论文标题
关于多凸功能的梯度流的非唯一性和不规则性
On nonuniqueness and nonregularity for gradient flows of polyconvex functionals
论文作者
论文摘要
我们提供了一些反例，以表明某些问题可以具有微不足道的经典溶液以及无限的许多弱解决方案，这些弱解决方案的弱解决方案对初始有限值问题的唯一性和规律性。此类多型函数函数已经在先前的工作中构建了，并且通过将梯度流作为一种时空的部分差分关系重新升级，然后使用凸集成方法来构建某些强烈收敛的亚物质序列，从而实现梯度流，从而实现了均匀收敛的序列，这些序列对其空间梯度的局部基本振荡具有统一的控制。
We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a trivial classical solution as well as infinitely many weak solutions that are nowhere smooth. Such polyconvex functions have been constructed in the previous work, and the nonuniqueness and nonregularity will be achieved by reformulating the gradient flow as a space-time partial differential relation and then using the convex integration method to construct certain strongly convergent sequences of subsolutions that have a uniform control on the local essential oscillations of their spatial gradients.
