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We study the mean curvature evolution of smooth, closed, twoconvex hypersurfaces in Rn+1 for n ≥ 3.Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in [HS3]—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain Lp-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

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We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De~Lellis and Sz\'{e}kelyhidi \cite{dLSz1,dLSz2} for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.