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Gradient flows with jumps associated with nonlinear Hamilton-Jacobi equations with jumps. (English) Zbl 1265.35048
For a given finite set of complete vector fields $$\{g_1,\dots,g_m\}\subseteq\mathcal C^\infty(\mathbb R^n;\mathbb R^n)$$, consider the corresponding local flows $\left\{G_1(t_1)[x],\dots ,G_m(t_m)[x] : | t_i|\leq a_i,\,\, x \in B(x^\ast, 3\gamma) \subseteq\mathbb R^n,\,\, 1\leq i\leq m\right\},$ generated by $$\{g_1, \dots , g_m\}$$ correspondingly and satisfying $\left| G_i(t_i)[x]-x\right|\leq\frac{\gamma}{2m},\quad x\in B(x^\ast, 3\gamma),\quad | t_i|\leq a_i,\quad 1\leq i\leq m,$ for some fixed constants $$a_i > 0$$ and $$\gamma> 0$$. Denote by $$\mathcal U_a$$ the set of admissible perturbations consisting of all piecewise right-continuous mappings (of $$t \geq 0$$) $$u(t, x) : [0,\infty) \times\mathbb R^n \to \bigsqcup =\prod_{i=1}^m[-a_i, a_i]$$ fulfilling $u(0,\lambda) = 0,\,\,u(t, \cdot) \in\mathcal C^1_b (\mathbb R^n;\mathbb R^n),\,\,\left| \partial_\lambda u_i(t,\lambda)\right| \leq K_1,\,\, t\geq 0,\,\,\lambda\in\mathbb R^n,\,\, 1\leq i\leq m,$ for some fixed constant $$K_1 > 0$$. For each admissible perturbation $$u\in\mathcal U_a$$, associate a piecewise right-continuous trajectory (for $$t\geq 0$$) $x^u(t,\lambda) = G(u(t, \lambda))[\lambda],\quad t\geq 0,\quad \lambda\in B(x^\ast, 2\gamma),$ where the smooth mapping $$G(p)[x] : \bigsqcup \times B(x^\ast, 2\gamma)\to B(x^\ast, 3\gamma)$$ is defined by $G(p)[x] = G_1(t_1) \circ \dots \circ G_m(t_m)[x],\quad p = (t_1, \dots , t_m)\in\bigsqcup, \quad x\in B(x^\ast, 2\gamma),$ verifying $$G(p)[x]\in B(x^\ast, 3\gamma)$$.
The authors introduce some nonlinear ODE with jumps fulfilled by the bounded flow $$\{x^u(t,\lambda) : t\in [0, T],\lambda \in B(x^\ast,2\gamma)\}$$, when $$u\in U_a$$ has a bounded variation property. In addition, the unique solution $\{\lambda =\psi (t, x)\in B(x^\ast, 2\gamma) : t\in [0, T], x\in B(x^\ast,\gamma)\}$ of the integral equation $$x^u(t,\lambda) = x\in B(x^\ast,\gamma)$$, $$t\in [0, T],$$ fulfills a quasilinear Hamilton-Jacobi equation on each continuity interval $$t\in [t_k, t_{k+1})\subseteq [0, T].$$ The vector fields $$\{g_1, \dots, g_m\}\in\mathcal C^1(\mathbb R^n;\mathbb R^n)$$ are supposed to be in involution over reals which lead to make use of algebraic representation for gradient systems in a finite-dimensional Lie algebra without involving a global nonsingularity or local times.
The analysis performed in this paper reveals the meaningful connection between dynamical systems and partial differential equations.
##### MSC:
 35F21 Hamilton-Jacobi equations 70H20 Hamilton-Jacobi equations in mechanics
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