Non-uniqueness of parabolic solutions for advection-diffusion equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911003954905088 |
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| author | Moerschell, Thérèse Sorella, Massimo |
| author_facet | Moerschell, Thérèse Sorella, Massimo |
| contents | We present a novel example of a divergence-free velocity field $b \in L^\infty ((0,1); L^p (\mathbb{T}^2))$ for $p<2$ arbitrary but fixed which leads to non-unique solutions of advection-diffusion in the class $L^\infty_{t,x} \cap L^2_t H^1_x$ while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and $L^2_{t,x}$ integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence-free velocity field $b \in L^p ((0,1); L^\infty (\mathbb{T}^2))$, for $p< 2$ fixed, but arbitrary, with non-unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for $d\geq 3$ divergence-free velocity fields $ b \in L^\infty((0,1);L^p (\mathbb{T}^d))$ with $p < \frac{2d}{d+2}$ could lead to non-unique solutions in the space $L^\infty_t L^{\frac{2d}{d-2}}_x \cap L^2_t H^1_x$. Our proof is based on a stochastic Lagrangian approach and does not rely on convex integration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of parabolic solutions for advection-diffusion equation Moerschell, Thérèse Sorella, Massimo Analysis of PDEs 35A02, 35Q49, 76F25 We present a novel example of a divergence-free velocity field $b \in L^\infty ((0,1); L^p (\mathbb{T}^2))$ for $p<2$ arbitrary but fixed which leads to non-unique solutions of advection-diffusion in the class $L^\infty_{t,x} \cap L^2_t H^1_x$ while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and $L^2_{t,x}$ integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence-free velocity field $b \in L^p ((0,1); L^\infty (\mathbb{T}^2))$, for $p< 2$ fixed, but arbitrary, with non-unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for $d\geq 3$ divergence-free velocity fields $ b \in L^\infty((0,1);L^p (\mathbb{T}^d))$ with $p < \frac{2d}{d+2}$ could lead to non-unique solutions in the space $L^\infty_t L^{\frac{2d}{d-2}}_x \cap L^2_t H^1_x$. Our proof is based on a stochastic Lagrangian approach and does not rely on convex integration. |
| title | Non-uniqueness of parabolic solutions for advection-diffusion equation |
| topic | Analysis of PDEs 35A02, 35Q49, 76F25 |
| url | https://arxiv.org/abs/2410.03308 |