The stochastic Zakharov system in dimension $d \geq 4$: Local well-posedness and regularization by noise for scattering
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arXiv
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| Natura: | Preprint |
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2026
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| author | Spitz, Martin Zhang, Deng Zhao, Zhenqi |
| author_facet | Spitz, Martin Zhang, Deng Zhao, Zhenqi |
| contents | In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the large data regime, where the global existence and long-time dynamics of the deterministic Zakharov system remain largely open. We prove the local well-posedness of the stochastic system in the full deterministic regularity regime and establish a blow-up alternative at the endpoint regularity, which implies the persistence of regularity in the full well-posedness regime. Furthermore, we prove that for any large initial data, with high probability, non-conservative noise yields global and scattering solutions. Our proof introduces a tailored functional framework. To establish local well-posedness, we employ a refinement of adapted Fourier restriction and lateral Strichartz spaces, which allows us to control both the nonlinear interactions and the critical first-order derivative perturbations arising from rescaling transforms. To achieve the noise-regularization effect, we augment this setting with maximal function spaces. We derive new trilinear estimates for the stochastic wave nonlinearity that are crucial for the global dynamics by fully exploiting the temporal regularity of geometric Brownian motions in scaling-(sub)critical Besov spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11787 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The stochastic Zakharov system in dimension $d \geq 4$: Local well-posedness and regularization by noise for scattering Spitz, Martin Zhang, Deng Zhao, Zhenqi Analysis of PDEs Probability In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the large data regime, where the global existence and long-time dynamics of the deterministic Zakharov system remain largely open. We prove the local well-posedness of the stochastic system in the full deterministic regularity regime and establish a blow-up alternative at the endpoint regularity, which implies the persistence of regularity in the full well-posedness regime. Furthermore, we prove that for any large initial data, with high probability, non-conservative noise yields global and scattering solutions. Our proof introduces a tailored functional framework. To establish local well-posedness, we employ a refinement of adapted Fourier restriction and lateral Strichartz spaces, which allows us to control both the nonlinear interactions and the critical first-order derivative perturbations arising from rescaling transforms. To achieve the noise-regularization effect, we augment this setting with maximal function spaces. We derive new trilinear estimates for the stochastic wave nonlinearity that are crucial for the global dynamics by fully exploiting the temporal regularity of geometric Brownian motions in scaling-(sub)critical Besov spaces. |
| title | The stochastic Zakharov system in dimension $d \geq 4$: Local well-posedness and regularization by noise for scattering |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2604.11787 |