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Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations. 

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.