AI 中文总结
本文研究二维具有库仑相互作用的Dean-Kawasaki方程，建立其整体适定性和小噪声大偏差原理，速率函数由骨架方程的二次控制问题给出。
AI 中文摘要
我们建立了具有库仑相互作用的二维Dean-Kawasaki方程的整体适定性和小噪声大偏差原理，方程为$\partial_t\rho^\varepsilon=\Delta\rho^\varepsilon-\nabla\cdot(\rho^\varepsilon(V*\rho^\varepsilon))-\sqrt{\varepsilon}\nabla\cdot(\sqrt{\rho^\varepsilon}\circ\xi^{K(\varepsilon)})$，其中$V=\lambda_V(\cos(\alpha_V)\nabla G+\sin(\alpha_V)J\nabla G)$。这里$\lambda_V\geq0$是相互作用强度，$\alpha_V$是相互作用角，$G$是$\mathbb{T}^2$上$-\Delta$的零均值格林函数，$J$是旋转$\pi/2$，$\xi^K$是时空白噪声的有限模态近似。对于满足$\lambda_V\max\{\cos(\alpha_V),0\}M<8\pi$的质量为$M$的有限熵初始数据，该方程在精确平方根系数下，在随机动力学解类中对每个有限傅里叶截断都是全局逐路径适定的。继Fehrman和Gess（arXiv:1910.11860）之后，我们在$K(\varepsilon)\to\infty$和$\varepsilon K(\varepsilon)^4\to0$的标度下，在$L^1((0,T)\times\mathbb{T}^2)$上证明了大偏差原理，其良好速率函数由骨架方程的二次控制问题给出。
英文摘要
We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction $\partial_tρ^\varepsilon=Δρ^\varepsilon-\nabla\cdot(ρ^\varepsilon(V*ρ^\varepsilon))-\sqrt{\varepsilon}\nabla\cdot(\sqrt{ρ^\varepsilon}\circξ^{K(\varepsilon)})$, where $V=λ_V(\cos(α_V)\nabla G+\sin(α_V)J\nabla G)$. Here $λ_V\geq0$ is the interaction strength, $α_V$ is the interaction angle, $G$ is the mean-zero Green function of $-Δ$ on $\mathbb{T}^2$, $J$ is rotation by $π/2$, and $ξ^K$ is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass $M$ satisfying $λ_V\max\{\cos(α_V),0\}M<8π$, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on $L^1((0,T)\times\mathbb{T}^2)$ under the scaling $K(\varepsilon)\to\infty$ and $\varepsilon K(\varepsilon)^4\to0$, with good rate function given by the quadratic control problem for the skeleton equation.
