最速下降曲线是一类退化对流-扩散方程的熵解
Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations
摘要：
考虑一类非线性退化对流扩散方程，证明了在$L^2$ -Wasserstein空间中，其Kružkov意义下的熵解是伴随梯度流的JKO变分逼近格式的后验唯一极限点.该方程缺乏必要的凸性质，这将允许推导出适定性的初值问题的抽象理论的度量梯度流。相反，我们直接用变分方法证明了熵不等式，并通过变量加倍得出唯一性结论。
英文摘要：
We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kružkov are obtained as the - a posteriori unique - limit points of the JKO variational approximation scheme for an associated gradient flow in the $L^2$-Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doubling of the variables.
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