Compactness methods for doubly nonlinear parabolic systems
Compactness methods for doubly nonlinear parabolic systems
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双非线性抛物线系统的紧致性方法
DOI：
10.1090/tran/6828
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Ryan Hynd
中科院分区：
文献类型：
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作者：
Ryan Hynd
研究了偏微分方程组D\psi（{\bfv}_t）=\text{div}DF（D{\bfv}）$的解，其中$\psi$和$F$是凸函数.这种类型的系统出现在相变的各种物理模型中。我们建立紧性的解决方案，使我们能够验证部分正则性时，$F$是二次和弱解的大时限的特点。特别考虑到系统是均匀的，它们与非线性特征值问题的连接。虽然这种系统的偏微分方程的弱解的唯一性仍然是一个悬而未决的问题，我们表明标量方程总是有一个首选的解决方案，也是唯一的粘性解决方案。
We study solutions of the system of PDE $D\psi({\bf v}_t)=\text{div}DF(D{\bf v})$, where $\psi$ and $F$ are convex functions. This type of system arises in various physical models for phase transitions. We establish compactness properties of solutions that allow us to verify partial regularity when $F$ is quadratic and characterize the large time limits of weak solutions. Special consideration is also given to systems that are homogeneous and their connections with nonlinear eigenvalue problems. While the uniqueness of weak solutions of such systems of PDE remains an open problem, we show scalar equations always have a preferred solution that is also unique as a viscosity solution.