The sampleability projection exists and is unique with a unique Brenier map, but porous medium flow cannot reach the bounded density-ratio class while preserving compact support due to vanishing essential infima.
Brenier,Polar factorization and monotone rearrangement of vector-valued functions, Comm
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Sampleability transport, nonlinear regularization, and the porous medium flow
The sampleability projection exists and is unique with a unique Brenier map, but porous medium flow cannot reach the bounded density-ratio class while preserving compact support due to vanishing essential infima.