Large Deviations for the Two-Dimensional Dean-Kawasaki Equation with Coulomb Interactions

Abstract

We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction ${\partial }_t{\rho }^{\epsilon }=\Delta {\rho }^{\epsilon }-\nabla \cdot ({\rho }^{\epsilon }(V\ast {\rho }^{\epsilon }))-\sqrt{\epsilon }\nabla \cdot (\sqrt{{\rho }^{\epsilon }}\circ {\xi }^{K(\epsilon )})$, where $V={\lambda }_V(\cos ({\alpha }_V)\nabla G+\sin ({\alpha }_V)J\nabla G)$. Here ${\lambda }_V\ge 0$ is the interaction strength, ${\alpha }_V$ is the interaction angle, $G$ is the mean-zero Green function of $-\Delta$ on ${\mathbb{T}}^2$, $J$ is rotation by $\pi /2$, and ${\xi }^K$ is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass $M$ satisfying ${\lambda }_V\max \{\cos ({\alpha }_V),0\}M<8\pi$, 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 $K(\epsilon )\to \infty$ and $\epsilon K(\epsilon {)}^4\to 0$, with good rate function given by the quadratic control problem for the skeleton equation.