带Wiener与跳噪声的随机驯化Navier--Stokes方程在$\mathbb R^3$上. II. 全局$L^p$适定性
Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on $\mathbb R^3$. II. Global $L^p$ Well-Posedness
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中文总结 AI 辅助
本文证明$\mathbb R^3$上带乘性Wiener与补偿Poisson噪声的随机驯化Navier--Stokes方程在无小性及初始$H^1$正则性条件下，对$p>3$的$L^p$初值具有全局适定性，并给出Serrin准则与$H^1$正则性增益。
中文摘要 AI 辅助
我们在$\mathbb R^3$上建立了由乘性柱状Wiener噪声和补偿Poisson噪声同时驱动的随机驯化Navier--Stokes方程的内在延拓准则和有限能量全局可解性。在局部系数假设下，当$p>3$时，极大局部$L^p$解满足独立于辅助截断的爆破替代以及时间指数为$2p/(p-3)$的Serrin准则。驯化项的额外强制性以及相容的$L^2$和梯度噪声界产生了对于无散度初值$u_0\in L^p(\Omega;L^p)\cap L^2(\Omega;L^2)$的全局适定性理论。不需要小性条件或初始$H^1$正则性。解具有càdlàg的$L^p\cap L^2$路径，并在正时刻获得$H^1$正则性，同时具有时间加权的$H^1$和$H^2$估计。证明结合了有限能量持久性与端点完备化，后者保留终端的Poisson跳跃并允许在原始唯一性类中重启。
英文摘要
We establish intrinsic continuation criteria and finite-energy global solvability for the stochastic tamed Navier--Stokes equations on $\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener and compensated Poisson noise. Under local coefficient hypotheses, the maximal local $L^p$ solution, $p>3$, satisfies a blow-up alternative independent of auxiliary cutoffs and a Serrin criterion with time exponent $2p/(p-3)$. Additional coercivity of the taming term and compatible $L^2$ and gradient noise bounds yield the global well-posedness theory for divergence-free initial data $u_0\in L^p(Ω;L^p)\cap L^2(Ω;L^2)$. No smallness or initial $H^1$ regularity is required. The solution has càdlàg $L^p\cap L^2$ paths and gains $H^1$ regularity at positive times, with time-weighted $H^1$ and $H^2$ estimates. The proof combines finite-energy persistence with endpoint completion that retains terminal Poisson jumps and permits restart in the original uniqueness class.
发表机构
- Department of Mathematics, University of Delhi（德里大学数学系）
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