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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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