Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation
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| Format: | Preprint |
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2025
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| _version_ | 1866911275783553024 |
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| author | Li, Tengyu Zhang, Lei |
| author_facet | Li, Tengyu Zhang, Lei |
| contents | In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus $\mathbb{T}^d$. First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces $H^s(\mathbb{T}^d)$ for $s>\frac{d}{2}+1$, under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation Li, Tengyu Zhang, Lei Probability Analysis of PDEs In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus $\mathbb{T}^d$. First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces $H^s(\mathbb{T}^d)$ for $s>\frac{d}{2}+1$, under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability. |
| title | Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2510.17673 |