One-Day Workshop on Elliptic and Parabolic PDEs III
时间：2020年12月26日
地点：北京师范大学后主楼 1124
邀请报告人：
敖微微 （武汉大学）
吕 勇 （南京大学）
王 伟 （浙江大学）
郑有泉 （天津大学）
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报告安排：
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敖微微
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9:00—10:00
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休 息
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30分钟
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吕 勇
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10:30—11:30
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午 餐
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王 伟
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14:00—15:00
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休 息
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30分钟
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郑有泉
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15:30—16:30
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组织者：熊金钢 Email: jx@bnu.edu.cn
报告题目和摘要
敖微微，武汉大学
Title: Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality
Abstract: In this talk, I will discuss about the following fractional version of the Caffarelli-Kohn-Nirenberg inequality
\[
\Lambda \left(\int_{\mathbb{R}^n}\frac{|u(x)|^{p}}{|x|^{{\beta} {p}}}\,dx\right)^{\frac{2}{p}}
\leq
\int_{\mathbb{R}^n }\int_{\mathbb{R}^n }\frac{(u(x)-u(y))^2}{|x-y|^{n+2\gamma}
|x|^{{\alpha}}|y|^{{\alpha}}}\,dy\,dx
\]
for $\gamma\in(0,1)$, $n>2\gamma$, and $\alpha,\beta\in \mathbb{R}$ satisfying
\[
\alpha\leq \beta\leq \alpha+\gamma, \ -2\gamma<\alpha<\frac{n-2\gamma}{2}
\]
and
\[
p=\frac{2n}{n-2\gamma+2(\beta-\alpha)}.
\]
We first study the existence and nonexistence of minimizers. Our next goal is to show some results for the symmetry and symmetry breaking region for the minimizers. In order to get these results we reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables and we provide a non-local ODE to find the radially symmetric extremals. We also get the non-degeneracy and uniqueness of minimizers in the radial symmetry class. This is joint work with Azahara DelaTorre and Maria del Mar Gonzalez.
吕勇，南京大学
Title: Laplace and Stokes equations in a domain with a shrinking hole
Abstract: We consider the Dirichlet problem of the Laplace and Stokes equations in a domain with a shrinking hole in $R^d$, $d\geq 2$. A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $p\neq 2$, the classical results indicate that the $W^{1,p}$ estimate of the solution may go to infinity as the size of the hole tends to zero. We give a complete description for the uniform $W^{1,p}$ estimate of the solution for all $1<p<\infty$. We show that the uniform $W^{1,p}$ estimate holds if and only if $d'<p<d$ ($p=2$ when $d=2$).
This work is motivated by the study of homogenization problems in fluid mechanics.
王伟，浙江大学
Title: Uniaxial limit from the Qian-Sheng theory to the inertial Ericksen-Leslie theory
Abstract: The Qian-Sheng inertial Q-tensor model and the full Ericksen-Leslie model for two important models for the liquid crystal flow. By using the Hilbert expansion method, we show that when the elastic coefficients tend to zero (also called the uniaxial limit), the solution to the Qian-Sheng inertial model will converge to the solution to the full inertial Ericksen-Leslie system. The main difficulty comes from the hyperbolic natures of the two systems, which cannot provide sufficient dissipative energy to control the singular terms.
郑有泉，天津大学
Title: Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions
Abstract: For a smooth bounded domain, considering the fast diffusion equation with critical Sobolev exponent under Dirichlet boundary condition, using the parabolic gluing method, we prove the existence of initial data such that the corresponding solution $w(x, t)$ has extinction rate
\[
\gamma_0 (T-t)^{\frac{n+2}{4}}|\ln (T-t)|^{\frac{n+2}{2(n-2)}}(1+o(1)) \quad \mbox{as }t\to T^-,
\]
which generalizes a result of Galaktionov and King for the radially symmetrical case. Here $T > 0$ is the finite extinction time of $w(x, t)$. This is a joint work with Professors Yannick Sire and Juncheng Wei.
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One-Day Workshop on Elliptic and Parabolic PDEs (II)
Date: June 17, 2019
Venue: Room 1220, New main building, BNU
Invited speakers：
Hongjie Dong (Brown)
Siyuan Lu (Rutgers)
Giuseppe Tinaglia (King’s College London)
Chao Wang (PKU)
Schedule:
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