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In this article, some elementary observations are made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggest the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity con- dition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type cri- terion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solu- tions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise. 

We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear continuous data assimilation algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate these algorithms in the context of the 1D Kuramoto-Sivashinsky equations. We observe that the nonlinear models experience super-exponential convergence in time, and converge to machine precision significantly faster than the linear AOT algorithm in our tests. For both simplicity and completeness, we provide the key analysis of the exponential-in-time convergence in the linear case. 

We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally. 

We derive numerical stability conditions and analyze convergence to analytical nonlocal solutions of 1D peridynamic models for transient diffusion with and without a moving interface. In heat transfer or oxidation, for example, one often encounters initial conditions that are discontinuous, as in thermal shock or sudden exposure to oxygen. We study the numerical error in these models with continuous and discontinuous initial conditions and determine that the initial discontinuities lead to lower convergence rates, but this issue is present at early times only. Except for the early times, the convergence rates of models with continuous and discontinuous initial conditions are the same. In problems with moving interfaces, we show that the numerical solution captures the exact interface location well, in time. These results can be used in simulating a variety of reaction-diffusion type problems, such as the oxidation-induced damage in zirconium carbide at high temperatures. 

Abstract 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.