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There are very important results by Enrique Zuazua on the subject of the convection-diffusion equation u t  −div( a ( x )∇  u  )=− d ⋅∇(| u |  q −1  u ), in (0,+∞) × ℝ  N  .  In some sense this paper deals with a linear ( i.e. q  = 1) elliptic counterpart of the above equation if  d  is not constant.  We prove regularizing results on the solutions, under assumptions of interplay between the datum and the coefficient of the zero order term or between the modulus of the drift and the coefficient of the zero order term.

Regularizing effect of the interplay between coefficients in Dirichlet problems with convection or drift terms