Rough linear transport equation with an irregular drift
Abstract
We study the linear transport equation \[ ∂∂ t u ( t,x ) +b ( t,x ) · ∇ u ( t,x ) + ∇ u ( t,x ) · ∂∂ t X ( t ) =0, 2em u ( 0,x ) =u0 ( x ) \] where b is a vectorfield of limited regularity and X a vector-valued H\"older continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the PDE and solve that equation for smooth vectorfields b. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed.
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