長馬閃教授邀請,俄羅斯科學(xué)院Keldysh應(yīng)用數(shù)學(xué)研究所Alexei Ilyin教授將于2024年9月1日至9月6日訪問我校并作系列學(xué)術(shù)報告。
報告題目一：Inequalities for orthonormal families and optimal bounds for the dimension of attractors of dissipative dynamical systems：a few classical and new results
時間：2024年9月2日（星期一）下午14: 30-15: 30
地點(diǎn)：大學(xué)生活動中心506報告廳
報告題目二：Inequalities for orthonormal families and optimal bounds for the dimension of attractors of dissipative dynamical systems：applications on weakly damped nonlinear hyperbolic system
時間：2024年9月3日（星期二）下午14: 30-15: 30
地點(diǎn)：大學(xué)生活動中心506報告廳
報告摘要：
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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報告人簡介
Alexei Ilyin教授，現(xiàn)任俄羅斯科學(xué)院Keldysh應(yīng)用數(shù)學(xué)研究所首席科學(xué)研究員，是國際無窮維動力系統(tǒng)領(lǐng)域的專家之一。1975至1980年在莫斯科國立大學(xué)攻讀學(xué)士學(xué)位；博士期間師從Andrey Tikhonov教授，并于1990年獲得博士學(xué)位；2006年在Steklov數(shù)學(xué)研究所獲得科學(xué)博士學(xué)位。Alexei Ilyin教授特別擅長偏微分方程中積分不等式以及譜不等式的最佳常數(shù)估計，在一些Sobolev不等式，吸引子分形維數(shù)上下界的估計等方面做出了突出的成果。對無窮維動力系統(tǒng)吸引子的存在性、正則性等相關(guān)問題也有深入的研究，尤其關(guān)于經(jīng)典Navier-Stokes方程以及Euler方程的吸引子問題上取得了一系列深刻結(jié)果。1994年主持國際科學(xué)基金項(xiàng)目（美國），先后主持多項(xiàng)俄羅斯基礎(chǔ)研究基金項(xiàng)目，并在Comm. Pure Appl. Math., J. London Math. Soc. Int. Math. Research Notices等國際重要學(xué)術(shù)期刊上發(fā)表論文六十余篇。
甘肅應(yīng)用數(shù)學(xué)中心
甘肅省高校應(yīng)用數(shù)學(xué)與復(fù)雜系統(tǒng)省級重點(diǎn)實(shí)驗(yàn)室
yl7703永利官網(wǎng)
萃英學(xué)院
2024年8月27日
