论文标题
非线性开放映射原则，并应用于雅各布方程和其他规模不变的PDE
Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs
论文作者
论文摘要
对于满足某些结构假设的非线性操作员$ t $，我们的主要定理指出，以下索赔是等效的：i）$ t $是围绕的；该定理适用于规则性制度中的众多规模不变PDE，在这些方程式下，在弱$^*$融合下方程式稳定。我们探索的两个特定示例是雅各布方程和不可压缩流体流的方程。 对于雅各布式而言，决定它是否进入关键的索博尔夫空间和耐寒空间之间是一个长期的开放问题。对于一个负面的答案，我们表明，如果雅各比亚人进入了，那么就足以排除出令人惊讶的表现良好的解决方案。 对于不可压缩的Euler方程，我们表明，对于任何$ p <\ infty $，在$ l^p_t l^2_x $中都有耗散弱解决方案的一组初始数据在螺线管$ l^2 $字段的空间中都是微薄的。其他不可压缩的流体动力学方程式的相似结果。
For a nonlinear operator $T$ satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) $T$ is surjective, ii) $T$ is open at zero, and iii) $T$ has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak$^*$ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow. For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions. For the incompressible Euler equations, we show that, for any $p<\infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal $L^2$ fields. Similar results hold for other equations of incompressible fluid dynamics.
