We study solutions of the system of PDE $D\psi({\bf v}_t)=\text{div}DF(D{\bfv})$, where $\psi$ and $F$ are convex functions. This type of system arises invarious physical models for phase transitions. We establish compactnessproperties of solutions that allow us to verify partial regularity when $F$ isquadratic and characterize the large time limits of weak solutions. Specialconsideration is also given to systems that are homogeneous and theirconnections with nonlinear eigenvalue problems. While the uniqueness of weaksolutions of such systems of PDE remains an open problem, we show scalarequations always have a preferred solution that is also unique as a viscositysolution.
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机译：我们研究PDE $ D \ psi（{\ bf v} _t）= \ text {div} DF（D {\ bfv}）$的系统的解，其中$ \ psi $和$ F $是凸函数。这种类型的系统会产生用于相变的各种物理模型。我们建立了解决方案的紧凑性属性，这些属性使我们能够验证$ F $等价时的局部正则性，并刻画弱解决方案的较大时限。还特别考虑了齐次系统及其与非线性特征值问题的联系。虽然此类PDE系统的弱溶液的唯一性仍然是一个悬而未决的问题，但我们证明标量方程始终具有一种首选溶液，该溶液也具有独特的粘度溶液。
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