On two-dimensional Hamiltonian transport equations with continuous coefficients. (English) Zbl 1028.35042

According to the theory of characteristics, the linear transport equation \[ \partial_t u(t, x)+ a(t, x)\cdot\nabla_x u(t, x)= 0,\quad u(0,x)= u^-(x),\tag{1} \] where \(t\in\mathbb{R}\), \(x\in\mathbb{R}^N\), \(u^0: \mathbb{R}^N\to \mathbb{R}\), \(a: \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N\) and \(u: \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}\), is related to the system of ordinary differential equations \[ {dX\over ds}(s)= a(s, X(s)),\quad X(t)= x, \] via the relation \(u(t,x)= u^0(X(t, x))\). This system admits a unique (local) solution of class \(C^1\) as soon as \(a\in C^1(\mathbb{R}\times \mathbb{R}^N; \mathbb{R}^N)\). The authors prove here that the regularity of \(a\) can even be relaxed (generically) to \(a\) only continuous when \(N= 2\) and \(a\) does not depend on \(t\) (and \(\text{div}_x a=0\)), that is, for autonomous two-dimensional Hamiltonian fields \[ a(x_1, x_2)= \Biggl(-{\partial H\over\partial x_2} (x_1, x_2), {\partial H\over\partial x_1} (x_1,x_2)\Biggr),\quad H\in C^1(\mathbb{R}^2; \mathbb{R}). \] This leads to the Hamiltonian system \[ \begin{aligned} &{dX_1\over dt}(t)= -{\partial H\over\partial x_2} (X_1(t), X_2(t)), {dX_2\over dt}(t)={\partial H\over\partial x_1} (X_1(t), X_2(t)),\\ &(X_1(0), X_2(0)))= (x^0_1, x^0_2).\end{aligned}\tag{2} \] In Section 2, under the assumption that \(H(x^0)\) is not a critical value of \(H\), the uniqueness of the solutions to system (2) is proved. Then, a generic condition on \(a\) such that there is uniqueness of the solutions to system (2) for almost every \(x^0\in\mathbb{R}^2\) is presented. Finally, an analogous result of uniqueness for the transport equation (1) is proved.


35F10 Initial value problems for linear first-order PDEs