中山大學(xué)羅果博士作了題為三維軸對(duì)稱歐拉方程的潛在奇異解的講座，中山大學(xué)是直屬的綜合性重點(diǎn)大學(xué)，國(guó)家“985工程”、“211工程”建設(shè)高校，同時(shí)是“珠峰計(jì)劃”、“111計(jì)劃”、“卓越法律人才教育培養(yǎng)計(jì)劃”、“卓越醫(yī)生教育培養(yǎng)計(jì)劃”實(shí)施高校。在職研究生講座的主要內(nèi)容是：

　　無(wú)論是3D不可壓歐拉方程可以開(kāi)發(fā)從光滑的初始數(shù)據(jù)在有限時(shí)間奇點(diǎn)是在數(shù)學(xué)流體力學(xué)中最具挑戰(zhàn)性的問(wèn)題之一。這項(xiàng)工作將嘗試從一個(gè)數(shù)值來(lái)看，通過(guò)提供一類潛在的奇異解歐拉方程計(jì)算的軸對(duì)稱的幾何形狀提供一個(gè)肯定的答案，這一長(zhǎng)期懸而未決的問(wèn)題。該解決方案滿足在固體壁沿軸向的周期性邊界條件和不流動(dòng)邊界條件。該公式使用混合6階伽和6階有限差分法，在專門設(shè)計(jì)的自適應(yīng)(移動(dòng))網(wǎng)格的動(dòng)態(tài)調(diào)整，以適應(yīng)不斷變化的解決方案離散空間。憑借超過(guò)(3 * 10 ^ 12)^ 2附近的奇異點(diǎn)的最大有效分辨率，我們能夠提前解決高達(dá)tau_2 = 0.003505，預(yù)測(cè)t滑的奇點(diǎn)時(shí)～0.0035056，同時(shí)實(shí)現(xiàn)逐點(diǎn)相對(duì)于在渦度矢量和觀測(cè)的最大渦一(3 * 10 ^ 8)倍增長(zhǎng) - 為O(4)10 ^()錯(cuò)誤。數(shù)值數(shù)據(jù)進(jìn)行比對(duì)所有主要的爆破(不爆破)的標(biāo)準(zhǔn)，其中包括比爾 - 加藤M(fèi)ajda，康斯坦丁 - Fefferman-Majda，鄧侯宇，確認(rèn)奇異的有效性。附近的奇異點(diǎn)當(dāng)?shù)氐姆治鲆脖砻髯韵嗨票�破的存在。我們還討論一個(gè)1D模型可以被看作是一個(gè)本地近似圓柱體的固體邊界附近的歐拉方程。這種一維模型的有限時(shí)間爆破證明一類的光滑初始數(shù)據(jù)，這基本上是在全3D爆破計(jì)算中使用的初始數(shù)據(jù)的限制。

　　羅果博士為中山大學(xué)學(xué)士，香港中文大學(xué)碩士，美國(guó)俄亥俄州立大學(xué)(Ohio State University)博士。畢業(yè)后在加州理工大學(xué)從事博士后研究，現(xiàn)在為香港城市大學(xué)數(shù)學(xué)系A(chǔ)ssistant Professor。他的主要研究方向?yàn)槠�微分方程的�?jì)算及理論。

　　原文：Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3*10^12)^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t_s ~ 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector and observing a (3*10^8)-fold increase in the maximum vorticity. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup. We also discuss a 1D model which can be viewed as a local approximation to the Euler equations near the solid boundary of the cylinder. The finite-time blowup of this 1D model is proved for a class of smooth initial data, which are essentially restrictions of the initial data used in the full 3D blowup calculations.