On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914216703688704 |
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| author | Jenssen, Helge Kristian |
| author_facet | Jenssen, Helge Kristian |
| contents | Explicit irrotational solutions, obtained via the Cole-Hopf transform from the multi-d heat
equation, give examples of non-uniqueness for the Cauchy problem in supercritical
$L^p$, $W^{1,p}$, and $W^{2,p}$ regimes.
We verify non-uniqueness of the trivial solution in the sense of $L^p(\RR^n)$, whenever $n\geq2$
and $1\leq p<n$. The same solutions give non-uniqueness in $W^{1,p}(\RR^n)$
and $W^{2,p}(\RR^n)$ for $1\leq p<\frac{n}{2}$ and $1\leq p<\frac{n}{3}$,
respectively. The main example provides solutions which are classical for
strictly positive times, and vanish in the stated norms, but explode in $L^\infty(\RR^n)$,
as $t\to0+$. The non-uniqueness is unrelated to the
Tikhonov non-uniqueness phenomenon for the heat equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$ Jenssen, Helge Kristian Analysis of PDEs 35A02, 35K40, 35K59 Explicit irrotational solutions, obtained via the Cole-Hopf transform from the multi-d heat equation, give examples of non-uniqueness for the Cauchy problem in supercritical $L^p$, $W^{1,p}$, and $W^{2,p}$ regimes. We verify non-uniqueness of the trivial solution in the sense of $L^p(\RR^n)$, whenever $n\geq2$ and $1\leq p<n$. The same solutions give non-uniqueness in $W^{1,p}(\RR^n)$ and $W^{2,p}(\RR^n)$ for $1\leq p<\frac{n}{2}$ and $1\leq p<\frac{n}{3}$, respectively. The main example provides solutions which are classical for strictly positive times, and vanish in the stated norms, but explode in $L^\infty(\RR^n)$, as $t\to0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation. |
| title | On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$ |
| topic | Analysis of PDEs 35A02, 35K40, 35K59 |
| url | https://arxiv.org/abs/2512.19878 |