外双调和热流弱解非唯一性及部分正则性

In the past few years, the biharmonic maps (intrinsic biharmonic maps and extrinsic biharmonic maps) and related issues have become a hot issue in mathematics.This project will focus on the extrinsic biharmonic heat flow. We study the nonuniqueness and the partial regularity of the weak solutions and the blow-up of the classical solutions to the extrinsic biharmonic heat flow. Firstly using time difference method constructed out of more than 5 d and 5 d extrinsic biharmonic sequence of weak solutions to the heat flow, exist initial value of the weak solution sequence converges to a solution of time dependent, and for any extrinsic biharmonic maps independent, is the weak solution of extrinsic biharmonic heat flow, so that we get the uniqueness of weak solution of extrinsic biharmonic heat flow. Secondly rewrite the right hand side of the extrinsic biharmonic heat flow into divergence form and anti-symmetric form. We get the C.B.Morrey decay estimate by these two kind of special forms, Littlewood-Paley decomposition (possible), energy estimate and the properties of high order equations. We get the regularity immediately after C.B.Morrey decay estimate. Finally prove that the classical solutions of 4 d extrinsic biharmonic heat flow are blow-up and study the order of the blow-up. The project investigation will inject new life into researches of 4-order parabolic equations and help us to understand the geometry and the topology of many model spaces.

双调和映射(分为内双调和映射和外双调和映射)及其相关问题，已经成为数学的一个研究热点。本项目主要研究外双调和热流。具体来说，研究5维及5维以上外双调和映射热流弱解的非唯一性、部分正则性及4维古典解爆破问题。首先，用时间差分法构造出5维及5维以上外双调和热流弱解序列，证明存在初值使得此弱解序列收敛到一个时间依赖的解，而对于任意非时间依赖的外双调和映射，也是外双调和热流弱解，这样我们就得到了弱解的非唯一性。其次，通过把外双调和热流方程右端项改写成散度形式与反对称形式之和，通过这两类特殊的结构，借助Littlewood-Paley分解(可能)，运用能量估计及高阶方程性质，得到 C.B.Morrey 衰减性估计，从而得到正则性。最后，证明 4 维外双调和热流经典解的爆破性，并估计解的爆破阶数。本项目研究将为4阶抛物方程注入新的活力，也有助于我们了解许多模空间的几何与拓扑结构。

双调和映射及其相关问题已经成为数学的一个研究热点。本项目研究外双调和热流的非唯一性和部分正则性及双调和热流在液晶Ericksen-Leslie模型中的应用（该问题在项目申请书摘要未写，在申请书内容中有详细写）。就外双调和映射热流的非唯一性及部分正则性，仍在寻找有效替代比较原理或能量单调的方法。就双调和热流在液晶中的应用，我们借用双调和热流，证明了带有高阶项及带有Leslie张量的液晶Ericksen-Leslie模型，初始密度含有真空的柯西问题的局部强解的存在唯一性，借助特殊的双调和热流—调和热流，我们证明了简化液晶Ericksen-Leslie模型的弱解的非唯一性，亦得到了液晶Q-张量模型的刘维尔定理及NSPNP模型的部分正则性。