Weak existence for adaptive porous flow with a negative drag jump

We prove weak existence for a stationary porous-flow model whose drag law changes at speed one and has a negative jump. At the threshold the drag is an arbitrary vector field, as in the free-transition formulation of the model. The result holds in every dimension $d\ge2$ on bounded connected Lipschitz domains, for all finite growth exponents $r\ge2$ and all the prescribed mixed pressure and flux data in their original Lebesgue and Sobolev spaces. The solution can be chosen with speed at least one almost everywhere. Finite divergence-free and gradient oscillations use the free unit sphere, while monotone auxiliary equations confine the only constitutive defects to bounded values. Comparing their exact solutions with the oscillatory targets gives summable corrections in the conjugate norms, so the argument also applies to rough data.