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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.01670 |
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| _version_ | 1866910433302020096 |
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| author | Bruè, Elia Colombo, Maria Kumar, Anuj |
| author_facet | Bruè, Elia Colombo, Maria Kumar, Anuj |
| contents | Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $ρ_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\infty_t L^r_x$ densities for $\frac{1}{p} + \frac{1}{r} \leq 1$. This range was later improved in [BCDL21] to $\frac{1}{p} + \frac{d-1}{dr} \leq 1$. We prove that this range is sharp by providing a counterexample to uniqueness when $\frac{1}{p} + \frac{d-1}{dr} > 1$.
To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field Bruè, Elia Colombo, Maria Kumar, Anuj Analysis of PDEs Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $ρ_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\infty_t L^r_x$ densities for $\frac{1}{p} + \frac{1}{r} \leq 1$. This range was later improved in [BCDL21] to $\frac{1}{p} + \frac{d-1}{dr} \leq 1$. We prove that this range is sharp by providing a counterexample to uniqueness when $\frac{1}{p} + \frac{d-1}{dr} > 1$. To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization. |
| title | Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.01670 |