论文标题
汉密尔顿 - 雅各比方程的度量熵，均匀方向凸汉密尔顿
Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian
论文作者
论文摘要
本文首先旨在研究Hamilton-Jacobi方程的粘度解决方案的BV型规律性\ [ u_t（t，x）+h \ big（d_ {x} u（t，x）\ big） $ h \ in \ mathcal {c}^{1}（\ mathbb {r}^d）$。更确切地说，我们在向后特性的斜率上建立了BV限制$ dh（u（t，\ cdot））$，从正时$ t> 0 $开始。依靠BV绑定，我们在$ {\ bf w}^{1,1} _ {\ Mathrm {loc}}}（\ Mathbb {r}^d）$中量化了公制熵lip} \ big（\ mathbb {r}^d \ big）$，相应的解决方案$ s_tu_0 $。最后，构建了一个反示例，以表明$ d_xu（t，\ cdot）$和$ dh（d_xu（t，\ cdot））$都无法在$ bv _ {\ mathrm {loc}} $中，对于一般的convex和cocive $ h \ in \ in \ mathcal $ n \ mathcal {c} c}^2（c}^2（c）
The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$ starting at a positive time $t>0$. Relying on the BV bound, we quantify the metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a counter example is constructed to show that both $D_xu(t,\cdot)$ and $DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.
