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We propose and prove several regularity criteria for the 2D and 3D Kuramoto–Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution$$\phi $$$\varphi$, the vector solution$$u\triangleq \nabla \phi $$$u\triangleq \nabla \varphi$, as well as the divergence$$\text {div}(u)=\Delta \phi $$$\text{div}\left(u\right)=\Delta \varphi$, and each component ofuand$$\nabla u$$$\nabla u$. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we supply a rigorous analytical proof that guarantees the convergence of this algorithm to the true parameter values when the system in question is the classic three-dimensional Lorenz system. Such a result appears to be the first of its kind for dynamical parameter estimation of nonlinear systems. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We moreover probe the limitations of the algorithm by identifying dynamical regimes under which certain parameters cannot be effectively inferred having only observed certain state variables. In such cases, modifications to the algorithm are proposed that ultimately result in recovery of the parameter. Lastly, computational evidence is provided that supports the efficacy of the algorithm well beyond the hypotheses specified by the theorem, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.