Convex integration solutions to the transport equation with full dimensional concentration
Abstract
We construct infinitely many incompressible Sobolev vector fields u ∈ Ct W1, px on the periodic domain Td for which uniqueness of solutions to the transport equation fails in the class of densities ∈ Ct Lpx, provided 1/p + 1/ p > 1 + 1/d. The same result applies to the transport-diffusion equation, if, in addition p'<d.
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