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A note on two-dimensional transport with bounded divergence. (English) Zbl 1122.35143
The authors prove uniqueness for two dimensional transport across a noncharacteristic curve, under hypothesis that the vector field is autonomous, bounded, and with bounded divergence. They study also the Cauchy problem for the transport equation in \(\mathbb R_t\times \mathbb R_x^2\), \[ \partial_tu(t,x)+b(x)\cdot \nabla_xu(t,x)=0, \qquad u(0,\cdot )=u_0. \] The uniqueness for the Cauchy problem in \(\mathbb R_t\times \mathbb R_x^2\) is obtained under a special condition, that is, the inequality \(b(y)\cdot \xi \geq \alpha \), where \(b\) is the vector field, \(\xi \in \mathbb S^1\), \(\alpha >0\), \(\varepsilon >0\), \(y\in B_{\varepsilon }(x)\).

MSC:
35Q72 Other PDE from mechanics (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
35F10 Initial value problems for linear first-order PDEs
82C70 Transport processes in time-dependent statistical mechanics
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