Harnack inequalities for nonlocal operators with supercritical drifts and their applications
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909907870023680 |
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| author | Chen, Zhen-Qing Zhang, Xicheng |
| author_facet | Chen, Zhen-Qing Zhang, Xicheng |
| contents | In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = Δ^{α/2} u + b \cdot \nabla u + f, $$ where $Δ^{α/2}$ denotes the fractional Laplacian, $b$ is a divergence-free vector field in a critical or supercritical regularity regime, and $f$ is a distribution in a fractional Sobolev space with negative indices. As applications of the analytical results obtained in this paper, we establish the well-posedness of critical stochastic quasi-geostrophic equations driven by additive Brownian noise, prove the existence of weak solutions to the two-dimensional fractional Navier--Stokes equations with measure-valued initial vorticity, and demonstrate the well-posedness of generalized martingale problems associated with critical stochastic differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harnack inequalities for nonlocal operators with supercritical drifts and their applications Chen, Zhen-Qing Zhang, Xicheng Analysis of PDEs Probability 35R09, 60H10 In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = Δ^{α/2} u + b \cdot \nabla u + f, $$ where $Δ^{α/2}$ denotes the fractional Laplacian, $b$ is a divergence-free vector field in a critical or supercritical regularity regime, and $f$ is a distribution in a fractional Sobolev space with negative indices. As applications of the analytical results obtained in this paper, we establish the well-posedness of critical stochastic quasi-geostrophic equations driven by additive Brownian noise, prove the existence of weak solutions to the two-dimensional fractional Navier--Stokes equations with measure-valued initial vorticity, and demonstrate the well-posedness of generalized martingale problems associated with critical stochastic differential equations. |
| title | Harnack inequalities for nonlocal operators with supercritical drifts and their applications |
| topic | Analysis of PDEs Probability 35R09, 60H10 |
| url | https://arxiv.org/abs/2511.13012 |