Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909502779949056 |
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| author | Miao, Changxing Zhao, Zhiwen |
| author_facet | Miao, Changxing Zhao, Zhiwen |
| contents | In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{α,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<α\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space Miao, Changxing Zhao, Zhiwen Analysis of PDEs In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{α,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<α\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$. |
| title | Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.09698 |