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Rough linear transport equation with an irregular drift

Abstract : We study the linear transport equation \[ \frac{\partial}{\partial t} u ( t,x ) +b ( t,x ) \cdot \nabla u ( t,x ) + \nabla u ( t,x ) \cdot \frac{\partial}{\partial t} X ( t ) =0, \hspace{2em} u ( 0,x ) =u_{0} ( x ) \] 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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https://hal.archives-ouvertes.fr/hal-01110491
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Submitted on : Wednesday, January 28, 2015 - 12:01:20 PM
Last modification on : Thursday, January 20, 2022 - 9:02:01 AM

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Rémi Catellier. Rough linear transport equation with an irregular drift. Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2016, 4 (3), pp.477-534. ⟨10.1007/s40072-016-0069-y⟩. ⟨hal-01110491⟩

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