Adapting the convex integration technique introduced in \cite{dLSz5} and subsequently developed in \cite{IsettVicol,IsettOh16,Buckmaster2013transporting}, we construct H\"{o}lder continuous weak solutions to the three dimensional Prandtl system and some other models with vertical viscosity.

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.

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The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth $n$-dimensional complex projective varieties by considering the sum of at least $n$ general sufficiently ample hypersurfaces.