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Nearly Lipschitzean divergence free transport propagates neither continuity nor BV regularity. (English) Zbl 1088.35015

Summary: We give examples of divergence free vector fields \[ {\mathbf a}(x,y) \in\cap_{1\leq p<\infty}W^{1,p}(\mathbb{R}^2). \] For such fields the Cauchy problem for the linear transport equation \[ \frac{\partial u} {\partial t}+ {\mathbf a}_1(x,y)\frac{\partial u}{\partial x}+{\mathbf a}_2(x,y)\frac{\partial u}{\partial y}=0,\;\text{div} \,{\mathbf a}:= \frac{\partial{\mathbf a}_1}{\partial x}+\frac{\partial{\mathbf a}_2}{\partial u}=0, \] has unique bounded solutions for \(u_0\in L^\infty(\mathbb{R}^2)\). The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics. In addition, there are smooth initial data \(u_0\in C_0^\infty(\mathbb{R}^2)\) so that the unique bounded solution is not continuous on any neighborhood of the origin.
The second example is a field of similar regularity and initial data in \(W^{1,1}\subset BV\) so that for no \(t>0\) is it true that \(u(t,\cdot)\) is of bounded variation.

MSC:

35F10 Initial value problems for linear first-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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