Vanishing diffusivity uniquely selects the solution of the advection equation for divergence-free BV vector fields singular only at the initial time, including Depauw's non-uniqueness example.
On the lack of selection for the transport equation over a dense set of vector fields
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abstract
We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qu\'e. 46 (2022), De Lellis and Giri] are generic.
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On vanishing diffusivity selection for the advection equation
Vanishing diffusivity uniquely selects the solution of the advection equation for divergence-free BV vector fields singular only at the initial time, including Depauw's non-uniqueness example.