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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.

##### MSC:
 35F10 Initial value problems for linear first-order PDEs
##### Keywords:
autonomous two-dimensional Hamiltonian fields